Applications of the Vector Dot Product, Math for Java Game Programmers

  public void actionPerformed(ActionEvent e){

    //Erase the off-screen image and draw the axes.
    setCoordinateFrame(g2D);

    //Create one ColMatrix3D object based on the user
    // input values.
    GM02.ColMatrix3D matrixA = new GM02.ColMatrix3D(
                  Double.parseDouble(vecX.getText()),
                  Double.parseDouble(vecY.getText()),
                  Double.parseDouble(vecZ.getText()));

    //Create ColMatrix3D objects that represent each of
    // the three axes.
    GM02.ColMatrix3D matrixX = 
                              new GM02.ColMatrix3D(1,0,0);
    GM02.ColMatrix3D matrixY = 
                              new GM02.ColMatrix3D(0,1,0);
    GM02.ColMatrix3D matrixZ = 
                              new GM02.ColMatrix3D(0,0,1);

    GM02.ColMatrix3D matrixXY = new GM02.ColMatrix3D(
                      Double.parseDouble(vecX.getText()),
                      Double.parseDouble(vecY.getText()),
                      0);

    GM02.ColMatrix3D matrixYZ = new GM02.ColMatrix3D(
                      0,
                      Double.parseDouble(vecY.getText()),
                      Double.parseDouble(vecZ.getText()));

    GM02.ColMatrix3D matrixZX = new GM02.ColMatrix3D(
                      Double.parseDouble(vecX.getText()),
                      0,
                      Double.parseDouble(vecZ.getText()));

    //Use the ColMatrix3D objects to create Vector3D
    // objects representing the user-specified vector and
    // each of the axes.
    GM02.Vector3D vecA = new GM02.Vector3D(matrixA);
    GM02.Vector3D vecX = new GM02.Vector3D(matrixX);
    GM02.Vector3D vecY = new GM02.Vector3D(matrixY);
    GM02.Vector3D vecZ = new GM02.Vector3D(matrixZ);

    //Create Vector3D objects that represent the
    // projection of the user-specified vector on each of
    // the planes.
    GM02.Vector3D vecXY = new GM02.Vector3D(matrixXY);
    GM02.Vector3D vecYZ = new GM02.Vector3D(matrixYZ);
    GM02.Vector3D vecZX = new GM02.Vector3D(matrixZX);

    //Draw the projection of the user specified vector on
    // each of the three planes.
    g2D.setColor(Color.MAGENTA);
    vecXY.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
    vecYZ.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
    vecZX.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));

    //Draw the user-specified vector with its tail at the
    // origin.
    g2D.setColor(Color.BLACK);
    vecA.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));

    //Compute and display the angle relative to the
    // x-axis.
    double angle = vecA.angle(vecX);
    angleX.setText("" +  prepareForDisplay(angle));

    //Compute and display the angle relative to the
    // y-axis.
    angle = vecA.angle(vecY);
    angleY.setText("" +  prepareForDisplay(angle));

    //Compute and display the angle relative to the
    // z-axis.
    angle = vecA.angle(vecZ);
    angleZ.setText("" +  prepareForDisplay(angle));


    //Compute and display the angle relative to the
    // XY plane
    angle = vecA.angle(vecXY);
    angleXY.setText("" +  prepareForDisplay(angle));

    //Compute and display the angle relative to the
    // YZ plane
    angle = vecA.angle(vecYZ);
    angleYZ.setText("" +  prepareForDisplay(angle));

    //Compute and display the angle relative to the
    // ZX plane
    angle = vecA.angle(vecZX);
    angleZX.setText("" +  prepareForDisplay(angle));


    //Compute and display the angle of the projection onto
    // the XY plane relative to the x-axis
    angle = vecXY.angle(vecX);
    anglePX.setText("" +  prepareForDisplay(angle));

    //Compute and display the angle of the projection onto
    // the YZ plane relative to the y-axis
    angle = vecYZ.angle(vecY);
    anglePY.setText("" +  prepareForDisplay(angle));

    //Compute and display the angle of the projection onto
    // the ZX plane relative to the z-axis
    angle = vecZX.angle(vecZ);
    anglePZ.setText("" +  prepareForDisplay(angle));    


    myCanvas.repaint();//Copy off-screen image to canvas.
    
  }//end actionPerformed

ot product = x1*x2 + y1*y2 + z1*z2

If the two vectors are perpendicular:

x1*x2 + y1*y2 + z1*z2 = 0.0
x1*x2 =  -(y1*y2 + z1*z2)

x2 = -(y1*y2 + z1*z2)/x1

or

y2 = -(x1*x2 + z1*z2)/y1

or

z2 = -(x1*x2 + z1*z2)/z1

  public void actionPerformed(ActionEvent e){

    //Code deleted for brevity.

    GM02.Vector3D tempVec;
    GM02.ColMatrix3D tempMatrix;
    g2D.setColor(Color.MAGENTA);

    if(Math.abs(zCoor) > 0.001){
      tempMatrix = new GM02.ColMatrix3D(
          xCoor,yCoor,-(xCoor*xCoor + yCoor*yCoor)/zCoor);

      tempVec = new GM02.Vector3D(tempMatrix);

      System.out.println(tempVec);
      //Normalize and scale the perpendicular vector.
      tempVec = tempVec.normalize().scale(50.0);
      tempVec.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
      tempVec.negate().draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
      System.out.println(vecA.angle(tempVec));
    }//end if

    if(Math.abs(yCoor) > 0.001){
      tempMatrix = new GM02.ColMatrix3D(
          xCoor,-(xCoor*xCoor + zCoor*zCoor)/yCoor,zCoor);
      tempVec = new GM02.Vector3D(tempMatrix);
      System.out.println(tempVec);
      //Normalize and scale the perpendicular vector.
      tempVec = tempVec.normalize().scale(50.0);
      tempVec.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
      tempVec.negate().draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
      System.out.println(vecA.angle(tempVec));
    }//end if

    if(Math.abs(xCoor) > 0.001){
      tempMatrix = new GM02.ColMatrix3D(
         -(yCoor*yCoor + zCoor*zCoor)/xCoor, yCoor,zCoor);
      tempVec = new GM02.Vector3D(tempMatrix);
      System.out.println(tempVec);
      //Normalize and scale the perpendicular vector.
      tempVec = tempVec.normalize().scale(50.0);
      tempVec.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
      tempVec.negate().draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
      System.out.println(vecA.angle(tempVec));
    }//end if


    myCanvas.repaint();//Copy off-screen image to canvas.
    System.out.println();//blank line
  }//end actionPerformed
  //----------------------------------------------------//

  private void drawTheCylinder(Graphics2D g2D){

    //Code deleted for brevity.

    //Draw a cylinder by stacking 20 circular disks on
    // the x-z plane. The disks are centered on the
    // y-axis and are parallel to the x-z plane. The
    // thickness of each disk is 5 vertical units.
    GM02.Point3D tempPointA;
    GM02.Point3D tempPointB;
    g2D.setColor(Color.BLACK);

    for(int y = 0;y < 105;y += 5){//iterate on disks
      //Define the starting point on the circle for this
      // disk.
      tempPointA = new GM02.Point3D(
              new GM02.ColMatrix3D(76,y - osiHeight/4,0));
      //Iterate on points on the circle that represents
      // this disk.
      for(int cnt = 0;cnt < 360;cnt++){//360 points
        //Compute the next point on the circle.
        tempPointB = 
            new GM02.Point3D(new GM02.ColMatrix3D(
              76*Math.cos(Math.toRadians(cnt*360/360)),
              y - osiHeight/4,
              76*Math.sin(Math.toRadians(cnt*360/360))));

        //Draw the line in 3D. Note that there is no
        // backface culling in this program.
        new GM02.Line3D(tempPointA,tempPointB).draw(g2D);

        //Save the point for use in drawing the next line.
        tempPointA = tempPointB;
      }//end for loop on points
    }//end for loop on disks

  }//end drawTheCylinder method

  private void drawTheCylinder(Graphics2D g2D){

    //Code deleted for brevity.

    //Get a vector that is approximately parallel to the
    // viewpoint of the viewer. The best values for this
    // vector were determined experimentally using this
    // program and the earlier program named DotProd3D02.
    GM02.Vector3D viewPoint = 
        new GM02.Vector3D(new GM02.ColMatrix3D(43,5,50));

    //Draw a cylinder by stacking 20 circular disks on
    // the x-z plane. The disks are centered on the
    // y-axis and are parallel to the x-z plane. The
    // thickness of each disk is 5 vertical units.
    GM02.Point3D tempPointA;
    GM02.Point3D tempPointB;
    g2D.setColor(Color.BLACK);

    for(int y = 0;y < 105;y += 5){//iterate on disks
      //Define the starting point on the circle for this
      // disk.
      tempPointA = new GM02.Point3D(new GM02.ColMatrix3D(
                                   76,y - osiHeight/4,0));
      //Iterate on points on the circle that represents
      // this disk.
      for(int cnt = 0;cnt < 360;cnt++){//360 points
        //Compute the next point on the circle.
        tempPointB = 
            new GM02.Point3D(new GM02.ColMatrix3D(
              76*Math.cos(Math.toRadians(cnt*360/360)),
              y - osiHeight/4,
              76*Math.sin(Math.toRadians(cnt*360/360))));

        //Do backface culling using the dot product of the
        // viewpoint vector and a vector that is almost
        // perpendicular to the line being drawn. If the
        // dot product is negative, or if the disk being
        // drawn is the top disk on the stack, draw the
        // line. Otherwise, don't draw the line.
        //The perpendicular vector used in the dot
        // product is the displacement vector from the
        // origin to a point that defines one end of the
        // line being drawn. Note that this vector is not
        // perfectly perpendicular to the line being 
        // drawn. Later in the series, we will learn about
        // and use the cross product to get perpendicular
        // vectors.
        if((tempPointB.getDisplacementVector(
                           new GM02.Point3D(
                             new GM02.ColMatrix3D(0,0,0)))
                                   .dot(viewPoint) < 0.0) 
                                           || (y == 100)){
          //Draw the line in 3D.
          new GM02.Line3D(
                         tempPointA,tempPointB).draw(g2D);
        }//end if

        //Save the point for use in drawing the next line.
        tempPointA = tempPointB;
      }//end for loop on points
    }//end for loop on disks

  }//end drawTheCylinder method

/*GM02.java
Copyright 2008, R.G.Baldwin
Revised 02/08/08

This is an update to the game-math library named GM01.

The main purpose of this update was to add vector dot
product and related capabilities to the library.

Please see the comments at the beginning of the library
class named GM01 for a general description of the library.

The following methods are new instance methods of the 
indicated static top-level classes belonging to the class
named GM02.

GM02.ColMatrix2D.dot - compute dot product of two
 ColMatrix2D objects.
GM02.Vector2D.dot - compute dot product of two
 Vector2D objects.
GM02.Vector2D.angle - compute angle between two Vector2D
 objects.

GM02.ColMatrix3D.dot - compute dot product of two
 ColMatrix3D objects
GM02.Vector3D.dot - compute dot product of two
 Vector3D objects.
GM02.Vector3D.angle - compute angle between two Vector3D
 objects.

Tested using JDK 1.6 under WinXP.
*********************************************************/
import java.awt.geom.*;
import java.awt.*;

public class GM02{
  //----------------------------------------------------//

  //This method converts a ColMatrix3D object to a
  // ColMatrix2D object. The purpose is to accept
  // x, y, and z coordinate values and transform those
  // values into a pair of coordinate values suitable for
  // display in two dimensions.
  //See http://local.wasp.uwa.edu.au/~pbourke/geometry/
  // classification/ for technical background on the
  // transform from 3D to 2D.
  //The transform equations are:
  // x2d = x3d + z3d * cos(theta)/tan(alpha)
  // y2d = y3d + z3d * sin(theta)/tan(alpha);
  //Let theta = 30 degrees and alpha = 45 degrees
  //Then:cos(theta) = 0.866
  //     sin(theta) = 0.5
  //     tan(alpha) = 1;
  //Note that the signs in the above equations depend
  // on the assumed directions of the angles as well as
  // the assumed positive directions of the axes. The
  // signs used in this method assume the following:
  //   Positive x is to the right.
  //   Positive y is up the screen.
  //   Positive z is protruding out the front of the
  //     screen.
  //   The viewing position is above the x axis and to the
  //     right of the z-y plane.
  public static GM02.ColMatrix2D convert3Dto2D(
                                   GM02.ColMatrix3D data){
    return new GM02.ColMatrix2D(
                  data.getData(0) - 0.866*data.getData(2),
                  data.getData(1) - 0.50*data.getData(2));
  }//end convert3Dto2D 
  //----------------------------------------------------//

  //This method wraps around the translate method of the
  // Graphics2D class. The purpose is to cause the
  // positive direction for the y-axis to be up the screen
  // instead of down the screen. When you use this method,
  // you should program as though the positive direction
  // for the y-axis is up.
  public static void translate(Graphics2D g2D,
                               double xOffset,
                               double yOffset){
    //Flip the sign on the y-coordinate to change the
    // direction of the positive y-axis to go up the
    // screen.
    g2D.translate(xOffset,-yOffset);
  }//end translate
  //----------------------------------------------------//

  //This method wraps around the drawLine method of the
  // Graphics class. The purpose is to cause the positive
  // direction for the y-axis to be up the screen instead
  // of down the screen. When you use this method, you
  // should program as though the positive direction for
  // the y-axis is up.
  public static void drawLine(Graphics2D g2D,
                              double x1,
                              double y1,
                              double x2,
                              double y2){
    //Flip the sign on the y-coordinate value.
    g2D.drawLine((int)x1,-(int)y1,(int)x2,-(int)y2);
  }//end drawLine
  //----------------------------------------------------//

  //This method wraps around the fillOval method of the
  // Graphics class. The purpose is to cause the positive
  // direction for the y-axis to be up the screen instead
  // of down the screen. When you use this method, you
  // should program as though the positive direction for
  // the y-axis is up.
  public static void fillOval(Graphics2D g2D,
                              double x,
                              double y,
                              double width,
                              double height){
    //Flip the sign on the y-coordinate value.
    g2D.fillOval((int)x,-(int)y,(int)width,(int)height);
  }//end fillOval
  //----------------------------------------------------//

  //This method wraps around the drawOval method of the
  // Graphics class. The purpose is to cause the positive
  // direction for the y-axis to be up the screen instead
  // of down the screen. When you use this method, you
  // should program as though the positive direction for
  // the y-axis is up.
  public static void drawOval(Graphics2D g2D,
                              double x,
                              double y,
                              double width,
                              double height){
    //Flip the sign on the y-coordinate value.
    g2D.drawOval((int)x,-(int)y,(int)width,(int)height);
  }//end drawOval
  //----------------------------------------------------//

  //This method wraps around the fillRect method of the
  // Graphics class. The purpose is to cause the positive
  // direction for the y-axis to be up the screen instead
  // of down the screen. When you use this method, you
  // should program as though the positive direction for
  // the y-axis is up.
  public static void fillRect(Graphics2D g2D,
                              double x,
                              double y,
                              double width,
                              double height){
    //Flip the sign on the y-coordinate value.
    g2D.fillRect((int)x,-(int)y,(int)width,(int)height);
  }//end fillRect
  //----------------------------------------------------//



  //An object of this class represents a 2D column matrix.
  // An object of this class is the fundamental building
  // block for several of the other classes in the
  // library.
  public static class ColMatrix2D{
    double[] data = new double[2];

    public ColMatrix2D(double data0,double data1){
      data[0] = data0;
      data[1] = data1;
    }//end constructor
    //--------------------------------------------------//

    //Overridden toString method.
    public String toString(){
      return data[0] + "," + data[1];
    }//end overridden toString method
    //--------------------------------------------------//

    public double getData(int index){
      if((index < 0) || (index > 1)){ 
        throw new IndexOutOfBoundsException();
      }else{
        return data[index];
      }//end else
    }//end getData method
    //--------------------------------------------------//

    public void setData(int index,double data){
      if((index < 0) || (index > 1)){ 
        throw new IndexOutOfBoundsException();
      }else{
        this.data[index] = data;
      }//end else
    }//end setData method
    //--------------------------------------------------//

    //This method overrides the equals method inherited
    // from the class named Object. It compares the values
    // stored in two matrices and returns true if the
    // values are equal or almost equal and returns false
    // otherwise.
    public boolean equals(Object obj){
      if(obj instanceof GM02.ColMatrix2D &&
         Math.abs(((GM02.ColMatrix2D)obj).getData(0) - 
                                 getData(0)) <= 0.00001 &&
         Math.abs(((GM02.ColMatrix2D)obj).getData(1) - 
                                  getData(1)) <= 0.00001){
        return true;
      }else{
        return false;
      }//end else
     
    }//end overridden equals method
    //--------------------------------------------------//

    //Adds one ColMatrix2D object to another ColMatrix2D
    // object, returning a ColMatrix2D object.
    public GM02.ColMatrix2D add(GM02.ColMatrix2D matrix){
      return new GM02.ColMatrix2D(
                            getData(0)+matrix.getData(0),
                            getData(1)+matrix.getData(1));
    }//end add
    //--------------------------------------------------//
    
    //Subtracts one ColMatrix2D object from another
    // ColMatrix2D object, returning a ColMatrix2D object.
    // The object that is received as an incoming
    // parameter  is subtracted from the object on which
    // the method is called.
    public GM02.ColMatrix2D subtract(
                                 GM02.ColMatrix2D matrix){
      return new GM02.ColMatrix2D(
                            getData(0)-matrix.getData(0),
                            getData(1)-matrix.getData(1));
    }//end subtract
    //--------------------------------------------------//
    
    //Computes the dot product of two ColMatrix2D
    // objects and returns the result as type double.
    public double dot(GM02.ColMatrix2D matrix){
      return getData(0) * matrix.getData(0) 
           + getData(1) * matrix.getData(1);
    }//end dot
    //--------------------------------------------------//
  }//end class ColMatrix2D
  //====================================================//


  //An object of this class represents a 3D column matrix.
  // An object of this class is the fundamental building
  // block for several of the other classes in the
  // library.
  public static class ColMatrix3D{
    double[] data = new double[3];
    
    public ColMatrix3D(
                  double data0,double data1,double data2){
      data[0] = data0;
      data[1] = data1;
      data[2] = data2;
    }//end constructor
    //--------------------------------------------------//
    
    public String toString(){
      return data[0] + "," + data[1] + "," + data[2];
    }//end overridden toString method
    //--------------------------------------------------//
    
    public double getData(int index){
      if((index < 0) || (index > 2)){ 
        throw new IndexOutOfBoundsException();
      }else{
        return data[index];
      }//end else
    }//end getData method
    //--------------------------------------------------//
    
    public void setData(int index,double data){
      if((index < 0) || (index > 2)){ 
        throw new IndexOutOfBoundsException();
      }else{
        this.data[index] = data;
      }//end else
    }//end setData method
    //--------------------------------------------------//
    
    //This method overrides the equals method inherited
    // from the class named Object. It compares the values
    // stored in two matrices and returns true if the
    // values are equal or almost equal and returns false
    // otherwise. 
    public boolean equals(Object obj){
      if(obj instanceof GM02.ColMatrix3D &&
         Math.abs(((GM02.ColMatrix3D)obj).getData(0) - 
                                 getData(0)) <= 0.00001 &&
         Math.abs(((GM02.ColMatrix3D)obj).getData(1) - 
                                 getData(1)) <= 0.00001 &&
         Math.abs(((GM02.ColMatrix3D)obj).getData(2) - 
                                  getData(2)) <= 0.00001){
        return true;
      }else{
        return false;
      }//end else
     
    }//end overridden equals method
    //--------------------------------------------------//

    //Adds one ColMatrix3D object to another ColMatrix3D
    // object, returning a ColMatrix3D object.
    public GM02.ColMatrix3D add(GM02.ColMatrix3D matrix){
      return new GM02.ColMatrix3D(
                            getData(0)+matrix.getData(0),
                            getData(1)+matrix.getData(1),
                            getData(2)+matrix.getData(2));
    }//end add
    //--------------------------------------------------//
    
    //Subtracts one ColMatrix3D object from another
    // ColMatrix3D object, returning a ColMatrix3D object.
    // The object that is received as an incoming
    // parameter is subtracted from the object on which
    // the method is called.
    public GM02.ColMatrix3D subtract(
                                 GM02.ColMatrix3D matrix){
      return new GM02.ColMatrix3D(
                            getData(0)-matrix.getData(0),
                            getData(1)-matrix.getData(1),
                            getData(2)-matrix.getData(2));
    }//end subtract
    //--------------------------------------------------//
    
    //Computes the dot product of two ColMatrix3D
    // objects and returns the result as type double.
    public double dot(GM02.ColMatrix3D matrix){
      return getData(0) * matrix.getData(0) 
           + getData(1) * matrix.getData(1)
           + getData(2) * matrix.getData(2);
    }//end dot
    //--------------------------------------------------//
  }//end class ColMatrix3D
  //====================================================//
  //====================================================//


  public static class Point2D{
    GM02.ColMatrix2D point;
    
    public Point2D(GM02.ColMatrix2D point){//constructor
      //Create and save a clone of the ColMatrix2D object
      // used to define the point to prevent the point
      // from being corrupted by a later change in the
      // values stored in the original ColMatrix2D object
      // through use of its set method.
      this.point = new ColMatrix2D(
                       point.getData(0),point.getData(1));
    }//end constructor
    //--------------------------------------------------//

    public String toString(){
      return point.getData(0) + "," + point.getData(1);
    }//end toString
    //--------------------------------------------------//
    
    public double getData(int index){
      if((index < 0) || (index > 1)){ 
        throw new IndexOutOfBoundsException();
      }else{
        return point.getData(index);
      }//end else
    }//end getData
    //--------------------------------------------------//
    
    public void setData(int index,double data){
      if((index < 0) || (index > 1)){ 
        throw new IndexOutOfBoundsException();
      }else{
        point.setData(index,data);
      }//end else
    }//end setData
    //--------------------------------------------------//

    //This method draws a small circle around the location
    // of the point on the specified graphics context.
    public void draw(Graphics2D g2D){
      drawOval(g2D,getData(0)-3,
                   getData(1)+3,6,6);
    }//end draw

    //--------------------------------------------------//
    
    //Returns a reference to the ColMatrix2D object that
    // defines this Point2D object.
    public GM02.ColMatrix2D getColMatrix(){
      return point;
    }//end getColMatrix
    //--------------------------------------------------//

    //This method overrides the equals method inherited
    // from the class named Object. It compares the values
    // stored in the ColMatrix2D objects that define two
    // Point2D objects and returns true if they are equal
    // and false otherwise. 
    public boolean equals(Object obj){
      if(point.equals(((GM02.Point2D)obj).
                                         getColMatrix())){
        return true;
      }else{
        return false;
      }//end else
     
    }//end overridden equals method
    //--------------------------------------------------//

    //Gets a displacement vector from one Point2D object
    // to a second Point2D object. The vector points from
    // the object on which the method is called to the
    // object passed as a parameter to the method. Kjell
    // describes this as the distance you would have to
    // walk along the x and then the y axes to get from
    // the first point to the second point.
    public GM02.Vector2D getDisplacementVector(
                                      GM02.Point2D point){
      return new GM02.Vector2D(new GM02.ColMatrix2D(
                            point.getData(0)-getData(0),
                            point.getData(1)-getData(1)));
    }//end getDisplacementVector
    //--------------------------------------------------//
    
    //Adds a Vector2D to a Point2D producing a
    // new Point2D.
    public GM02.Point2D addVectorToPoint(
                                      GM02.Vector2D vec){
      return new GM02.Point2D(new GM02.ColMatrix2D(
                          getData(0) + vec.getData(0),
                          getData(1) + vec.getData(1)));
    }//end addVectorToPoint
    //--------------------------------------------------//
    
    //Returns a new Point2D object that is a clone of
    // the object on which the method is called.
    public Point2D clone(){
      return new Point2D(
                  new ColMatrix2D(getData(0),getData(1)));
    }//end clone
    //--------------------------------------------------//
    
    //The purpose of this method is to rotate a point
    // around a specified anchor point in the x-y plane.
    //The rotation angle is passed in as a double value
    // in degrees with the positive angle of rotation
    // being counter-clockwise.
    //This method does not modify the contents of the
    // Point2D object on which the method is called.
    // Rather, it uses the contents of that object to
    // instantiate, rotate, and return a new Point2D
    // object.
    //For simplicity, this method translates the
    // anchorPoint to the origin, rotates around the
    // origin, and then translates back to the
    // anchorPoint.
    /*
    See http://www.ia.hiof.no/~borres/cgraph/math/threed/
    p-threed.html for a definition of the equations
    required to do the rotation.

    x2 = x1*cos - y1*sin
    y2 = x1*sin + y1*cos
    */ 
    public GM02.Point2D rotate(GM02.Point2D anchorPoint,
                               double angle){
      GM02.Point2D newPoint = this.clone();
      
      double tempX ;
      double tempY;
 
      //Translate anchorPoint to the origin
      GM02.Vector2D tempVec = 
            new GM02.Vector2D(anchorPoint.getColMatrix());
      newPoint = 
              newPoint.addVectorToPoint(tempVec.negate());
      
      //Rotate around the origin.
      tempX = newPoint.getData(0);
      tempY = newPoint.getData(1);
      newPoint.setData(//new x coordinate
                      0,
                      tempX*Math.cos(angle*Math.PI/180) -
                      tempY*Math.sin(angle*Math.PI/180));

      newPoint.setData(//new y coordinate
                      1,
                      tempX*Math.sin(angle*Math.PI/180) +
                      tempY*Math.cos(angle*Math.PI/180));

      //Translate back to anchorPoint
      newPoint = newPoint.addVectorToPoint(tempVec);
      
      return newPoint;

    }//end rotate
    //--------------------------------------------------//
    
    //Multiplies this point by a scaling matrix received
    // as an incoming parameter and returns the scaled
    // point.
    public GM02.Point2D scale(GM02.ColMatrix2D scale){
      return new GM02.Point2D(new ColMatrix2D(
                          getData(0) * scale.getData(0),
                          getData(1) * scale.getData(1)));
    }//end scale
    //--------------------------------------------------//
  }//end class Point2D
  //====================================================//


  public static class Point3D{
    GM02.ColMatrix3D point;
    
    public Point3D(GM02.ColMatrix3D point){//constructor
      //Create and save a clone of the ColMatrix3D object
      // used to define the point to prevent the point
      // from being corrupted by a later change in the
      // values stored in the original ColMatrix3D object
      // through use of its set method.
      this.point = 
         new ColMatrix3D(point.getData(0),
                         point.getData(1),
                         point.getData(2));
    }//end constructor
    //--------------------------------------------------//

    public String toString(){
      return point.getData(0) + "," + point.getData(1) 
                                 + "," + point.getData(2);
    }//end toString
    //--------------------------------------------------//
    
    public double getData(int index){
      if((index < 0) || (index > 2)){ 
        throw new IndexOutOfBoundsException();
      }else{
        return point.getData(index);
      }//end else
    }//end getData
    //--------------------------------------------------//
    
    public void setData(int index,double data){
      if((index < 0) || (index > 2)){ 
        throw new IndexOutOfBoundsException();
      }else{
        point.setData(index,data);
      }//end else
    }//end setData
    //--------------------------------------------------//

    //This method draws a small circle around the location
    // of the point on the specified graphics context.
    public void draw(Graphics2D g2D){
      
      //Get 2D projection coordinate values.
      ColMatrix2D temp = convert3Dto2D(point);
      drawOval(g2D,temp.getData(0)-3,
                   temp.getData(1)+3,
                   6,
                   6);
    }//end draw
    //--------------------------------------------------//
    
    //Returns a reference to the ColMatrix3D object that
    // defines this Point3D object.
    public GM02.ColMatrix3D getColMatrix(){
      return point;
    }//end getColMatrix
    //--------------------------------------------------//

    //This method overrides the equals method inherited
    // from the class named Object. It compares the values
    // stored in the ColMatrix3D objects that define two
    // Point3D objects and returns true if they are equal
    // and false otherwise. 
    public boolean equals(Object obj){
      if(point.equals(((GM02.Point3D)obj).
                                         getColMatrix())){
        return true;
      }else{
        return false;
      }//end else
     
    }//end overridden equals method
    //--------------------------------------------------//

    //Gets a displacement vector from one Point3D object
    // to a second Point3D object. The vector points from
    // the object on which the method is called to the
    // object passed as a parameter to the method. Kjell
    // describes this as the distance you would have to
    // walk along the x and then the y axes to get from
    // the first point to the second point.
    public GM02.Vector3D getDisplacementVector(
                                      GM02.Point3D point){
      return new GM02.Vector3D(new GM02.ColMatrix3D(
                            point.getData(0)-getData(0),
                            point.getData(1)-getData(1),
                            point.getData(2)-getData(2)));
    }//end getDisplacementVector
    //--------------------------------------------------//
    
    //Adds a Vector3D to a Point3D producing a
    // new Point3D.
    public GM02.Point3D addVectorToPoint(
                                      GM02.Vector3D vec){
      return new GM02.Point3D(new GM02.ColMatrix3D(
                          getData(0) + vec.getData(0),
                          getData(1) + vec.getData(1),
                          getData(2) + vec.getData(2)));
    }//end addVectorToPoint
    //--------------------------------------------------//
    
    //Returns a new Point3D object that is a clone of
    // the object on which the method is called.
    public Point3D clone(){
      return new Point3D(new ColMatrix3D(getData(0),
                                         getData(1),
                                         getData(2)));
    }//end clone
    //--------------------------------------------------//
    
    //The purpose of this method is to rotate a point
    // around a specified anchor point in the following
    // order:
    // Rotate around z - rotation in x-y plane.
    // Rotate around x - rotation in y-z plane.
    // Rotate around y - rotation in x-z plane.
    //The rotation angles are passed in as double values
    // in degrees (based on the right-hand rule) in the
    // order given above, packaged in an object of the
    // class GM02.ColMatrix3D. (Note that in this case,
    // the ColMatrix3D object is simply a convenient
    // container and it has no significance from a matrix
    // viewpoint.)
    //The right-hand rule states that if you point the
    // thumb of your right hand in the positive direction
    // of an axis, the direction of positive rotation
    // around that axis is given by the direction that
    // your fingers will be pointing.
    //This method does not modify the contents of the
    // Point3D object on which the method is called.
    // Rather, it uses the contents of that object to
    // instantiate, rotate, and return a new Point3D
    // object.
    //For simplicity, this method translates the
    // anchorPoint to the origin, rotates around the
    // origin, and then translates back to the
    // anchorPoint.
    /*
    See http://www.ia.hiof.no/~borres/cgraph/math/threed/
    p-threed.html for a definition of the equations
    required to do the rotation.
    z-axis 
    x2 = x1*cos - y1*sin
    y2 = x1*sin + y1*cos
    
    x-axis
    y2 = y1*cos(v) - z1*sin(v)
    z2 = y1*sin(v) + z1* cos(v) 
    
    y-axis
    x2 = x1*cos(v) + z1*sin(v)
    z2 = -x1*sin(v) + z1*cos(v)
    */ 
    public GM02.Point3D rotate(GM02.Point3D anchorPoint,
                               GM02.ColMatrix3D angles){
      GM02.Point3D newPoint = this.clone();
      
      double tempX ;
      double tempY;
      double tempZ;
 
      //Translate anchorPoint to the origin
      GM02.Vector3D tempVec = 
            new GM02.Vector3D(anchorPoint.getColMatrix());
      newPoint = 
              newPoint.addVectorToPoint(tempVec.negate());

      double zAngle = angles.getData(0);
      double xAngle = angles.getData(1);
      double yAngle = angles.getData(2);
      
      //Rotate around z-axis
      tempX = newPoint.getData(0);
      tempY = newPoint.getData(1);
      newPoint.setData(//new x coordinate
                      0,
                      tempX*Math.cos(zAngle*Math.PI/180) -
                      tempY*Math.sin(zAngle*Math.PI/180));

      newPoint.setData(//new y coordinate
                      1,
                      tempX*Math.sin(zAngle*Math.PI/180) +
                      tempY*Math.cos(zAngle*Math.PI/180));
      
      //Rotate around x-axis
      tempY = newPoint.getData(1);
      tempZ = newPoint.getData(2);
      newPoint.setData(//new y coordinate
                      1,
                      tempY*Math.cos(xAngle*Math.PI/180) -
                      tempZ*Math.sin(xAngle*Math.PI/180));

      newPoint.setData(//new z coordinate
                      2,
                      tempY*Math.sin(xAngle*Math.PI/180) +
                      tempZ*Math.cos(xAngle*Math.PI/180));
      
      //Rotate around y-axis
      tempX = newPoint.getData(0);
      tempZ = newPoint.getData(2);
      newPoint.setData(//new x coordinate
                      0,
                      tempX*Math.cos(yAngle*Math.PI/180) +
                      tempZ*Math.sin(yAngle*Math.PI/180));

      newPoint.setData(//new z coordinate
                     2,
                     -tempX*Math.sin(yAngle*Math.PI/180) +
                     tempZ*Math.cos(yAngle*Math.PI/180));
      
      //Translate back to anchorPoint
      newPoint = newPoint.addVectorToPoint(tempVec);
      
      return newPoint;

    }//end rotate
    //--------------------------------------------------//
    
    //Multiplies this point by a scaling matrix received
    // as an incoming parameter and returns the scaled
    // point.
    public GM02.Point3D scale(GM02.ColMatrix3D scale){
      return new GM02.Point3D(new ColMatrix3D(
                          getData(0) * scale.getData(0),
                          getData(1) * scale.getData(1),
                          getData(2) * scale.getData(2)));
    }//end scale
    //--------------------------------------------------//
  }//end class Point3D
  //====================================================//
  //====================================================//

  
  public static class Vector2D{
    GM02.ColMatrix2D vector;
    
    public Vector2D(GM02.ColMatrix2D vector){//constructor
      //Create and save a clone of the ColMatrix2D object
      // used to define the vector to prevent the vector
      // from being corrupted by a later change in the
      // values stored in the original ColVector2D object.
      this.vector = new ColMatrix2D(
                     vector.getData(0),vector.getData(1));
    }//end constructor
    //--------------------------------------------------//

    public String toString(){
      return vector.getData(0) + "," + vector.getData(1);
    }//end toString
    //--------------------------------------------------//
    
    public double getData(int index){
      if((index < 0) || (index > 1)){
        throw new IndexOutOfBoundsException();
      }else{
        return vector.getData(index);
      }//end else
    }//end getData
    //--------------------------------------------------//
    
    public void setData(int index,double data){
      if((index < 0) || (index > 1)){ 
        throw new IndexOutOfBoundsException();
      }else{
        vector.setData(index,data);
      }//end else
    }//end setData
    //--------------------------------------------------//
    
    //This method draws a vector on the specified graphics
    // context, with the tail of the vector located at a
    // specified point, and with a small filled circle at
    // the head.
    public void draw(Graphics2D g2D,GM02.Point2D tail){

      drawLine(g2D,
               tail.getData(0),
               tail.getData(1),
               tail.getData(0)+vector.getData(0),
               tail.getData(1)+vector.getData(1));

      fillOval(g2D,
               tail.getData(0)+vector.getData(0)-3,
               tail.getData(1)+vector.getData(1)+3,
               6,
               6);
    }//end draw
    //--------------------------------------------------//
    
    //Returns a reference to the ColMatrix2D object that
    // defines this Vector2D object.
    public GM02.ColMatrix2D getColMatrix(){
      return vector;
    }//end getColMatrix
    //--------------------------------------------------//

    //This method overrides the equals method inherited
    // from the class named Object. It compares the values
    // stored in the ColMatrix2D objects that define two
    // Vector2D objects and returns true if they are equal
    // and false otherwise. 
    public boolean equals(Object obj){
      if(vector.equals((
                     (GM02.Vector2D)obj).getColMatrix())){
        return true;
      }else{
        return false;
      }//end else
     
    }//end overridden equals method
    //--------------------------------------------------//
    
    //Adds this vector to a vector received as an incoming
    // parameter and returns the sum as a vector.
    public GM02.Vector2D add(GM02.Vector2D vec){
      return new GM02.Vector2D(new ColMatrix2D(
                       vec.getData(0)+vector.getData(0),
                       vec.getData(1)+vector.getData(1)));
    }//end add
    //--------------------------------------------------//
    
    //Returns the length of a Vector2D object.
    public double getLength(){
      return Math.sqrt(
           getData(0)*getData(0) + getData(1)*getData(1));
    }//end getLength
    //--------------------------------------------------//
    
    //Multiplies this vector by a scale factor received as
    // an incoming parameter and returns the scaled
    // vector.
    public GM02.Vector2D scale(Double factor){
      return new GM02.Vector2D(new ColMatrix2D(
                                    getData(0) * factor,
                                    getData(1) * factor));
    }//end scale
    //--------------------------------------------------//
    
    //Changes the sign on each of the vector components
    // and returns the negated vector.
    public GM02.Vector2D negate(){
      return new GM02.Vector2D(new ColMatrix2D(
                                            -getData(0),
                                            -getData(1)));
    }//end negate
    //--------------------------------------------------//
    
    //Returns a new vector that points in the same
    // direction but has a length of one unit.
    public GM02.Vector2D normalize(){
      double length = getLength();
      return new GM02.Vector2D(new ColMatrix2D(
                                      getData(0)/length,
                                      getData(1)/length));
    }//end normalize
    //--------------------------------------------------//
    
    //Computes the dot product of two Vector2D
    // objects and returns the result as type double.
    public double dot(GM02.Vector2D vec){
      GM02.ColMatrix2D matrixA = getColMatrix();
      GM02.ColMatrix2D matrixB = vec.getColMatrix();
      return matrixA.dot(matrixB);
    }//end dot
    //--------------------------------------------------//
    
    //Computes and returns the angle between two Vector2D
    // objects. The angle is returned in degrees as type
    // double.
    public double angle(GM02.Vector2D vec){
      GM02.Vector2D normA = normalize();
      GM02.Vector2D normB = vec.normalize();
      double normDotProd = normA.dot(normB);
      return Math.toDegrees(Math.acos(normDotProd));
    }//end angle
    //--------------------------------------------------//
  }//end class Vector2D
  //====================================================//


  public static class Vector3D{
    GM02.ColMatrix3D vector;
    
    public Vector3D(GM02.ColMatrix3D vector){//constructor
      //Create and save a clone of the ColMatrix3D object
      // used to define the vector to prevent the vector
      // from being corrupted by a later change in the
      // values stored in the original ColMatris3D object.
      this.vector = new ColMatrix3D(vector.getData(0),
                                    vector.getData(1),
                                    vector.getData(2));
    }//end constructor
    //--------------------------------------------------//

    public String toString(){
      return vector.getData(0) + "," + vector.getData(1) 
                                + "," + vector.getData(2);
    }//end toString
    //--------------------------------------------------//
    
    public double getData(int index){
      if((index < 0) || (index > 2)){
        throw new IndexOutOfBoundsException();
      }else{
        return vector.getData(index);
      }//end else
    }//end getData
    //--------------------------------------------------//
    
    public void setData(int index,double data){
      if((index < 0) || (index > 2)){ 
        throw new IndexOutOfBoundsException();
      }else{
        vector.setData(index,data);
      }//end else
    }//end setData
    //--------------------------------------------------//
    
    //This method draws a vector on the specified graphics
    // context, with the tail of the vector located at a
    // specified point, and with a small circle at the
    // head.
    public void draw(Graphics2D g2D,GM02.Point3D tail){
      
      //Get a 2D projection of the tail
      GM02.ColMatrix2D tail2D = convert3Dto2D(tail.point);
      
      //Get the 3D location of the head
      GM02.ColMatrix3D head = 
                      tail.point.add(this.getColMatrix());
      
      //Get a 2D projection of the head
      GM02.ColMatrix2D head2D = convert3Dto2D(head);
      drawLine(g2D,tail2D.getData(0),
                   tail2D.getData(1),
                   head2D.getData(0),
                   head2D.getData(1));      

      //Draw a small filled circle to identify the head.
      fillOval(g2D,head2D.getData(0)-3,
                   head2D.getData(1)+3,
                   6,
                   6);

    }//end draw
    //--------------------------------------------------//
    
    //Returns a reference to the ColMatrix3D object that
    // defines this Vector3D object.
    public GM02.ColMatrix3D getColMatrix(){
      return vector;
    }//end getColMatrix
    //--------------------------------------------------//

    //This method overrides the equals method inherited
    // from the class named Object. It compares the values
    // stored in the ColMatrix3D objects that define two
    // Vector3D objects and returns true if they are equal
    // and false otherwise. 
    public boolean equals(Object obj){
      if(vector.equals((
                     (GM02.Vector3D)obj).getColMatrix())){
        return true;
      }else{
        return false;
      }//end else
     
    }//end overridden equals method
    //--------------------------------------------------//
    
    //Adds this vector to a vector received as an incoming
    // parameter and returns the sum as a vector.
    public GM02.Vector3D add(GM02.Vector3D vec){
      return new GM02.Vector3D(new ColMatrix3D(
                       vec.getData(0)+vector.getData(0),
                       vec.getData(1)+vector.getData(1),
                       vec.getData(2)+vector.getData(2)));
    }//end add
    //--------------------------------------------------//
    
    //Returns the length of a Vector3D object.
    public double getLength(){
      return Math.sqrt(getData(0)*getData(0) + 
                       getData(1)*getData(1) + 
                       getData(2)*getData(2));
    }//end getLength
    //--------------------------------------------------//
    
    //Multiplies this vector by a scale factor received as
    // an incoming parameter and returns the scaled
    // vector.
    public GM02.Vector3D scale(Double factor){
      return new GM02.Vector3D(new ColMatrix3D(
                                    getData(0) * factor,
                                    getData(1) * factor,
                                    getData(2) * factor));
    }//end scale
    //--------------------------------------------------//
    
    //Changes the sign on each of the vector components
    // and returns the negated vector.
    public GM02.Vector3D negate(){
      return new GM02.Vector3D(new ColMatrix3D(
                                            -getData(0),
                                            -getData(1),
                                            -getData(2)));
    }//end negate
    //--------------------------------------------------//
    
    //Returns a new vector that points in the same
    // direction but has a length of one unit.
    public GM02.Vector3D normalize(){
      double length = getLength();
      return new GM02.Vector3D(new ColMatrix3D(
                                      getData(0)/length,
                                      getData(1)/length,
                                      getData(2)/length));
    }//end normalize
    //--------------------------------------------------//
    
    //Computes the dot product of two Vector3D
    // objects and returns the result as type double.
    public double dot(GM02.Vector3D vec){
      GM02.ColMatrix3D matrixA = getColMatrix();
      GM02.ColMatrix3D matrixB = vec.getColMatrix();
      return matrixA.dot(matrixB);
    }//end dot
    //--------------------------------------------------//
    
    //Computes and returns the angle between two Vector3D
    // objects. The angle is returned in degrees as type
    // double.
    public double angle(GM02.Vector3D vec){
      GM02.Vector3D normA = normalize();
      GM02.Vector3D normB = vec.normalize();
      double normDotProd = normA.dot(normB);
      return Math.toDegrees(Math.acos(normDotProd));
    }//end angle
    //--------------------------------------------------//
  }//end class Vector3D
  //====================================================//
  //====================================================//

  
  //A line is defined by two points. One is called the
  // tail and the other is called the head. Note that this
  // class has the same name as one of the classes in
  // the Graphics2D class. Therefore, if the class from
  // the Graphics2D class is used in some future upgrade
  // to this program, it will have to be fully qualified.
  public static class Line2D{
    GM02.Point2D[] line = new GM02.Point2D[2];
    
    public Line2D(GM02.Point2D tail,GM02.Point2D head){
      //Create and save clones of the points used to
      // define the line to prevent the line from being 
      // corrupted by a later change in the coordinate
      // values of the points.
      this.line[0] = new Point2D(new GM02.ColMatrix2D(
                        tail.getData(0),tail.getData(1)));
      this.line[1] = new Point2D(new GM02.ColMatrix2D(
                        head.getData(0),head.getData(1)));
    }//end constructor
    //--------------------------------------------------//

    public String toString(){
      return "Tail = " + line[0].getData(0) + "," 
             + line[0].getData(1) + "nHead = " 
             + line[1].getData(0) + "," 
             + line[1].getData(1);
    }//end toString
    //--------------------------------------------------//

    public GM02.Point2D getTail(){
      return line[0];
    }//end getTail
    //--------------------------------------------------//
    
    public GM02.Point2D getHead(){
      return line[1];
    }//end getHead
    //--------------------------------------------------//
    
    public void setTail(GM02.Point2D newPoint){
      //Create and save a clone of the new point to
      // prevent the line from being corrupted by a
      // later change in the coordinate values of the
      // point.
      this.line[0] = new Point2D(new GM02.ColMatrix2D(
                newPoint.getData(0),newPoint.getData(1)));
    }//end setTail
    //--------------------------------------------------//
    
    public void setHead(GM02.Point2D newPoint){
      //Create and save a clone of the new point to
      // prevent the line from being corrupted by a
      // later change in the coordinate values of the
      // point.
      this.line[1] = new Point2D(new GM02.ColMatrix2D(
                newPoint.getData(0),newPoint.getData(1)));
    }//end setHead
    //--------------------------------------------------//
    
    public void draw(Graphics2D g2D){
      drawLine(g2D,getTail().getData(0),
                   getTail().getData(1),
                   getHead().getData(0),
                   getHead().getData(1));
    }//end draw
    //--------------------------------------------------//
  }//end class Line2D
  //====================================================//


  //A line is defined by two points. One is called the
  // tail and the other is called the head.
  public static class Line3D{
    GM02.Point3D[] line = new GM02.Point3D[2];
    
    public Line3D(GM02.Point3D tail,GM02.Point3D head){
      //Create and save clones of the points used to
      // define the line to prevent the line from being 
      // corrupted by a later change in the coordinate
      // values of the points.
      this.line[0] = new Point3D(new GM02.ColMatrix3D(
                                        tail.getData(0),
                                        tail.getData(1),
                                        tail.getData(2)));
      this.line[1] = new Point3D(new GM02.ColMatrix3D(
                                        head.getData(0),
                                        head.getData(1),
                                        head.getData(2)));
    }//end constructor
    //--------------------------------------------------//

    public String toString(){
      return "Tail = " + line[0].getData(0) + "," 
                       + line[0].getData(1)  + "," 
                       + line[0].getData(2) 
                       + "nHead = " 
                       + line[1].getData(0) + "," 
                       + line[1].getData(1) + ","      
                       + line[1].getData(2);
    }//end toString
    //--------------------------------------------------//

    public GM02.Point3D getTail(){
      return line[0];
    }//end getTail
    //--------------------------------------------------//
    
    public GM02.Point3D getHead(){
      return line[1];
    }//end getHead
    //--------------------------------------------------//

    public void setTail(GM02.Point3D newPoint){
      //Create and save a clone of the new point to
      // prevent the line from being corrupted by a
      // later change in the coordinate values of the
      // point.
      this.line[0] = new Point3D(new GM02.ColMatrix3D(
                                    newPoint.getData(0),
                                    newPoint.getData(1),
                                    newPoint.getData(2)));
    }//end setTail
    //--------------------------------------------------//
    
    public void setHead(GM02.Point3D newPoint){
      //Create and save a clone of the new point to
      // prevent the line from being corrupted by a
      // later change in the coordinate values of the
      // point.
      this.line[1] = new Point3D(new GM02.ColMatrix3D(
                                    newPoint.getData(0),
                                    newPoint.getData(1),
                                    newPoint.getData(2)));
    }//end setHead
    //--------------------------------------------------//

    public void draw(Graphics2D g2D){
      
      //Get 2D projection coordinates.
      GM02.ColMatrix2D tail = 
                           convert3Dto2D(getTail().point);
      GM02.ColMatrix2D head = 
                           convert3Dto2D(getHead().point);

      drawLine(g2D,tail.getData(0),
                   tail.getData(1),
                   head.getData(0),
                   head.getData(1));
    }//end draw
    //--------------------------------------------------//
  }//end class Line3D
  //====================================================//

}//end class GM02
//======================================================//

/*DotProd3D05.java 
Copyright 2008, R.G.Baldwin
Revised 03/06/08

The purpose of this program is to demonstrate how the dot 
product can be used to compute nine different angles of 
interest that a vector makes with various elements in 3D 
space.

First, the program computes and displays the angle between
a user-specified vector and each of the X, Y, and Z axes.
These values are displayed with the labels Angle X, 
Angle Y, and Angle Z.

Then the program computes and displays the angle between
the vector and each of the XY, YZ, and ZX planes. In
this case, the program computes the smallest possible 
angle by projecting the vector onto the plane and then 
computing the angle between the vector and its projection.
These values are displayed with the labels Angle XY, 
Angle YZ, and Angle ZX.

Finally, the program computes and displays the angle 
between the projection of the vector on each of the three 
planes and one of the axes that defines each plane. 
Obviously, the angle between the projection and the other 
axis that defines the plane is 90 degrees less the 
computed angle.  Specifically the values that are computed
and displayed are:

Projection onto the XY plane relative to the x-axis,
displayed with the label Angle PX.

Projection onto the YZ plane relative to the y-axis,
displayed with the label Angle PY.

Projection onto the ZX plane relative to the z-axis,
displayed with the label Angle PZ.

All angles are reported as positive angles in degrees.
  
Study Kjell through Chapter 10, Angle between 3D Vectors.

A GUI is provided that allows the user to enter three
double values that define a GM02.Vector3D object.  The GUI
also provides an OK button as well as nine text fields 
used to display the computed results described above.

In addition, the GUI provides a 3D drawing area.

When the user clicks the OK button, the program draws the
user-specified vector in black with the tail located at 
the origin in 3D space. It also draws the projection of
that vector in magenta on each of the XY, YZ, AND ZX
planes

Tested using JDK 1.6 under WinXP.
*********************************************************/
import java.awt.*;
import javax.swing.*;
import java.awt.event.*;

class DotProd3D05{
  public static void main(String[] args){
    GUI guiObj = new GUI();
  }//end main
}//end controlling class DotProd3D05
//======================================================//

class GUI extends JFrame implements ActionListener{
  //Specify the horizontal and vertical size of a JFrame
  // object.
  int hSize = 400;
  int vSize = 400;
  Image osi;//an off-screen image
  int osiWidth;//off-screen image width
  int osiHeight;//off-screen image height
  MyCanvas myCanvas;//a subclass of Canvas 
  Graphics2D g2D;//off-screen graphics context.

  //User input components.
  JTextField vecX = new JTextField("50.0");
  JTextField vecY = new JTextField("100.0");
  JTextField vecZ = new JTextField("0.0");

  JTextField angleX = new JTextField("0");
  JTextField angleY = new JTextField("0");
  JTextField angleZ = new JTextField("0");
  
  JTextField angleXY = new JTextField("0");
  JTextField angleYZ = new JTextField("0");
  JTextField angleZX = new JTextField("0");
  
  JTextField anglePX = new JTextField("0");
  JTextField anglePY = new JTextField("0");
  JTextField anglePZ = new JTextField("0");

  JButton button = new JButton("OK");
  
  //----------------------------------------------------//
  
  GUI(){//constructor

    //Set JFrame size, title, and close operation.
    setSize(hSize,vSize);
    setTitle("Copyright 2008,R.G.Baldwin");
    setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
    
    //Instantiate a JPanel that will house the user input
    // components and set its layout manager.
    JPanel controlPanel = new JPanel();
    controlPanel.setLayout(new GridLayout(0,6));

    //Add the user input components and appropriate labels
    // to the control panel.
    controlPanel.add(new JLabel(" Vec X "));    
    controlPanel.add(vecX);

    controlPanel.add(new JLabel(" Vec Y "));    
    controlPanel.add(vecY);
    
    controlPanel.add(new JLabel(" Vec Z "));    
    controlPanel.add(vecZ);
    
    controlPanel.add(new JLabel(" Angle X "));    
    controlPanel.add(angleX);
    
    controlPanel.add(new JLabel(" Angle Y "));    
    controlPanel.add(angleY);
    
    controlPanel.add(new JLabel(" Angle Z "));    
    controlPanel.add(angleZ);
    
    controlPanel.add(new JLabel(" Angle XY "));    
    controlPanel.add(angleXY);
    
    controlPanel.add(new JLabel(" Angle YZ "));    
    controlPanel.add(angleYZ);
    
    controlPanel.add(new JLabel(" Angle ZX "));
    controlPanel.add(angleZX);
    
    controlPanel.add(new JLabel(" Angle PX "));    
    controlPanel.add(anglePX);
    
    controlPanel.add(new JLabel(" Angle PY "));    
    controlPanel.add(anglePY);
    
    controlPanel.add(new JLabel(" Angle PZ "));
    controlPanel.add(anglePZ);
    
    controlPanel.add(button);

    //Add the control panel to the SOUTH position in the
    // JFrame.
    this.getContentPane().add(
                        BorderLayout.SOUTH,controlPanel);

    
    //Create a new drawing canvas and add it to the
    // CENTER of the JFrame above the control panel.
    myCanvas = new MyCanvas();
    this.getContentPane().add(
                            BorderLayout.CENTER,myCanvas);

    //This object must be visible before you can get an
    // off-screen image.  It must also be visible before
    // you can compute the size of the canvas.
    setVisible(true);
    
    //Make the size of the off-screen image match the
    // size of the canvas.
    osiWidth = myCanvas.getWidth();
    osiHeight = myCanvas.getHeight();
    
    //Create an off-screen image and get a graphics
    // context on it.
    osi = createImage(osiWidth,osiHeight);
    g2D = (Graphics2D)(osi.getGraphics());
    

    //Translate the origin to the center.
    GM02.translate(g2D,0.5*osiWidth,-0.5*osiHeight);

    
    //Register this object as an action listener on the
    // button.
    button.addActionListener(this);

    //Cause the overridden paint method belonging to
    // myCanvas to be executed.
    myCanvas.repaint();
    
  }//end constructor
  //----------------------------------------------------//
  
  //This method is used to draw orthogonal 3D axes on the
  // off-screen image that intersect at the origin.
  private void setCoordinateFrame(Graphics2D g2D){

    //Erase the screen
    g2D.setColor(Color.WHITE);
    GM02.fillRect(g2D,-osiWidth/2,osiHeight/2,
                                      osiWidth,osiHeight);

    //Draw x-axis in RED
    g2D.setColor(Color.RED);
    GM02.Point3D pointA = new GM02.Point3D(
                   new GM02.ColMatrix3D(-osiWidth/2,0,0));
    GM02.Point3D pointB = new GM02.Point3D(
                    new GM02.ColMatrix3D(osiWidth/2,0,0));
    new GM02.Line3D(pointA,pointB).draw(g2D);
    
    //Draw y-axis in GREEN
    g2D.setColor(Color.GREEN);
    pointA = new GM02.Point3D(
                  new GM02.ColMatrix3D(0,-osiHeight/2,0));
    pointB = new GM02.Point3D(
                   new GM02.ColMatrix3D(0,osiHeight/2,0));
    new GM02.Line3D(pointA,pointB).draw(g2D);
    
    //Draw z-axis in BLUE. Make its length the same as the
    // length of the x-axis.
    g2D.setColor(Color.BLUE);
    pointA = new GM02.Point3D(
                   new GM02.ColMatrix3D(0,0,-osiWidth/2));
    pointB = new GM02.Point3D(
                    new GM02.ColMatrix3D(0,0,osiWidth/2));
    new GM02.Line3D(pointA,pointB).draw(g2D);

  }//end setCoordinateFrame method
  //----------------------------------------------------//
  
  //This method is called to respond to a click on the
  // button.
  public void actionPerformed(ActionEvent e){
    
    //Erase the off-screen image and draw the axes.
    setCoordinateFrame(g2D);
    
    //Create one ColMatrix3D object based on the user
    // input values.
    GM02.ColMatrix3D matrixA = new GM02.ColMatrix3D(
                  Double.parseDouble(vecX.getText()),
                  Double.parseDouble(vecY.getText()),
                  Double.parseDouble(vecZ.getText()));
                  
    //Create ColMatrix3D objects that represent each of
    // the three axes.
    GM02.ColMatrix3D matrixX = 
                              new GM02.ColMatrix3D(1,0,0);
    GM02.ColMatrix3D matrixY = 
                              new GM02.ColMatrix3D(0,1,0);
    GM02.ColMatrix3D matrixZ = 
                              new GM02.ColMatrix3D(0,0,1);
    
    //Create ColMatrix3D objects that represent the
    // projection of the user-specified vector onto each
    // of the three planes.
    GM02.ColMatrix3D matrixXY = new GM02.ColMatrix3D(
                      Double.parseDouble(vecX.getText()),
                      Double.parseDouble(vecY.getText()),
                      0);
                  
    GM02.ColMatrix3D matrixYZ = new GM02.ColMatrix3D(
                      0,
                      Double.parseDouble(vecY.getText()),
                      Double.parseDouble(vecZ.getText()));
                  
    GM02.ColMatrix3D matrixZX = new GM02.ColMatrix3D(
                      Double.parseDouble(vecX.getText()),
                      0,
                      Double.parseDouble(vecZ.getText()));
    


    //Use the ColMatrix3D objects to create Vector3D
    // objects representing the user-specified vector and
    // each of the axes.
    GM02.Vector3D vecA = new GM02.Vector3D(matrixA);
    GM02.Vector3D vecX = new GM02.Vector3D(matrixX);
    GM02.Vector3D vecY = new GM02.Vector3D(matrixY);
    GM02.Vector3D vecZ = new GM02.Vector3D(matrixZ);
    
    //Create Vector3D objects that represent the
    // projection of the user-specified vector on each of
    // the planes.
    GM02.Vector3D vecXY = new GM02.Vector3D(matrixXY);
    GM02.Vector3D vecYZ = new GM02.Vector3D(matrixYZ);
    GM02.Vector3D vecZX = new GM02.Vector3D(matrixZX);
    	                    
    //Draw the projection of the user specified vector on
    // each of the three planes.
    g2D.setColor(Color.MAGENTA);
    vecXY.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
    vecYZ.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
    vecZX.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
    
    //Draw the user-specified vector with its tail at the
    // origin.
    g2D.setColor(Color.BLACK);
    vecA.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
    
    	                    

    //Compute and display the angle relative to the
    // x-axis.
    double angle = vecA.angle(vecX);
    angleX.setText("" +  prepareForDisplay(angle));

    //Compute and display the angle relative to the
    // y-axis.
    angle = vecA.angle(vecY);
    angleY.setText("" +  prepareForDisplay(angle));
    
    //Compute and display the angle relative to the
    // z-axis.
    angle = vecA.angle(vecZ);
    angleZ.setText("" +  prepareForDisplay(angle));
    
    
    //Compute and display the angle relative to the
    // XY plane
    angle = vecA.angle(vecXY);
    angleXY.setText("" +  prepareForDisplay(angle));
    
    //Compute and display the angle relative to the
    // YZ plane
    angle = vecA.angle(vecYZ);
    angleYZ.setText("" +  prepareForDisplay(angle));
    
    //Compute and display the angle relative to the
    // ZX plane
    angle = vecA.angle(vecZX);
    angleZX.setText("" +  prepareForDisplay(angle));
    
    
    //Compute and display the angle of the projection onto
    // the XY plane relative to the x-axis
    angle = vecXY.angle(vecX);
    anglePX.setText("" +  prepareForDisplay(angle));
    
    //Compute and display the angle of the projection onto
    // the YZ plane relative to the y-axis
    angle = vecYZ.angle(vecY);
    anglePY.setText("" +  prepareForDisplay(angle));
    
    //Compute and display the angle of the projection onto
    // the ZX plane relative to the z-axis
    angle = vecZX.angle(vecZ);
    anglePZ.setText("" +  prepareForDisplay(angle));    
    

    myCanvas.repaint();//Copy off-screen image to canvas.
    
    
  }//end actionPerformed
  //----------------------------------------------------//
  
  //The code in this method prepares a double value for
  // display in a text field by eliminating exponential
  // format for very small values and setting the number
  // of decimal digits to four.
  private double prepareForDisplay(double data){
      //Eliminate exponential notation in the display.
      if(Math.abs(data) < 0.001){
        data = 0.0;
      }//end if
      
      //Convert to four decimal digits.
      return ((int)(10000*data))/10000.0;
  }//end prepareForDisplay
  
  //====================================================//
  
  
  //This is an inner class of the GUI class.
  class MyCanvas extends Canvas{
    //Override the paint() method. This method will be
    // called when the JFrame and the Canvas appear on the
    // screen or when the repaint method is called on the
    // Canvas object.
    //The purpose of this method is to display the
    // off-screen image on the screen.
    public void paint(Graphics g){
      g.drawImage(osi,0,0,this);
    }//end overridden paint()
    
  }//end inner class MyCanvas
    
}//end class GUI
//======================================================//

/*DotProd3D06.java 
Copyright 2008, R.G.Baldwin
Revised 03/09/08

This program demonstrates how the dot product can be used
to find vectors that are perpendicular to a given vector.

The program computes and displays normalized and scaled
versions of six of the infinite set of vectors that are 
perpendicular to a user specified vector.

If the user specifies one of the coordinates to be zero
or close to zero, the program only computes and displays 
four of the possible vectors in order to avoid performing 
division by a near-zero value. For a value of zero, the
orientation of two of the vectors will overlay the 
orientation of the other two. Because they are the same
length, and occupy the same space, you will only see two
vectors.

If the user specifies two of the coordinates to be zero
or close to zero, the program doesn't produce a valid
result.  Instead, it displays the coordinates for a 
perpendicular vector where all of the coordinates are
zero and displays NaN for the angle.

Study Kjell through Chapter 10, Angle between 3D Vectors.

A GUI is provided that allows the user to enter three
double values that define a GM02.Vector3D object.  The GUI
also provides an OK button.

In addition, the GUI provides a 3D drawing area.

When the user clicks the OK button, the program draws the
user-specified vector in black with the tail located at 
the origin in 3D space. It also draws normalized versions
of the perpendicular vectors in magenta with their tails 
located at the origin.  Each normalized vector is scaled
by a factor of 50 before it is drawn.

The program also displays the values of three of the
perpendicular vectors on the command-line screen along
with the angle between the perpendicular vector and the
user-specified vector. The angle should be 90 degrees or
at least very very close to 90 degrees. The other three
perpendicular vectors are simply negated versions of the
three for which the values are displayed.

Tested using JDK 1.6 under WinXP.
*********************************************************/
import java.awt.*;
import javax.swing.*;
import java.awt.event.*;

class DotProd3D06{
  public static void main(String[] args){
    GUI guiObj = new GUI();
  }//end main
}//end controlling class DotProd3D06
//======================================================//

class GUI extends JFrame implements ActionListener{
  //Specify the horizontal and vertical size of a JFrame
  // object.
  int hSize = 400;
  int vSize = 400;
  Image osi;//an off-screen image
  int osiWidth;//off-screen image width
  int osiHeight;//off-screen image height
  MyCanvas myCanvas;//a subclass of Canvas 
  Graphics2D g2D;//off-screen graphics context.

  //User input components.
  JTextField vecX = new JTextField("50.0");
  JTextField vecY = new JTextField("50.0");
  JTextField vecZ = new JTextField("50.0");

  JButton button = new JButton("OK");
  
  //----------------------------------------------------//
  
  GUI(){//constructor

    //Set JFrame size, title, and close operation.
    setSize(hSize,vSize);
    setTitle("Copyright 2008,R.G.Baldwin");
    setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
    
    //Instantiate a JPanel that will house the user input
    // components and set its layout manager.
    JPanel controlPanel = new JPanel();
    controlPanel.setLayout(new GridLayout(0,6));

    //Add the user input components and appropriate labels
    // to the control panel.
    controlPanel.add(new JLabel(" Vec X "));    
    controlPanel.add(vecX);

    controlPanel.add(new JLabel(" Vec Y "));    
    controlPanel.add(vecY);
    
    controlPanel.add(new JLabel(" Vec Z "));    
    controlPanel.add(vecZ);

    controlPanel.add(button);

    //Add the control panel to the SOUTH position in the
    // JFrame.
    this.getContentPane().add(
                        BorderLayout.SOUTH,controlPanel);

    
    //Create a new drawing canvas and add it to the
    // CENTER of the JFrame above the control panel.
    myCanvas = new MyCanvas();
    this.getContentPane().add(
                            BorderLayout.CENTER,myCanvas);

    //This object must be visible before you can get an
    // off-screen image.  It must also be visible before
    // you can compute the size of the canvas.
    setVisible(true);
    
    //Make the size of the off-screen image match the
    // size of the canvas.
    osiWidth = myCanvas.getWidth();
    osiHeight = myCanvas.getHeight();
    
    //Create an off-screen image and get a graphics
    // context on it.
    osi = createImage(osiWidth,osiHeight);
    g2D = (Graphics2D)(osi.getGraphics());
    

    //Translate the origin to the center.
    GM02.translate(g2D,0.5*osiWidth,-0.5*osiHeight);

    
    //Register this object as an action listener on the
    // button.
    button.addActionListener(this);

    //Cause the overridden paint method belonging to
    // myCanvas to be executed.
    myCanvas.repaint();
    
  }//end constructor
  //----------------------------------------------------//
  
  //This method is used to draw orthogonal 3D axes on the
  // off-screen image that intersect at the origin.
  private void setCoordinateFrame(Graphics2D g2D){

    //Erase the screen
    g2D.setColor(Color.WHITE);
    GM02.fillRect(g2D,-osiWidth/2,osiHeight/2,
                                      osiWidth,osiHeight);

    //Draw x-axis in RED
    g2D.setColor(Color.RED);
    GM02.Point3D pointA = new GM02.Point3D(
                   new GM02.ColMatrix3D(-osiWidth/2,0,0));
    GM02.Point3D pointB = new GM02.Point3D(
                    new GM02.ColMatrix3D(osiWidth/2,0,0));
    new GM02.Line3D(pointA,pointB).draw(g2D);
    
    //Draw y-axis in GREEN
    g2D.setColor(Color.GREEN);
    pointA = new GM02.Point3D(
                  new GM02.ColMatrix3D(0,-osiHeight/2,0));
    pointB = new GM02.Point3D(
                   new GM02.ColMatrix3D(0,osiHeight/2,0));
    new GM02.Line3D(pointA,pointB).draw(g2D);
    
    //Draw z-axis in BLUE. Make its length the same as the
    // length of the x-axis.
    g2D.setColor(Color.BLUE);
    pointA = new GM02.Point3D(
                   new GM02.ColMatrix3D(0,0,-osiWidth/2));
    pointB = new GM02.Point3D(
                    new GM02.ColMatrix3D(0,0,osiWidth/2));
    new GM02.Line3D(pointA,pointB).draw(g2D);

  }//end setCoordinateFrame method
  //----------------------------------------------------//
  
  //This method is called to respond to a click on the
  // button.
  public void actionPerformed(ActionEvent e){
    
    //Erase the off-screen image and draw the axes.
    setCoordinateFrame(g2D);
    
    //Get and save the user specified coordinate values.
    double xCoor = Double.parseDouble(vecX.getText());
    double yCoor = Double.parseDouble(vecY.getText());
    double zCoor = Double.parseDouble(vecZ.getText());
    
    //Create a ColMatrix3D object based on the user input
    // values.
    GM02.ColMatrix3D matrixA = 
                  new GM02.ColMatrix3D(xCoor,yCoor,zCoor);
    
    //Use the ColMatrix3D object to create a Vector3D
    // object representing the user-specified vector.
    GM02.Vector3D vecA = new GM02.Vector3D(matrixA);
    
    //Draw the user-specified vector with its tail at the
    // origin.
    g2D.setColor(Color.BLACK);
    vecA.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
    
    //Create and draw the perpendicular vectors.  However,
    // if a coordinate value is near zero, don't attempt
    // to create and draw the perpendicular vector that
    // would require division by the near-zero value.
    GM02.Vector3D tempVec;
    GM02.ColMatrix3D tempMatrix;
    g2D.setColor(Color.MAGENTA);
    
    if(Math.abs(zCoor) > 0.001){
      tempMatrix = new GM02.ColMatrix3D(
          xCoor,yCoor,-(xCoor*xCoor + yCoor*yCoor)/zCoor);
      tempVec = new GM02.Vector3D(tempMatrix);
      System.out.println(tempVec);
      //Normalize and scale the perpendicular vector.
      tempVec = tempVec.normalize().scale(50.0);
      tempVec.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
      tempVec.negate().draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
      System.out.println(vecA.angle(tempVec));
    }//end if
    
    if(Math.abs(yCoor) > 0.001){
      tempMatrix = new GM02.ColMatrix3D(
          xCoor,-(xCoor*xCoor + zCoor*zCoor)/yCoor,zCoor);
      tempVec = new GM02.Vector3D(tempMatrix);
      System.out.println(tempVec);
      //Normalize and scale the perpendicular vector.
      tempVec = tempVec.normalize().scale(50.0);
      tempVec.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
      tempVec.negate().draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
      System.out.println(vecA.angle(tempVec));
    }//end if
    
    if(Math.abs(xCoor) > 0.001){
      tempMatrix = new GM02.ColMatrix3D(
         -(yCoor*yCoor + zCoor*zCoor)/xCoor, yCoor,zCoor);
      tempVec = new GM02.Vector3D(tempMatrix);
      System.out.println(tempVec);
      //Normalize and scale the perpendicular vector.
      tempVec = tempVec.normalize().scale(50.0);
      tempVec.draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
      tempVec.negate().draw(g2D,new GM02.Point3D(
                            new GM02.ColMatrix3D(0,0,0)));
      System.out.println(vecA.angle(tempVec));
    }//end if


    myCanvas.repaint();//Copy off-screen image to canvas.
    System.out.println();//blank line
  }//end actionPerformed
  //----------------------------------------------------//
 
  //====================================================//
  
  //This is an inner class of the GUI class.
  class MyCanvas extends Canvas{
    //Override the paint() method. This method will be
    // called when the JFrame and the Canvas appear on the
    // screen or when the repaint method is called on the
    // Canvas object.
    //The purpose of this method is to display the
    // off-screen image on the screen.
    public void paint(Graphics g){
      g.drawImage(osi,0,0,this);
    }//end overridden paint()
    
  }//end inner class MyCanvas
    
}//end class GUI
//======================================================//

/*DotProd3D04.java 
Copyright 2008, R.G.Baldwin
Revised 03/07/08

The purpose of this program is serve as a counterpoint to 
the program named Prob3D03, which demonstrates backface 
culling.

This program draws the same 3D object as the one drawn in 
DotProd3D03 but without the benefit of backface culling.

Study Kjell through Chapter 10, Angle between 3D Vectors.

The program draws a 3D circular cylinder by stacking 20 
circular disks on the x-z plane. The disks are centered on
the y-axis and are parallel to the x-z plane. The 
thickness of each disk is 5 vertical units.

There is no backface culling in this program, so all of 
the lines that should be hidden show through.


Tested using JDK 1.6 under WinXP.
*********************************************************/
import java.awt.*;
import javax.swing.*;

class DotProd3D04{
  public static void main(String[] args){
    GUI guiObj = new GUI();
  }//end main
}//end controlling class DotProd3D04
//======================================================//

class GUI extends JFrame{
  //Specify the horizontal and vertical size of a JFrame
  // object.
  int hSize = 230;
  int vSize = 250;
  Image osi;//an off-screen image
  int osiWidth;//off-screen image width
  int osiHeight;//off-screen image height
  MyCanvas myCanvas;//a subclass of Canvas 
  Graphics2D g2D;//off-screen graphics context.
  //----------------------------------------------------//

  GUI(){//constructor

    //Set JFrame size, title, and close operation.
    setSize(hSize,vSize);
    setTitle("Copyright 2008,R.G.Baldwin");
    setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);

    //Create a new drawing canvas and add it to the
    // CENTER of the JFrame above the control panel.
    myCanvas = new MyCanvas();
    this.getContentPane().add(
                            BorderLayout.CENTER,myCanvas);

    //This object must be visible before you can get an
    // off-screen image.  It must also be visible before
    // you can compute the size of the canvas.
    setVisible(true);

    //Make the size of the off-screen image match the
    // size of the canvas.
    osiWidth = myCanvas.getWidth();
    osiHeight = myCanvas.getHeight();

    //Create an off-screen image and get a graphics
    // context on it.
    osi = createImage(osiWidth,osiHeight);
    g2D = (Graphics2D)(osi.getGraphics());

    //Call a method that sets the axes and draws the 
    // cylinder.
    drawTheCylinder(g2D);

    //Cause the overridden paint method belonging to
    // myCanvas to be executed.
    myCanvas.repaint();

  }//end constructor
  //----------------------------------------------------//

  //This method is used to set the axes to the center of
  // the off-screen image and to draw a 3D cylinder that
  // is centered on the y-axis.
  private void drawTheCylinder(Graphics2D g2D){

    //Translate the origin to the center of the off-screen
    // image.
    GM02.translate(g2D,0.5*osiWidth,-0.5*osiHeight);

    //Erase the screen
    g2D.setColor(Color.WHITE);
    GM02.fillRect(g2D,-osiWidth/2,osiHeight/2,
                                      osiWidth,osiHeight);
    //Draw a cylinder by stacking 20 circular disks on
    // the x-z plane. The disks are centered on the
    // y-axis and are parallel to the x-z plane. The
    // thickness of each disk is 5 vertical units.
    GM02.Point3D tempPointA;
    GM02.Point3D tempPointB;
    g2D.setColor(Color.BLACK);

    for(int y = 0;y < 105;y += 5){//iterate on disks
      //Define the starting point on the circle for this
      // disk.
      tempPointA = new GM02.Point3D(
              new GM02.ColMatrix3D(76,y - osiHeight/4,0));
      //Iterate on points on the circle that represents
      // this disk.
      for(int cnt = 0;cnt < 360;cnt++){//360 points
        //Compute the next point on the circle.
        tempPointB = 
            new GM02.Point3D(new GM02.ColMatrix3D(
              76*Math.cos(Math.toRadians(cnt*360/360)),
              y - osiHeight/4,
              76*Math.sin(Math.toRadians(cnt*360/360))));

        //Draw the line in 3D. Note that there is no
        // backface culling in this program.
        new GM02.Line3D(tempPointA,tempPointB).draw(g2D);

        //Save the point for use in drawing the next line.
        tempPointA = tempPointB;
      }//end for loop on points
    }//end for loop on disks

  }//end drawTheCylinder method
  //====================================================//


  //This is an inner class of the GUI class.
  class MyCanvas extends Canvas{
    //Override the paint() method. This method will be
    // called when the JFrame and the Canvas appear on the
    // screen or when the repaint method is called on the
    // Canvas object.
    //The purpose of this method is to display the
    // off-screen image on the screen.
    public void paint(Graphics g){
      g.drawImage(osi,0,0,this);
    }//end overridden paint()

  }//end inner class MyCanvas

}//end class GUI
//======================================================//

/*DotProd3D03.java
Copyright 2008, R.G.Baldwin
Revised 03/07/08

The purpose of this program is to demonstrate a practical
use of the vector dot product - backface culling.

Study Kjell through Chapter 10, Angle between 3D Vectors.

The program draws a 3D circular cylinder by stacking 20
circular disks on the x-z plane. The disks are centered on
the y-axis and are parallel to the x-z plane. The
thickness of each disk is 5 vertical units.

Backface culling is done using the dot product between a
vector that is parallel to the viewpoint of the viewer and
a vector that is perpendicular to the line being drawn to
form the outline of a disk.  The backface culling 
process is good but not perfect.  There is some leakage 
around the back on the right and left sides of the 
cylinder and some short lines segments are visible that 
should not be visible.


Tested using JDK 1.6 under WinXP.
*********************************************************/
import java.awt.*;
import javax.swing.*;

class DotProd3D03{
  public static void main(String[] args){
    GUI guiObj = new GUI();
  }//end main
}//end controlling class DotProd3D03
//======================================================//

class GUI extends JFrame{
  //Specify the horizontal and vertical size of a JFrame
  // object.
  int hSize = 230;
  int vSize = 250;
  Image osi;//an off-screen image
  int osiWidth;//off-screen image width
  int osiHeight;//off-screen image height
  MyCanvas myCanvas;//a subclass of Canvas 
  Graphics2D g2D;//off-screen graphics context.
  //----------------------------------------------------//

  GUI(){//constructor

    //Set JFrame size, title, and close operation.
    setSize(hSize,vSize);
    setTitle("Copyright 2008,R.G.Baldwin");
    setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);

    //Create a new drawing canvas and add it to the
    // CENTER of the JFrame above the control panel.
    myCanvas = new MyCanvas();
    this.getContentPane().add(
                            BorderLayout.CENTER,myCanvas);

    //This object must be visible before you can get an
    // off-screen image.  It must also be visible before
    // you can compute the size of the canvas.
    setVisible(true);

    //Make the size of the off-screen image match the
    // size of the canvas.
    osiWidth = myCanvas.getWidth();
    osiHeight = myCanvas.getHeight();

    //Create an off-screen image and get a graphics
    // context on it.
    osi = createImage(osiWidth,osiHeight);
    g2D = (Graphics2D)(osi.getGraphics());

    //Call a method that sets the axes and draws the 
    // cylinder.
    drawTheCylinder(g2D);

    //Cause the overridden paint method belonging to
    // myCanvas to be executed.
    myCanvas.repaint();

  }//end constructor
  //----------------------------------------------------//

  //This method is used to set the axes to the center of
  // the off-screen image and to draw a 3D cylinder that
  // is centered on the y-axis.
  private void drawTheCylinder(Graphics2D g2D){

    //Translate the origin to the center of the off-screen
    // image.
    GM02.translate(g2D,0.5*osiWidth,-0.5*osiHeight);

    //Erase the screen
    g2D.setColor(Color.WHITE);
    GM02.fillRect(g2D,-osiWidth/2,osiHeight/2,
                                      osiWidth,osiHeight);

    //Get a vector that is approximately parallel to the
    // viewpoint of the viewer. The best values for this
    // vector were determined experimentally using this
    // program and the earlier program named DotProd3D02.
    GM02.Vector3D viewPoint = 
        new GM02.Vector3D(new GM02.ColMatrix3D(43,5,50));

    //Draw a cylinder by stacking 20 circular disks on
    // the x-z plane. The disks are centered on the
    // y-axis and are parallel to the x-z plane. The
    // thickness of each disk is 5 vertical units.
    GM02.Point3D tempPointA;
    GM02.Point3D tempPointB;
    g2D.setColor(Color.BLACK);

    for(int y = 0;y < 105;y += 5){//iterate on disks
      //Define the starting point on the circle for this
      // disk.
      tempPointA = new GM02.Point3D(new GM02.ColMatrix3D(
                                   76,y - osiHeight/4,0));
      //Iterate on points on the circle that represents
      // this disk.
      for(int cnt = 0;cnt < 360;cnt++){//360 points
        //Compute the next point on the circle.
        tempPointB = 
            new GM02.Point3D(new GM02.ColMatrix3D(
              76*Math.cos(Math.toRadians(cnt*360/360)),
              y - osiHeight/4,
              76*Math.sin(Math.toRadians(cnt*360/360))));

        //Do backface culling using the dot product of the
        // viewpoint vector and a vector that is almost
        // perpendicular to the line being drawn. If the
        // dot product is negative, or if the disk being
        // drawn is the top disk on the stack, draw the
        // line. Otherwise, don't draw the line.
        //The perpendicular vector used in the dot
        // product is the displacement vector from the
        // origin to a point that defines one end of the
        // line being drawn. Note that this vector is not
        // perfectly perpendicular to the line being 
        // drawn. Later in the series, we will learn about
        // and use the cross product to get perpendicular
        // vectors.
        if((tempPointB.getDisplacementVector(
                           new GM02.Point3D(
                             new GM02.ColMatrix3D(0,0,0)))
                                   .dot(viewPoint) < 0.0) 
                                           || (y == 100)){
          //Draw the line in 3D.
          new GM02.Line3D(
                         tempPointA,tempPointB).draw(g2D);
        }//end if

        //Save the point for use in drawing the next line.
        tempPointA = tempPointB;
      }//end for loop on points
    }//end for loop on disks

  }//end drawTheCylinder method
  //====================================================//


  //This is an inner class of the GUI class.
  class MyCanvas extends Canvas{
    //Override the paint() method. This method will be
    // called when the JFrame and the Canvas appear on the
    // screen or when the repaint method is called on the
    // Canvas object.
    //The purpose of this method is to display the
    // off-screen image on the screen.
    public void paint(Graphics g){
      g.drawImage(osi,0,0,this);
    }//end overridden paint()

  }//end inner class MyCanvas

}//end class GUI
//======================================================//