Homework 7, due Tuesday, Nov 23.

Splines:

In class I talked about parametric cubic splines, and briefly mentioned parametric bicubic surface patches. This assignment has two parts:

  1. Implement an interactive Java applet that shows its user some cool picture made out of spline curves, and allows the user to drag around control points to modify that picture.

  2. Implement an interactive Java applet that allows the user to manipulate bicubic spline patches. You can use Bezier, BSpline, Catmull-Rom or Hermite patches, or the option for more than one of these if you're feeling ambitious. At the very least, allow the user to edit the geometry of a single bicubic patch. Then when you get that working, try to allow the user to edit the geometry of multiple bicubic patches that fit smoothly together.

For this assignment you'll want to go back to using the java.awt.graphics support class that you used earlier in the semester, since you'll mostly be drawing lines and circles - you will be making curves by drawing lots of little successive lines end to end.

For both parts of this assignment, feel free to use the Cubic.java class that provides cubic spline support fo the interactive cubic curve applet that I showed in class. This week I really want you to concentrate on a nice interactive tool for curve and surface editing.

Your first program should behave as follows:

Helpful notes:

As we covered in class, the way you define a cubic curve is to treat the x geometry and the y geometry independently, creating a different parametric cubic curve for each coordinate dimension.

For every type of spline there is a unique matrix that transforms the four geometry values in each dimension to the cubic polynomial at3+bt2+ct+d. For example, you can construct Bezier splines by:

   
   Cubic xSpline = new Cubic Spline(Cubic.BEZIER, GX);
   Cubic ySpline = new Cubic Spline(Cubic.BEZIER, GY);

where double GX[], GY[] are each arrays containing the four geometry values for their respective coordinate.

This constructor uses the matrix to transform the four geometry values into (a,b,c,d), which will let you evaluate the cubic polynomials:

X(t) = axt3 + bxt2 + cxt + dx
Y(t) = ayt3 + byt2 + cyt + dy

where each evaluation is implemented by xSpline.eval(t) and ySpline.eval(t).

Once you know the cubic polynomials that define X(t) and Y(t) for any individual parametric cubic curve, the simplest way to draw the curve is to loop through values of t, stepping from 0.0 to 1.0, and draw short lines between successive values. For example, if you have already defined methods double X(double t) and double Y(double t), then you can use code structured something like:

   for (double t = 0 ; t <= 1 ; t += ε)
      g.drawLine((int)X(t), (int)Y(t), (int)X(t+ε), (int)Y(t+ε));

Similarly, you can define the coordinates at (u,v) on a bicubic patch by X(u,v),Y(u,v),Z(u,v) by using the bicubic constructors in the Cubic class. For example, you can construct Bezier surface patches by:

   
   Cubic xSpline = new Cubic Spline(Cubic.BEZIER, GX);
   Cubic ySpline = new Cubic Spline(Cubic.BEZIER, GY);
   Cubic zSpline = new Cubic Spline(Cubic.BEZIER, GZ);

where double GX[][],GY[][],GZ[][] are each arrays containing the sixteen geometry values for their respective coordinate.

You can then fill up a geometric mesh (sort of like you did when you made sphere meshes) by:

   for (int j = 0 ; j < nRows ; j++)
   for (int i = 0 ; i < nCols ; i++) {
      double u = (double)i / nCols;
      double v = (double)j / nRows;
      defineVertex(i, j, X(u,v), Y(u,v), Z(u,v));
   }

where X(u,v), Y(u,v) and Z(u,v) are implemented by xSpline.eval(u,v), ySpline.eval(u,v) and zSpline.eval(u,v), respectively.