Splines:
In class I talked about parametric cubic splines, and briefly mentioned parametric bicubic surface patches. This assignment has two parts:
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:
fillPolygon
method,
rather than just draw curves via the drawLine
method.
fillOval
method)
to show that those control points are the ones
which mark the beginning and end of the individual Bezier
cubic spline curves.
An important thing to keep in mind is that if you want two successive Bezier curves that share some key point P_{3n} to join together without a sharp bend between them, then you must make sure that the slope from P_{3n} to the next key point P_{3n+1} is equal to the slope from the previous key point P_{3n-1} to P_{3n}. In order to constrain the key points so that successive spline segments maintain the same slope at their common point when the user drags control points around, you should do something like the following:
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 at^{3}+bt^{2}+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) = a_{x}t^{3} + b_{x}t^{2} + c_{x}t + d_{x}
Y(t) = a_{y}t^{3} + b_{y}t^{2} + c_{y}t + d_{y}where each evaluation is implemented by
xSpline.eval(t)
andySpline.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)
anddouble 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)
andzSpline.eval(u,v)
, respectively.