Splines:
In class I talked about parametric cubic splines, and briefly mentioned parametric bicubic surface patches. What I would like you to do for this assignment is to implement an interactive Java applet that allows the user to drag around control points to modify a picture that's been created out of Bezier splines. Your program should behave as follows:
drawLine
method.
In that way
your drawing can contain shaded areas bounded
by smooth curves.
g.fillOval
method
for this)
to show that those control points are the ones
which mark the beginning and end of the individual Bezier curves.
An important thing to keep in mind is that if you want two successive Bezier curves that share some key point P_{n} to join together without a sharp bend between them, then you must make sure that the slope from P_{n} to the next key point P_{n+1} is equal to the slope from the previous key point P_{n-1} to P_{n}. In order to constrain the key points so that successive spline segments still have the same slope where they join, even as 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 Bezier curve from four control points A,B,C,D is to treat the set of x coordinates {A_{x}, B_{x}, C_{x}, D_{x}} and the set of y coordinates {A_{y}, B_{y}, C_{y}, D_{y}} independently.As I said in class, for every type of spline there is a unique matrix that transforms the control point values to the (a,b,c,d) values of the cubic polynomial at^{3}+bt^{2}+ct+d.
For Bezier curves, the particular matrix to use is:
-1 3 -3 1 3 -6 3 0 -3 3 0 0 1 0 0 0 To get the cubic polynomial equation for the x coordinates and y coordinates, respectively, you need to use this matrix to transform the two geometry column vectors:
into the two column vectors of cubic coefficients:
A_{x}
B_{x}
C_{x}
D_{x}A_{y}
B_{y}
C_{y}
D_{y}
a_{x}
b_{x}
c_{x}
d_{x}a_{y}
b_{y}
c_{y}
d_{y}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}Once you know the cubic polynomials that define X(t) and Y(t) for any individual Bezier 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));