Computer Graphics: Programming, Problem Solving, and Visual Communication

1. Computer Graphics:Programming, Problem Solving, and Visual Communication Steve Cunningham California State University Stanislaus and Grinnell College PowerPoint Instructor’s Resource

2. Interpolation and Spline Modeling Creating curves and surfaces with just a few control points

3. Interpolations • If we have just a few points, we can define curves or surfaces from polygons of relatively low degree that go through the points OR: • We could modify this requirement to have the curves or surfaces influenced by the points

4. Interpolations (2) • For example, with any three points P0, P1, P2 we can define a quadratic function of one variable f0(t)P0+f1(t)P1+f2(t)P2 with the quadratic functions f0(t)=(1-t)2, f1(t)=2t(1-t), and f2(t)=t2 with domain [0,1] that goes through P0 and P2 and goes towards P1

5. Interpolations (3) • These three functions are called the quadratic Bernstein basis for the quadratic Bézier curves, and an example curve is:

6. Interpolations (4) • There are Bernstein basis functions of all degrees; the general form for the n+1 functions of degree n on [0,1] is Notice that these are the terms in expanding (1-t)n

7. Interpolations (5) • The cubic Bézier curve is the most commonly used of these. It takes four control points, goes through the two endpoints, and interpolates the others. • An example curve is:

8. Other Interpolations • Catmull-Rom spline: a cubic spline that uses four control points and goes through the second and third of them in directions determined by the first and fourth

9. Another Way to Compute Splines • Another way to compute spline functions uses matrices and may be more efficient • For the Bézier cubic spline, this is:

10. Using P0-P1-P2-P3 then P3-P4-P5-P6 is not smooth Adding midpoints Q0 and Q1 between points P2-P3 and P4-P5: Extending Spline Curves

11. Spline Curves in 3D • Everything said about spline curves is valid in 3D as well as 2D, and we have:

12. Spline Surfaces • Simplest surface is a patch, defined by a square array of control points • The interpolating functions are used for both variables (u,v) in the square domain [0,1]x[0,1] • If the functions are cubic in each variable, the result is a bicubic whose coefficients depend on the control points:

13. Spline Surface (2) • An example of a single bicubic patch of a Bézier surface, with control points:

14. Spline Surface (3) • OpenGL gives you some tools that let you avoid the details of programming all the basis functions • These are the evaluator functions, and they let you get normals and texture coordinates automatically. • The downside is that you can only use the kind of splines that OpenGL gives you.

15. Spline Surface (4) • Examples of an extended surface with automatic normals (left) and a patch with automatic texture coordinates (right)