# CS 426 Exercises   Curves

1. For each of the following properties, give an example of a curve representation that guarantees it: a) local control, b) interpolates control points, c) C1 continuity, d) C2 continuity, d) curve lies with convex hull of control points. <\li>
2. Why do computer graphics applications use piecewise polynomial curves of degree 3 rather than curves of higher-order, say degree 100? Which representation provides more continuity guarantees?
3. How many control points are required to specify a Bezier curve of degree d?
4. From the Bernstein polynomials (the Bezier blending functions), prove that a cubic Bezier curve interpolates V0 and V3.
5. What property of the Bernstein polynomials guarantees that a Bezier curve lies within the convex hull of its control points?
6. Draw a Bezier curve for which recursive subdivision would be a more efficient method for rendering.
7. What is C1 continuity? How is it different than G1 continuity? How is it different that C2 continuity?
8. Draw a spline curve comprising two Bezier curve segments in which the derivatives at the joint are in OPPOSITE directions.
9. How many degrees of freedom are available for a spline with m cubic segments? How many constraints (degrees of freedom) are required to specify C2 continuity at each interior joint of a spline with m cubic segments? How many constraints at the endpoints of the spline? How many degrees of freedom are left?
10. Which of the following properties are guaranteed by C2 interpolating splines a) C2 continuity, b) interpolation of control points, c) local control, d) convex hull.
11. Which of the following properties are guaranteed by cubic B-Splines: a) C2 continuity, b) interpolation of control points, c) local control, d) convex hull.
12. Which of the following properties are guaranteed by cubic Catmull-Rom Splines: a) C2 continuity, b) interpolation of control points, c) local control, d) convex hull.