Curves

- 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>
- 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?
- How many control points are required to specify a Bezier curve of degree d?
- From the Bernstein polynomials (the Bezier blending functions), prove that a cubic Bezier curve interpolates V0 and V3.
- What property of the Bernstein polynomials guarantees that a Bezier curve lies within the convex hull of its control points?
- Draw a Bezier curve for which recursive subdivision would be a more efficient method for rendering.
- What is C1 continuity? How is it different than G1 continuity? How is it different that C2 continuity?
- Draw a spline curve comprising two Bezier curve segments in which the derivatives at the joint are in OPPOSITE directions.
- 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?
- 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.
- 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.
- 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.