CS 426 Exercises
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.