Tutorial
Parametric Cubic Curves 2

1. A B-spline vector is defined by four points, namely a geometry vector

  G = [ P1 P2 P3 P4 ]^t.

See your course notes for the B-spline basis matrix.
1. What is the effect of choosing P1=P2, i.e, what does the resulting curve look like? Does it pass through any of the control points? How are the tangents at either end of the curve related to the control points?
2. What is the effect of choosing P1=P2=P3?
2. Show that two B-spline curve segments, defined by Ga = [ P1 P2 P3 P4 ] and Gb = [ P2 P3 P4 P5 ] are C2 continuous.
3. One of the simplest ways to render a parametric surface patch is to subdivide the patch into (possibly non-planar) quadrilaterals, as determined by a grid in terms of the parameters s and t. Describe how one might clip patches rendered using subdivision in an efficient way.