1. Announcements
Saturday, September 29 at 9:30 in Cummings 308: Botanical Illustration: The Marriage of Art and Science by Wendy Hollender
Wednesday class is in Cummings 133 with Del Harrow, visiting artist
Del Harrow will give a talk at 6:30
Thursday open house with Del Harrow
Friday class is in New London 214
New York City bus trip on Nov 3: depart 7:30 AM and return in the evening: Botanical Garden, I-Beam, etc.

2. Curves: Bezier
Matrix equations for cubic Bezier curves:

3. Properties of Bezier Curves
Goes through endpoints
Invariant under affine transformations (includes trans, rot, scaling)
Curve lies within the convex hull
If control points on a straight line then the curve is a straight line
Tangents at end points dependent on proximate points (p1 determines tangent at p0, and p2 determines tangent at p3)

4. Cubic B-splines
One problem with Bezier curves is that they aren’t “smooth” where they join – the derivatives aren’t the same
Cubic B-splines are also composed of cubic polynomials but exhibit C2 continuity: continuous, first and second derivatives all coincide
In general, B-splines are constructed from polynomials of degree k; have k-1 continuity

5. Cubic B-splines (con’t)
The matrix equations are

6. Other Curves
Rational curves have blending functions that can be the ratio of two polynomial curves (Bezier and B-splines use a third degree polynomial for the blending functions); with rational curves get a wider variety of curves (eg. circles)
NURBS: Non-uniform Rational B-Splines: rational means can have ratios of polynomials and non-uniform means the curve sections may not have t always ranging from [0,1]; more variety; one advantage is that perspective holds (not an affine transformation)

7. Surfaces
Bezier patches or surfaces: 16 control points pij in a grid with 4 control points on each side: interpolates the 4 corner points; defined by Bezier curves
B-spline surfaces
Can also use curves to sweep out 3D objects: in a circle, line, etc.
Rendering can be done through planar patches or directly through the equations