Announcements Saturday, September 29 at 9:30 in Cummings 308: Botanical Illustration: The Marriage of Art and Science by Wendy Hollender Wednesday class.

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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.

Curves: Bezier Matrix equations for cubic Bezier curves:

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)

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

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

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)

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