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: BezierMatrix equations for cubic Bezier curves:

0

12 3

2

3

0

13 2 2 3

2

3

1 0 0 0

3 3 0 0( ) 1

3 6 3 0

1 3 3 1

( ) (1 ) 3 (1 ) 3 (1 )

p

pC t t t t

p

p

p

pC t t t t t t t

p

p

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

1

2 3

1

1

1

3 3 2 3 2 3 2

1

1

1 3 3 1

3 6 3 01( ) 1

3 0 3 06

1 4 1 0

1( ) ( 3 3 3 1) 3 6 4 3 3 1

6

i

ii

i

i

i

ii

i

i

p

pf t t t t

p

p

p

pf t t t t t t t t t t

p

p

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