CPSC 645/VIZA 675

Homework 3 due 4/9/09

Similar to Homework 1, you may use a computer algebra package such as Mathematica or MatLab to perform simple calculations such as matrix multiplication, inversion or even to find eigenvalues/vectors. However, you should show all of your work.

1. [10 points] Given a set of control points (1,0,0), (0,1,0), (0,0,1), (0,0,2), (1,2,3) associated with parameter values 0, 1, 1.5, 3, 4 respectively for a C¹ Catmull-Rom curve, evaluate the curve at parameter value t=2.

2. Imagine you have a tensor-product Bezier patch of bi-degree 3 with control points

(0,0,4.5) (1,0,2.5) (2,0,2.5) (3,0,4.5)

(0,1,2.5) (1,1,0.5) (2,1,0.5) (3,1,2.5)

(0,2,2.5) (1,2,0.5) (2,2,0.5) (3,2,2.5)

(0,3,4.5) (1,3,2.5) (2,3,2.5) (3,3,4.5)

a. [7 points] Evaluate this surface at (s,t)=(.3,.5).

b. [7 points] Calculate the direction of the s, t derivatives at (s,t)=(.3,.5).

c. [7 points] Calculate the Gaussian curvature of the surface at (s,t)=(.3,.5). (Warning: the magnitude of the derivative matters here!)

d. [7 points] Calculate the Mean curvature of the surface at (s,t)=(.3,.5). (Warning: the magnitude of the derivative matters here!)

3. [10 points] Given the control points of a cubic, triangular Bezier patch in a triangular array

{{(0,0,1), (1,0,2),(2,0,1),(3,0,4)}, {(0,1,0),(1,1,3),(2,1,0)}, {(0,2,1),(1,2,1)}, {(0,3,2)}},

evaluate the patch at the parametric point (1/3, 1/3, 1/3).

4. [10 points] Given the following 4 curves:

,

find a parametric patch that interpolates all four curves.

5. Assume you have a curve with control points

(0,1,0), (1,2,1), (3,5,4), (2,2,1), (5,6,3), (5,3,2)

a. [7 points] Subdivide the curve once using uniform cubic B-spline subdivision and give the resulting control points.

b. [7 points] Subdivide the original curve once using the four-point method and give the resulting control points.

c. [7 points] Compute the limit position associated with the point (2,2,1) when the curve is subdivided using uniform cubic B-spline subdivision.

d. [7 points] Compute the direction of the derivative associated with the control point (2,2,1) on the limit curve defined using four-point subdivision.

6. [14 points] Use blossoming to define the rules for a subdivision scheme for uniform quartic B-splines.