Homework 4
CSCI-4967: Three-Dimensional Computer Graphics
Fall 2004
Due: Thursday, November 4, 2004
Homeworks are due at the beginning of lecture on Thursday, November 4. Late homeworks
will receive no credit. Homeworks are to be done individually and will be graded on the basis of
correctness, clarity, and legibility. Show the steps in your work where appropriate. Each question
is worth 10 points, for a total of 50 points.
Be sure to write your name,section number, and RPI email address on your homework
submission.
1. Given an array pj,k of control points, 0 ≤j≤m, 0 ≤k≤n, the Bezier surface patch is given
by
P(u, v) =
m
X
j=0
n
X
k=0
pj,kB ezj,m(u)Bezk,n(v)
for 0 ≤u, v ≤1, where Bezj,m(u) and Bezk ,n(v) denote Bezier blending functions of degree m
and nrespectively. Consider a bicubic Bezier surface patch, that is, a patch with m=n= 3.
(a) Given any fixed u0, 0 ≤u0≤1, define the u0-slice to be the curve c(v) = P(u0, v). Show
that c(v) is a Bezier curve of degree 3.
(b) What are the four control points for the curve c(v)?
(c) What is the normal vector to the Bezier surface at P(u= 0, v = 0)?
2. (a) What are the advantages of using a spline based representation of objects such as the
Utah teapot instead of a polygonal mesh representation?
(b) You want to know whether a given point P= (x, y, z) lies on a surface. With which
surface representation is it easier to answer this query: parametric or implicit? Justify
your answer.
(c) Give an implicit surface representation of a (right circular) cone with its vertex at the
origin and with its central axis along the positive Zaxis. That is, give an implicit
function f(x, y, z) = 0 that describes the surface of the cone. Assume the cone makes
an angle of αwith the Zaxis.
