Graphics Reference
In-Depth Information
problem was solved when Gordon and Riesenfeld ([GorR74a]) suggested using B-
splines as described by Schoenberg ([Scho67]). The term “NURBS” was coined at
Boeing in 1985 ([BloK02]). See [Roge01] for other historical tidbits.
When the design and manufacturing process started to be computerized, one tried
to imitate what was done by hand. Objects were defined by collections of curves. There
were lots of possible curves, but mathematically one tried to define and manipulate
them in terms of controlling features such as endpoints and tangents at the endpoints.
Until the early 1980s most ship-fairing systems were based on fairing curves. Poly-
nomial curves gave way to spline functions because, being piecewise polynomial, they
avoided the global oscillation of large degree global polynomials and also allowed
one to proceed in the same way that it was done with the physical splines by moving
control points.
12.20
E
XERCISES
Section 12.2
Consider the surface of revolution
S
obtained by revolving a space curve g : [a,b] Æ
R
3
about a line
L
through a point
p
and direction vector
v
.
12.2.1
(a)
Find a parameterization p(u,v) for
S
.
(b)
Find a parameterization p(u,v) for
S
when g(u) = (0,0,u), [a,b] = [1,3],
p
= (1,2,0),
and
v
= (2,-1,3). Also find ∂p/∂u, ∂p/∂v, and a normal vector at the point (0,0,1).
Section 12.3
12.3.1
Find an implicit equation f(x,y,z) = 0 for the surface
S
in Exercise 12.2.1(b). Use the
gradient of f to find a normal to S at (0,0,1) and verify that it is parallel to the normal
you got in Exercise 12.2.1(b).
Section 12.4
Consider the curves f, g : [1,3] Æ
R
3
, where f(u) = (u + 1,u
2
- 4u + 5,0) and g(u) = (0,u,u
3
).
Let p(u,v) be the lofted surface defined by f and g. Sketch the surface and find the equa-
tion of the tangent plane at p(2,0.5).
12.4.1
Section 12.5
Consider the spiral f : [0,p] Æ
R
3
, f(t) = (3 cos t,3 sin t,t), and its Frenet frame
(T(t),N(t),B(t)). Let p(u,v) be the sweep surface obtained by sweeping the segment
[(0,0,-1),(0,0,1)] along f(t) rotating the segment in the N(t)-B(t) plane in a uniform
counter-clockwise manner so that we have rotated through an angle of p when we reach
f(p). Find the formula for p(u,v).
12.5.1
