Surfaces - Geometric Algorithms for Modeling, Motion, and Animation

04/24/2006
Surfaces
• Locally a 2D manifold: i.e. approximating a plane in the ngbd. of
each point.
• A 2-parameter family of points
• Surface representations: à
used to construct, evaluate, analyze points
and curves, to reveal special properties, demonstrate the relationship
to other geometric à
objects
• Commonly used surface representations: explicit (parametric) or
implicit (algebraic)
• Explicit formulations are bivariate parametric equations: tensor
product, triangular patches or n-sided patches
• Procedural formulations: Generalized implicits, subdivision
surfaces
Dinesh Manocha, COMP236

04/24/2006
Implicit Surfaces
• Implicit formulation:F(x,y,z) = 0, where F() is a polynomial in x,y
and z of the form:
P
aijkx i,j,k
à
iyjz k =0
• If F() is irreducible à polynomial, the degree of the surface is: i+j+k
• Geometrically the degree refers to the maximum number of
intersections any line can have the surface (assuming finite
intersections)
• Any rational parametric surface can be converted into algebraic
(implicit) surface: implicitization
Dinesh Manocha, COMP236

04/24/2006
Quadric Surfaces
• Given as F(x,y,z) = 0, where F() is a quadratic polynomial in x,y and
z of the form:
Ax
à
2 + By2 + Cz2 +2Dxy +2Eyz +2Fxz +2Gx +2Hy +2Jz + K =0
If A = B = C = -K = 1 & D = E = F = G = H = J = 0, then it produces a
unit sphere at
à
the origin.
• Also given in matrix form as: P Q P T = 0, where P = [x y z 1] &
Q =
A D F G
D B E H
F E C J
G H J K
Dinesh Manocha, COMP236

04/24/2006
Quadric Surfaces
• The coefficients of Q many have no direct physical or geometric
meaning
• A rigid body transformation, can be directly applied to Q as:
à
Q’ = T –1 Q [T –1 ] T
à
• Certain properties of the matrix of quadric equation are invariant
under rigid transformation, including the determinants |Q| and
|Qu |,where Qu is the matrix corresponding to the gradient (or normal)
vectors
Dinesh Manocha, COMP236

04/24/2006
Parametric Surfaces
• The most common mathematical element used to model a surface is
a patch (equivalent to a segment of spline curve).
• A patch is a curve-bounded collection of points whose coordinates
are given àby
continuous, bivariate, single-valued polynomials of the
form P(u,w) = (x y z), where:
x = X(u,w) y = Y(u,w) z = Z(u,w),
à
where the parametric variables u and w are typically constrained to the
intervals, u,w ∈
[0,1].
• This generates a rectangular patch, though there are other topological
variations (e.g. triangular or n-sided).
• Fixing the value of one of the parametric variables results in a curve
on the patch in terms of the other variable. Or generate a curve net.
Dinesh Manocha, COMP236

04/24/2006
Parametric Surfaces: Patches
• Each patch has a set of boundary conditions associated with it:
four corner points (P(0,0), P(0,1), P(1,0), P(1,1)),
four curves defining its edges (P(u,0), P(u,1), P(0,w), P(1,w)),
tangent vectors or planes (Pu(u,w), P
à
w(u,w))
normal vectors (Pu(u,w) X Pw(u,w)) twist vectors defined at the corner points
à
• In practice, composite arrays are put together or assembled to represent complex
surfaces (with some appropriate continuity conditions at the boundary)
• Commonly used patches: Hermite patch, Bezier patch, B-spline patch (or NURBS)
Dinesh Manocha, COMP236

04/24/2006
Parametric Patches
• Tensor product or rectangular patches are of the form:
P
P(u,w) = u,w [0,1].
i,j
P i,jB m i (u)Bn j (w) ∈
à
The number of control points is (m+1)(n+1)
• Triangular patches à have triangular domain. They are of the form:
Pn
nài P
P(r,s,t) = r,s,t 0
Pi,jr i=0 j=0
isjtnàiàj , r+ s + t =1; õ
It has (n+1)(n+2)/2 control points
Dinesh Manocha, COMP236

04/24/2006
Tensor Product Bezier Patches
• Earlier work by de Casteljau (1959-63)
• Later work by Bezier made them popular for CAGD
• The simplest tensor product patch is a bilinear patch, based on the concept of
bilinear interpolation (fitting simplest surface among 4 points)
à
• Bilinear Patch: Given 4 points, b 00, b 01 ,b 10 ,b 11, the patch is defined as:
àP(u, w) = P1
P
i=0
1
j=0
bijB1 i (u)B1j (w)
or in matrix form, it can be expressed as:
[1 à u u] â b00 b01 b10 b11 ã â 1 à w
w
Dinesh Manocha, COMP236
ã

04/24/2006
Bilinear Tensor Product Bezier
Patches
• They are called a hyperbolic paraboloid
• Consider the surface, z = xy. It is the bilinear interpolant of 4 points
àâ
à
0
0
0
ã , â 1
0
0
ã , â 0
1
0
ã , â 1
1
1
• If we intersect with a plane parallel to the x,y plane, the resulting
curve is a hyperbola. If we intersect it with a plane containing the zaxis,
the resulting curve is a parabola.
ã
Dinesh Manocha, COMP236

04/24/2006
Direct de Casteljau Algorithm
• Extension of linear interpolation (or convex combination) algorithm for curves to
surfaces
• Given a rectangular array of points, bij, 0 ≤ i,j ≤ n and parameter value (u,w).
Compute a point on a surface determined by this array of points by setting:
b r,r
i,j =
r = 1,….,n
à
à
[1 à u u]
i,j = 0,….,n-r and
b n,n
0,0
â b rà1,rà1
i,j
b rà1,rà1
i+1,j
b 0,0
i,j = b i,j
b rà1,rà1
i,j+1
b rà1,rà1
i+1,j+1
is the point with parameter values (u,w) on the surface
ã
Dinesh Manocha, COMP236
â 1 à w
w
ã

04/24/2006
Direct de Casteljau Algorithm
• The net of b ij, is called the Bezier net or the control net.
• Tensor product formulation: A surface is the locus of a curve that is
moving thru space and thereby changing its shape
• The same àsurface
can also be represented using Bernstein basis as:
P(u, w) =
à
Pn
i=0
Pn
j=0
bijB n
i (u)Bnj
(w)
• de Casteljau’s be easily extended when the control points along u and
w directions are different (say m X n)
– Apply the recursive formulation till b is computed, where
k = min(m,n)
k,k
i,j
– After that use the univariate de Casteljau’s algorithm
Dinesh Manocha, COMP236

• Affine invariance
04/24/2006
Properties of Bezier Patches
• Convex hull property
• Boundary curves: are Bezier curves
• Variation àdimishing
property: Not clear whether it holds for surfaces
– Easy to come with an counter example
à
Dinesh Manocha, COMP236

04/24/2006
Degree Elevation
• Goal: To raise the degree from (m,n) to (m+1,n)
• Compute new coefficients, b , such that the surface can be
expressed as:
1,0
i,j
à
P(u, w) =
à
• The n+1 terms in square brackets represent n+1 univariate degree
elevation
P â n Pm+1
(1,0)
j=0 i=0 b ij Bm+1
i (u) ã B n j (w)
• The new control points can be computed, by using the univariate
formula:
b 1,0
i,j = i i
(1 à )bi,j + bià1,j
m+1 m+1
è i =0,...,m+1
j =0,...n
Dinesh Manocha, COMP236

04/24/2006
Derivatives
• Goal: To compute partials along u or w directions
• A partial derivative is the tangent vector of an isoparametric curve:
P(u, w)
à
=
• It can be expressed as:
à
Pn
â
∂
∂
j=0 ∂u
∂u
P
∂ Pn
P(u, w) =m j=0
∂u
P
i=0
where
Dinesh Manocha, COMP236
m
i=0
bijB m i (u)ãB n j (w)
mà1 (1,0) Δ bijB mà1
i (u)B n j (w)
Δ 1,0 b i,j = b i+1,j à b i,j
,

04/24/2006
Trimmed Patches
• Arise in applications involving surface intersections, visibility
(silhouettes), illumination etc.
• The domain is irregular
• Boundary àor
trimming curves are used to delimit a subset of points on
the patch
• In most applications, à
trimming curves correspond to high degree
algebraic curves
– Evaluate points on these curves using numerical methods
– Fit spline curve(s) to these points
• Trimmed domain is represented using piecewise spline curves
• Point Classification:Check whether a point is in the trimmed domain,
compute number of intersections with a line
Dinesh Manocha, COMP236