Surfaces
¤ Surfaces of revolution
• Sweep surfaces
• Parametric surfaces
• Quadric surfaces
• Bilinear surface
• Ruled surface
• Linear Coons surface
• Coons bicubic surface
• Bezier surface
• B-spline surface
• Rational B-spline surface
Surfaces
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東吳資訊科學 江清水
Surfaces of Revolution
A very simple family of surfaces is obtained
by rotating a plane curve around an axis.
For example, let C(t) be a curve on XZ
Plane, a surface can be generated by
rotating C(t) about Z-axis. Any point on
the surface of revolution is a function of
two parameters, t and . The parameter t
describes the entity to be rotated, and 
is the angle of rotation around the Z-axis.
So,
C(t) = [x(t), 0, z(t)]
S(t,) = [x(t)cos , x(t)sin , z(t)]
In matrix form:
/
\
S(t,) = [x(t) 0 z(t) 1] |
|
\
/
11.4
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東吳資訊科學 江清水
Surface of revolution generated by:
(Assume the curve are on the firsst quadrant of the
XZ plane)
(a) line segment parallel to Z-axis (Cylinder).
(b) line segment which is not parallel to Z-axis
(truncated cone).
(c) line setment parallel to X-axis (Hollowed disc).
(d) a circle (torus).
(e) cubic Bezier curve.
(f) others.
11.2, 11.3(a),(b), 11.5, 11.6
Picture from Anand's book.
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東吳資訊科學 江清水
6.1
Cylindrical surface of revolution. (a) Schematic; (b) result.
6.2
Conical surface of revolution. (a) Schematic; (b) result.
6.3
A disc as surface of revolution. (a) Schematic; (b) result.
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東吳資訊科學 江清水
6.4
A surface of revolution from a closed polygon. (a)
Schematic; (b) result.
6.5
Biparametric surface of revolution.
6.8
Surfaces of revolution. (a) Paraboloid; (b) hyperboloid.
Surfaces
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東吳資訊科學 江清水
6.7
Tori. (a) Circular cross section; (b) elliptical cross section.
6.9
Parabolically blended surface of revolution. (a) Generating
curve; (b) surface.
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東吳資訊科學 江清水
6.10
B-spline surface of revolution. (a) Polygon
vertices; (b) B-spline curves; (c) surface.
Picture from Rogers and Adams' book.
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東吳資訊科學 江清水
Example: Generate the surface obtained by
rotation of the line segment AB with
A=(1,0,1) and B=(3,0,1) around the Zaxis.
Y
X
Z
What the surface look alike if the line
segment AB intersect the Z-axis?
Y
X
Z
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東吳資訊科學 江清水
Example: Generate the surface obtained by
rotation of a circle contained in XZ plane
around the Z-axis. The center of the
circle has coordinates (a,0,0) and the
radius of the circle is r (a > r).
Y
X
Z
What the surface look alike if a <= r?
What the surface look alike if the circle
centered at the origin.
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東吳資訊科學 江清水
For any curve C(t) on XZ plane, the surface
of revolution is represented in matrix form
as:
/ cos  sin  0 0 \
S(t,) = [C(t) 1] | 0
0
0 0 |
| 0
0
1 0 |
\ 0
0
0 1/
Example: Consider the surface of
revplution generated by a cubic Bezier
curve defined by the control points
P1=(1,0,2), P2=(3,0,4), P3=(2,0,6),
P4=(5,0,7) rotating about the Z-axis.
Calculate the point on the surface at
t=0.5, =/4.
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東吳資訊科學 江清水
How to rotated a parametric curve about an
arbitrary axis?
6.11 and 6.12
Picture from Rogers and Adams' book.
Surfaces
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東吳資訊科學 江清水
Surfaces
• Surfaces of revolution
¤ Sweep surfaces
• Parametric surfaces
• Quadric surfaces
• Bilinear surface
• Ruled surface
• Linear Coons surface
• Coons bicubic surface
• Bezier surface
• B-spline surface
• Rational B-spline surface
Surfaces
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東吳資訊科學 江清水
Sweep Surface
A sweep surface is obtained by traversing
an entity, e.g., a line, polygon, or curve,
along a path in space.
The sweep surface is represented in
parametric form as:
S(t,s) = C(t)T(s)
where C(t) is the parametric equation of
the line or curve and T(s) is the sweep
transformation based on the shape of the
path.
11.7
Translational sweep and rotational sweep.
Picture from Anand's book.
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東吳資訊科學 江清水
If the path is a straight line starting at the
origin along the Z-axis, what T(s) looks
alike?
T(s) =
If the path is a straight line starting at the
origin, and the path can be represented
in parametric form as [X(s) Y(s) Z(s)] =
[as bs cs], what T(s) looks alike?
T(s) =
If the path is an origin-centered circle in a
z=constant plane, what T(s) looks alike?
T(s) =
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東吳資訊科學 江清水
Example: Sweep the normalized cubic
spline curve segment defined by P1 =
(0,3,0), P2 = (3,0,0), P1' = [3 0 0], P2' = [3
0 0] 10 units along the z-axis.
6.14
From Rogers and Adams' book.
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東吳資訊科學 江清水
Example: Consider the line segment in the
XY plane and parallel to the y-axis
defined by end points P1=(0,0,0) and
P2=(0,3,0). Determine the point at t=0.5,
s=0.5 on the sweep surface generated by
simultaneously translating the line 10
units along the axis and rotating it
through 2 about the x-axis.
6.13
From Roger and Adams' book.
Surfaces
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東吳資訊科學 江清水
Surfaces
• Surfaces of revolution
• Sweep surfaces
¤ Parametric surfaces
• Quadric surfaces
• Bilinear surface
• Ruled surface
• Linear Coons surface
• Coons bicubic surface
• Bezier surface
• B-spline surface
• Rational B-spline surface
Surfaces
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東吳資訊科學 江清水
Parametric Surfaces
The simplest mathematical element we use
to model a surface is a patch. A patch is a
curve bounded collection of points whose
coordinates are given by continuous,
two-parameter, single-valued
mathematical function of the form: (0 <=
s,t <= 1)
S(s,t) = [x(s,t), y(s,t), z(s,t)]
The following surface is created by a mesh
of curve along the s and t parametric
directions.
t
1
11.1 anand
s
0
1
Compare with the definition for curve
segment: (0 <= t <= 1)
C(1)
C(t) = [x(t), y(t)]
0
t
1
Surfaces
C(0)
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東吳資訊科學 江清水
Example: Find the surface patch for the
rectangular segment of the X-Y plane
with (a,b,0), (a,d,0),(c,b,0) and (c,d,0) as
vertices.
Y
t=1
d
s=0
s=1
b
t=0
a
c
Example: Find the parametrize
respresentation for the sphere centered
at (a,b,c) with radius r.
(From Mortenson's book)
3.3
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東吳資訊科學 江清水
X
We have the following questions for the
parametric form of surface patches:
1. How to find the normal of the surface
patch at a point? Why we want find the
suface normal?
2. How to display the surface patch?
3. Is it possible to have degenerate
patches? If yes, give examples for the
degenerate patches corresponding to a
point and a line.
Surfaces
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東吳資訊科學 江清水
Surfaces
• Surfaces of revolution
• Sweep surfaces
• Parametric surfaces
¤ Quadric surfaces
• Bilinear surface
• Ruled surface
• Linear Coons surface
• Coons bicubic surface
• Bezier surface
• B-spline surface
• Rational B-spline surface
Surfaces
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東吳資訊科學 江清水
Quadirc Surfaces
The natural quadric surfaces, the sphere,
cone, cylinder, ellipsoid, paraboloid of
revolution and the hyperboloid of
revolution, are also surfaces of
revolution. These natural quadric
surfaces, especially the sphere, cone and
cylinder, play an important part in the
manufacture of mechanical parts.
Given the examples of physical object
whose shape has part of natural quadric
surfaces.
•
•
•
•
basketball --------------funnels ------------------beer cans --------------satellite antennas -----
There are two methods for representing
quadric surfaces, algebraically and
geometrically.
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東吳資訊科學 江清水
Algebraic Descriptions of Quadirc Surfaces:
The general quadric surface is given by:
a200x2+a020y2+a002z2+a110xy+a011yz+a011xy+
a100x+a010y+a001z+a000=0
This general quadric equation can be
written in matrix form as [X][A][X]T = 0
where
[X] = [x y z 1]
[A] =
Let's try to eliminate the linear tems.
Assume a translation can eliminate the
linear terms. After the translation, the
matrix form becomes:
[X][Tr][A][Tr]T[X]T = [X][A'][X]T = 0
where the translation matrix is:
[Tr] =
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東吳資訊科學 江清水
The transformed matrix [A'] is:
[A'] =
Eliminating the linear terms, yields
[M][L] = [G]
where
[M] =
[L] =
[G] =
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東吳資訊科學 江清水
If [L] is invertible, a solution exists and the
quadric is central (sphere, cone, cylinder,
ellipsoid, or hyperboloid). Otherwise, the
quadric is noncentral (paraboloid).
Let's extend the idea we have learned in
conics and assume that:
If del[L]>0, then the quadric is ellipsoid.
If del[L]=0, then the quadric is paraboloid.
If del[L]>0, then the quadric is hyperboloid.
What's wrong with this assumption?
Hint: f(x,y,z) and -f(x,y,z) should represent the same surface,
but their corresponding del[L] have different sign.
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東吳資訊科學 江清水
How can we determine whether a quadric
is a ellipsoid, paraboloid, or hyperboloid?
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東吳資訊科學 江清水
As with the conic sections whether [L] is
singular or not, the axes of a quadric can
be made parallel to the coordinate axes
by rotation. Here, three rotations are
required, two to make one of the axes of
the quadric parallel to a coordinate axis
and a final rotation about the axis to
make the other two axes of the quadric
parallel to the other coordinate axes.
Now, a110, a101 and a011 will be zero.
Similarly, if the quadric is noncentral, i.e., a
paraboloid, as mentioned previously the
linear terms cannot all be eliminated.
However, two of the three linear terms
can be eliminated.
The result of these operations is to place
the quadric in the standard form. For the
central quadric the standard form is with
the center at the origin and the axis
aliged with the coordinate axes. In matrix
form the result standard form equation is:
[X][A][X]T = 0
where
[A] =
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東吳資訊科學 江清水
Quadric Surfaces
Fig 6-18 Rogers
from Rogers
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東吳資訊科學 江清水
Example: Transform the following quadric
into standard form and determine the
type of the quadric:
2xy - x - y + z = 0
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東吳資訊科學 江清水
Geometric Descriptions of Quadric Surfaces:
Every quadric surface is represented by a
point, two orthogonal (unit) vectors and
three scalars. The point, either center or
vertex, fix the location of the surface. The
vectors and their cross product determine
its axes or orientation. The scalars
determine its dimensions or size.
How to represent each quadric surfaces?
point
vectors
scalars
Ellipsoid
Hyperboloid
of one sheet
Hyperboloid
of two sheets
Elliptic paraboloid
Hyperbolic paraboloid
Elliptic cone
Sphere
Right circular cylinder
Right circular cone
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東吳資訊科學 江清水
Let's consider a surface patches S which
has boundary curves Cs0,Cs1,C0t ,C1t,
and coner points P00, P10, P01, P11, as
shown in the following picture.
p11
Cs1
p01
C0t
S
C1t
p00
Cs0
p10
There are infinite many surfaces S for given
coner points and/or boundary curves. We
have to specify the way to construct the
unknown information such as the interior
of the surface. Different ways of
constructing the unknown parts produce
different surfaces. Notice that we need
two parameters for surfaces S(s,t) and
one parameter for curves Cs0(t),
Cs1(t),C0t(s) and C1t(s).
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東吳資訊科學 江清水
Surfaces
• Surfaces of revolution
• Sweep surfaces
• Parametric surfaces
• Quadric surfaces
¤ Bilinear surface
• Ruled surface
• Linear Coons surface
• Coons bicubic surface
• Bezier surface
• B-spline surface
• Rational B-spline surface
Surfaces
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東吳資訊科學 江清水
Bilinear Surfaces
A bilinear surface is constructed from the
four corner points of the unit square in
parametric space. Any point in the interior
of the surface is specified by linearly
interpolating between opposite boundaries
of the unit square domain.
p11
p01
S
t
(1,1)
p00
(0,1)
s
(0,0)
p10
(1,0)
The linear interpolation between the points
P00 and P10 is:
Cs0(s) = S(s,0) = sP10+(1-s)P00
The linear interpolation between the points
P01 and P11 is:
Cs1(s) = S(s,1)=sP11+(1-s)P01
So, the surface S(s,t) can be generate by:
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東吳資訊科學 江清水
In matrix form:
S(s,t) = [
]
[
][ ]
Verify that S(i,j)=Pij for i,j {0,1} .
Example:
(1) Draw the picture for the bilinear surface
whose coner points are P00=(0,0,0),
P10=(0,1,1), P01=(1,0,1) and P11=(1,1,0).
(2) Determine the point on the bilinear
surface corresponding to s=t=1/2 in
parametric space.
(3) What kind of the surface it is?
Y
X
Z
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東吳資訊科學 江清水
Example: Derive the bilinear surface patch
for the hyperboloid paraboloid:
z=(x2/a2)-(y2/b2), -a<=x<=a and -b<=y<= b
Y
X
Z
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東吳資訊科學 江清水
Surfaces
• Surfaces of revolution
• Sweep surfaces
• Parametric surfaces
• Quadric surfaces
• Bilinear surface
¤ Ruled surface
• Linear Coons surface
• Coons bicubic surface
• Bezier surface
• B-spline surface
• Rational B-spline surface
Surfaces
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東吳資訊科學 江清水
Ruled Surfaces
A ruled surface is obtained by linearly
interpolating between two known
boundary curves associated with the
opposite side of the unit square in
parametric space.
Let Cs0 and Cs1 are two known boundary
curves and S(s,0)=Cs0(s),S(s,1)=Cs1(s),
the ruled surface is given by:
S(s,t) = (1-t)Cs0(s)+tCs1(s)
or S(s,t) = [1-t t]
[ ]
Alternatively, the curves C0t and C1t are
known and S(0,t)=C0t(t), S(1,t)=C1t(t), the
ruled surface is given by:
S(s,t) =
or S(s,t) = [
Surfaces
]
[ ]
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東吳資訊科學 江清水
Example: You are hired by a company and
your first assignment has to design a
basket which has a square bottom with a
circular top. Can you design this basket
for the company?
Y
X
(r,0,0)
(l,0,h)
Z
Notice how the bottom square, which
contains cusps, is smoothly blended into
the continuous top circle.
Surfaces
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東吳資訊科學 江清水
Example: you are fired by the basket
company and find a new job for an
airplane company. The manager think
you are experienced designer and ask
you to design the shape for aircraft
wings. How to do it?
Surfaces
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東吳資訊科學 江清水
Surfaces
• Surfaces of revolution
• Sweep surfaces
• Parametric surfaces
• Quadric surfaces
• Bilinear surface
• Ruled surface
¤ Linear Coons surface
• Coons bicubic surface
• Bezier surface
• B-spline surface
• Rational B-spline surface
Surfaces
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東吳資訊科學 江清水
Linear Coons Surface
If the four boundary curves Cs0,Cs1,C0t and
C1t are known, a linear coons surface is
obtained by bilinear interpolate the interior
of the surface patch.
Let the surface be defined by:
S(s,t) = (1-t)Cs0(s)+tCs1(s)+(1-s)C0t(t)+sC1t(t)
Let's check the boundary curves S(s,0) and
the corner point S(0,0) of the surface S:
S(s,0) =
S(0,0) =
What is wrong with the surface?
We have to modify the definition to satisfied
the following condition:
for i,j in {0,1}, S(i,j) = Pij and
S(s,j) = Csj(s) and S(i,t) = Cit(t)
So, the correct definition for linear Coons
surface is:
S(s,t) =
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東吳資訊科學 江清水
In matrix form, the surface is defined by:
S(s,t) =
or more compactly as:
S(s,t) =
The functions (1-t),t,(1-s), and s are called
blending functions because they blend
the boundary curves to produce the
internal shape of the surface. The linear
Coons surface is the simplest of the
Coons surfaces. A more general Coons
surface will be discussed in the future.
Surfaces
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東吳資訊科學 江清水
Example: Find the point on a linear Coons
surface at s=t=0.5 when the four
boundary curves are given by open thirdorder (k=3) B-spline curves defined by:
(Points for each B-spline curves are in
order).
Cs0: [0 0 0] [1 1 0] [2 1 0] [3 0 0] [3 0 3]
Cs1: [0 0 3] [1 1 3] [2 1 3] [2 1 3]
C0t: [0 0 3] [0 1 2] [0 1 1] [0 0 0]
C1t: [3 0 3] [3 1 2] [3 1 2] [3 1 1] [3 0 0]
Fig 6-30
from Rogers
Surfaces
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東吳資訊科學 江清水
Surfaces
• Surfaces of revolution
• Sweep surfaces
• Parametric surfaces
• Quadric surfaces
• Bilinear surface
• Ruled surface
• Linear Coons surface
¤ Coons bicubic surface
• Bezier surface
• B-spline surface
• Rational B-spline surface
Surfaces
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東吳資訊科學 江清水
Coons Bicubic Surface
The reason to use the Coons bicubic
surface --- flexibility.
The Coons bicubic surface patch uses
normalize cubic splines for all four
boundary curves. Cubic blending
functions are used to define the interior of
the patch.
Cs1(s)
p11
p01
C0t(t)
p00
Surfaces
Cs0(s)
S(s,t)
C1t(t)
p10
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東吳資訊科學 江清水
Thus, the four boundary curves are:
S(s,0) = Cs0(s) = [S][N][
]T
S(s,1) = Cs1(s) = [S][N][
]T
S(0,t) = C0t(t) = [
][N]T[T]T
S(1,t) = C1t(t) = [
][N]T[T]T
where
[S] = [s3 s2 s 1], [T] = [t3 t2 t 1]
2 -2
1
1
[N] =
-3 3 -2 -1
0 0 1 0
1 0 0 0
The cubic blending function used for both
parametric directions is:
[F] = [F1(s) F2(s) F3(s) F4(s)]
= [S] [N]
or
[F]T = [F1(t) F2(t) F3(t) F4(t)]T
= [N]T[T]T
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東吳資訊科學 江清水
The definition for a Coons bicubic patch is:
S(s,t) = [F1(s) F2(s) F3(s) F4(s)][Pc]
[F1(t) F2(t) F3(t) F4(t)]T
= [S][N][Pc][N]T[T]T
where
[Pc] =
What informations does the [P] matrix
contains?
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東吳資訊科學 江清水
The parametric derivative at any point on a
bicubic surface are obtained by:
Ss(s,t) = [S'][N][Pc][N]T[T]T
St(s,t) = [S][N][Pc][N]T[T']T
Sst(s,t) = [S'][N][Pc][N]T[T']T
Sss(s,t) = [S''][N][Pc][N]T[T]T
Stt(s,t) = [S][N][Pc][N]T[T'']T
where [S'] = [
], [S''] = [
]
[T'] = [
], [T''] = [
]
How to find the normal of the bicubic
surface?
One of the difficulities in using bicubic
surface patches is the disparate
magnitudes of position, tangent and twist
vectors.
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東吳資訊科學 江清水
Example: The Coons bicubic surface S has
the following geometric information:
(1) position vectors
P00=[-100 0 100]
P10=[100 -100 100]
(2) tangent vectors
Cs0s(0)=[100 100 0]
Cs1s(0)=[1 -1 0]
C0tt(0)=[0 10 -10]
C1tt(0)=[0 1 -1]
(3) twist vectors
Sst(0,0)=[0 0 0]
Sst(1,0)=[0.1 -0.1 -0.1]
P01=[ -100 -100 -100]
P11=[100 0 -100]
Cs0s(1)=[1 1 0]
Cs1s(1)=[1 -1 0]
C0tt(1)=[0 -1 -1]
C1tt(1)=[0 -1 -1]
Sst(0,1)=[0.1 0.1 0.1]
Sst(1,1)=[0 0 0]
Find the normal vector at s=t=1/2.
Surfaces
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東吳資訊科學 江清水
Surfaces
• Surfaces of revolution
• Sweep surfaces
• Parametric surfaces
• Quadric surfaces
• Bilinear surface
• Ruled surface
• Linear Coons surface
• Coons bicubic surface
¤ Bezier surface
• B-spline surface
• Rational B-spline surface
Surfaces
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東吳資訊科學 江清水
Bezier Surfaces
A Cartesian (tensor product) Bezier surface
is given by:
S(s,t) = i=0mj=0nPi,jBm,i(s)Bn,j(t)
where Bm,i(s) and Bn,j(t) are the Bernstein
basis functions in the s and t parametric
directions. That is:
Bm,i(s) =
Bn,j(t) =
The Bi,j's are the control points for the
Bezier surface. The (m+1)x(n+1) control
points define a polygon net.
The Bezier surfaces can be represented by
matrix form:
S(s,t) = [S][B(m)][Pb][B(n)][T]T
where [S] = [s3 s2 s 1], [T] = [t3 t2 t 1]
[Pb]=
[B(m)]=
mxm
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東吳資訊科學 江清水
The properities of the Bezier curve:
(1) The degree of the surface in each
parametric direction: m-1 and n-1.
(2) Convex hull property: The surface is
contained within the convex hull of the
defining polygon net.
(3) End points interpolation: The surface
passing through the corner points of the
defining polygon net.
(4) The surface is invariant under an affine
transformation.
(5) The surface generally follows the shape
of the defining polygon net. The user can
control the shape of the surface patch
without an intimate knowledge of tangent
or twist vector.
(6) Each boundary curves of the Bezier
surface is a Bezier curve.
(7) Tangent vectors at the corner:
(8) Continuity at the corner points and
boundary curves:
Surfaces
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東吳資訊科學 江清水
Consider the extension of (6), given a
specific value t0, where 0 <= t0 <= 1.
Intitively, the curve S(s,t0) should be a
Bezier curve. Find the control points for
the curve.
Prove that the control points for the curve
S(s,0) and S(s,1) are Pi,0 and Pi,n,
0<=i<=m, respectively.
Surfaces
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東吳資訊科學 江清水
The relationship between a bicubic Bezier
and a bicubic Coons surface is:
SCoons(s,t) = SBezier(s,t)
That is:
[S][N][Pc][N]T[T]T = [S][B(3)][Pb][B(3)][T]T
So,
[Pc] =
[Pb] =
Example: From the above equation, find
the relation for the control points of SBezier
and the geometric properities of SCoons.
Surfaces
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東吳資訊科學 江清水
Example: Consider a plane patch on z=0,
its Bezier control points are:
Pi,j = [i j 0]
(1) Find [Pb].
(2) Find [Pc].
(3) Let's change the point P3,3 to [3,3,3],
find new [Pb] and [Pc].
Surfaces
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東吳資訊科學 江清水
The parametric derivative at any point on a
bicubic surface are obtained by:
Ss(s,t) = [S'][N][Pc][N]T[T]T
St(s,t) = [S][N][Pc][N]T[T']T
Sst(s,t) = [S'][N][Pc][N]T[T']T
Sss(s,t) = [S''][N][Pc][N]T[T]T
Stt(s,t) = [S][N][Pc][N]T[T'']T
where [S'] = [
], [S''] = [
]
[T'] = [
], [T''] = [
]
How to find the normal of the bicubic
surface?
One of the difficulities in using bicubic
surface patches is the disparate
magnitudes of position, tangent and twist
vectors.
Surfaces
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東吳資訊科學 江清水
Surfaces
• Surfaces of revolution
• Sweep surfaces
• Parametric surfaces
• Quadric surfaces
• Bilinear surface
• Ruled surface
• Linear Coons surface
• Coons bicubic surface
• Bezier surface
¤ B-spline surface
• Rational B-spline surface
Surfaces
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東吳資訊科學 江清水
Surfaces
• Surfaces of revolution
• Sweep surfaces
• Parametric surfaces
• Quadric surfaces
• Bilinear surface
• Ruled surface
• Linear Coons surface
• Coons bicubic surface
• Bezier surface
• B-spline surface
¤ Rational B-spline surface
Surfaces
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東吳資訊科學 江清水