Surfaces
• Boundary representation of objects
• Smooth surfaces
> Implicit representation
f(x, y, z) = 0
> Parametric representation
P(u,v) = (x(u,v) y(u,v) z(u,v))
Surfaces
Parametric Surfaces
P(u,v) = (x(u,v) y(u,v) z(u,v))
Surfaces
Surface of Revolution
• One of the simplest method to generate surfaces
• Obtained by rotating 2D entity about an axis
Point
Line parallel to X
axis
y
y
x
Circle
x
Cylinder
Surfaces
Surface of Revolution
Parametric Form
Parametric equation of the entity to be rotated
P (t ) = [x (t ) y(t) z(t)] 0 ≤ t ≤ t max
Rotation angle Φ
The surface is now a bi-parametric function of
two parameters t and Φ
Example: Rotation about X – axis of an entity in XY plane
Q( t , Φ) = [x(t)
y(t)cosΦ y(t)sinΦ]
Surfaces
Surface of Revolution
Sphere is generated by rotating a semi-circle centered
at origin and lying in XY plane about X axis
Circle: x = rcosθ, y = rsinθ
Q (θ,Φ) = [x ycosΦ ysinΦ]
= [rcosθ rsinθcosΦ rsinθsinΦ]
Similarly ellipsoid is generated by rotating semi ellipse
about X axis
Surfaces
Surface of Revolution
Torus is generated by rotating a circle lying in XY plane
but whose center does not lie on the axis of rotation.
Circle: x = h + rcosθ y = k+ rsinθ
Q(θ,Φ)=[h+rcosθ (k+rsinθ)cosΦ
(k+rsinθ)sinΦ]
(h, k) is the center of the circle
Surfaces
Surface of Revolution
Matrix Form (X axis rotation)
0
1
0 cos ö
S=
0
0
0
0

0
sin ö
0
0
0
0

0
1
(x, ycosΦ, ysinΦ)
(x,y)
Φ
If parametric curve P(t) = FG
Then surface of revolution Q(t, Φ)=FGS
Can be extended for rotation about any arbitrary axis
Surfaces
Sweep Surfaces
Sweep surfaces are the surfaces generated by traversing
an entity along a path in space
Q(t,s) = P(t) T(s)
T(s) is called the sweep transformation
Surfaces
Sweep Surfaces
Translation sweep of a circle generates a cylinder
Translation sweep of a circle accompanied by
scaling generates cone
Surfaces
Sweep Surfaces
Example
y
y
Cubic Spline
Sweep Surface
x
1
0
T (s ) = 
0
0

x
0
0

0
0 ns 1
0
1
0
0
0
1
z
Translation in Z
Surfaces
Sweep Surfaces
Normal to the polygon or closed curve as the path is
swept can be either kept fixed or it can be made the
instantaneous tangent of the curve of sweep.
Fixed normal
Tangential normal
Surfaces
Bilinear Interpolation
Linear Interpolation fits the simplest curve between
two end points
P2
P1
P(t) = (1-t) P1+ t P2
Surfaces
Bilinear Interpolation
Bilinear interpolation fits the
simplest surface for the
four corner points
v
b01
b11
u
b00
v
u
b10
Surfaces
Bilinear Interpolation
Two Stage Process
b01
b11
01
b00
= (1 − v )b00 + vb01
01
b10
= (1 − v )b10 + vb11
11
01
01
(u,v ) = (1 − u )b00
X (u,v ) = b00
+ ub10
b00 b01  1 − v 
= [1 − u u ] 


b10 b11   v 
b01
b01
1
1
X (u,v ) = ∑ ∑ bij Bi1(u )B1j (v )
i =0 j =0
Surfaces
Ruled Surfaces
Given two space curves (C1 & C2) defined in parametric
range [0,1], Find a surface X that contains both curves as
boundary curves.
X ( u,0) = C1 ( u)
C2
X ( u,1) = C 2 ( u)
v
u
Many solutions are possible
C1
Surfaces
Ruled Surfaces
A simple solution is to do a linear interpolation in v
C2
v
u
C1
X ( u, v ) = (1 − v )C 1 ( u) + vC 2 ( u)
= (1 − v ) X ( u,0) + vX ( u,1)
For constant u iso-parametric
curve is a line
Surfaces
Coon’s Patch
Given Four Boundaries C1(u),C2(u),D1(v),D2(v)
C2
X ( u,1) = C 2 ( u)
D1
v
u
X ( u,0) = C 1 ( u)
C1
D2
X (0, v ) = D1 (v )
X (1, v ) = D2 (v )
Surfaces
Coon’s Patch
Ruled surface between C1(u),C2(u)
C2
rC
C1
Surfaces
Coon’s Patch
Ruled surface between
C1(u),C2(u)
Ruled surface between
D1(v),D2(v)
C2
D2
C1
rC
D1
rD
Surfaces
Coon’s Patch
Ruled surface between
C1(u),C2(u)
Ruled surface between
D1(v),D2(v)
C2
+
C1
rC
D2
D1
rD
Surfaces
Coon’s Patch
Ruled surface between
C1(u),C2(u)
Ruled surface between
D1(v),D2(v)
C2
+
C1
rC + rD
D2
D1
Surfaces
Coon’s Patch
Something is extra which is the bilinear patch
formed by the vertices
rCD
Surfaces
Coon’s Patch
C2
C1
+
Coon’s Patch = rC + rD - rCD
D1
D2
Surfaces
Coon’s Patch
Mathematically
rC (u,v ) = (1 − v ) X (u,0) + vX (u,1)
rD (u,v ) = (1 − u ) X (0,v ) + uX (1,v )
 X (0,0) X (0,1) 1 − v 
rCD (u,v ) = [1 − u u ]
 v 
X
(
1
,
0
)
X
(
1
,
1
)



Surfaces
Coon’s Patch
Mathematically
X(u,v)= rC + rD - rCD
 X (0,v ) 1 − v 
+
[ X (u,0)
X (u,v ) = [1 − u u ] 


 X (1,v )   v 
 X (0,0) X (0,1) 1 − v 
− [1 − u u ] 


 X (1,0) X (1,1)   v 
One can generalize with
f1(u)=1-u ,f2(u)=u and g1(v)=1-v,g2(v)=v
X (u,1)]
Surfaces
Parametric Surfaces
P(u,v) = (x(u,v) y(u,v) z(u,v))
Surfaces
Bezier Surface
Given control points:
b03
b00 b01
….
b33
b33
v
u
b00
b30
Surfaces
Bezier Surface
Bezier Curve (Revisit)
b2
b1
P(t)
b0
Mathematically
n
P (t ) = ∑ bi J in (t )
i =0
b3
b0 b1 b2 b3 : Control Polygon
0 ≤ t ≤1
Surfaces
Bezier Surface
Bezier Curve (Revisit)
The de Casteljau Algorithm
b01 (t ) = (1 − t )b0 + tb1
b1
b11
b11(t ) = (1 − t )b1 + tb2
b02 (t ) = (1 − t )b01 (t ) + tb11(t )
b01
= (1 − t )2 b0 + 2t (t − 1)b1 + t 2b2
b02
b2
b0
0
t
1
Surfaces
Bezier Surface
Bezier Curve (Revisit)
The de Casteljau Algorithm
Bezier Curve is constructed using successive linear
interpolation
Bezier Surface can be constructed using successive
bi-lineear interpolation
Surfaces
Bezier Surface
Bilinear Interpolation(Revisit)
Two Stage Process
b01
b00
b11
01
b00
= (1 − v )b00 + vb01
01
b10
= (1 − v )b10 + vb11
11
01
01
(u,v ) = (1 − u )b00
X (u,v ) = b00
+ ub10
b00 b01  1 − v 
= [1 − u u ] 
 v 
b
b

 10
11  
v
u
b10
1
1
X (u,v ) = ∑ ∑ bij Bi1(u )B1j (v )
i =0 j =0
Surfaces
Bezier Surface
De Castelejau Algorithm
b
11
b00
= [1 − u u ]  00
 b10
b03
b01  1 − v 
b11   v 
b33
v
u
b00
b30
Surfaces
Bezier Surface
De Castelejau Algorithm
b03
b33
v
u
b00
b30
Surfaces
Bezier Surface
De Castelejau Algorithm
11
11
b00
 1 − v 
b01
= [1 − u u ]  11

11  
v
b
b


 10
11 
b03
b33
22
b00
v
u
b00
b30
Surfaces
Bezier Surface
De Castelejau Algorithm
b03
b33
v
u
b00
b30
Surfaces
Bezier Surface
De Castelejau Algorithm
33
b00
b03
22
b00
= [1 − u u ]  22
b10
22
 1 − v 
b01

22  
b11
 v 
b33
v
u
33
b00
b00
b30
Surfaces
Bezier Surface
De Castelejau Algorithm
Mathematically
r −1,r −1
r −1,r −1

 1 − v 
b
b
i, j
i , j +1
r ,r
bi , j = [1 − u u ]  r −1,r −1
r −1,r −1  

v
b
b


+
+
+
i
1
,
j
i
1
,
j
1


r = 1, 2, L n
i , j = 0,1,L(n − r)
Surfaces
Bezier Surface
De Castelejau Algorithm
Problem: If degree in u is not equal to the degree in v
b03
b23
v
u
b00
b20
Surfaces
Bezier Surface
Tensor Product
A surface can be thought of as being swept out by a
moving and deforming curve
Surfaces
Bezier Surface
Tensor Product
Let the sweep curve be:
b ( u) =
m
m
m
∑ bi Bi
i =0
( u)
Each control point bi traverses a Bezier curve
bi = bi (v ) =
n
n
∑ bi , j B j
j=0
(v )
Surfaces
Bezier Surface
Tensor Product
Combining
b
m ,n
( u, v ) =
m
n
m
i
∑ ∑ bi , j B
i =0 j=0
n
j
( u) B ( v )
Surfaces
Bezier Surface
Tensor Product
Surfaces
Bezier Surface
Tensor Product
Bezier control net
Surfaces
Bezier Surface
Matrix Form
b
m,n
m n
(u,v ) = ∑ ∑ bij Bim (u )B nj (v )
i =0 j =0
n
 b00 L b0 n  B0 (v )


m
m


M  M 
= B0 (u )LBm (u ) M
 n

bm 0 L bmn  Bn (v )
[
]
Surfaces
Bezier Surface
Properties
• Affine invariance
• Convex hull
• Boundary curve and end-point interpolation
• Change in control point position changes surface
shape
Surfaces
Bezier Surface
Degree Elevation
Degree in u = 2
in v = 3
v
u
Surfaces
Bezier Surface
Degree Elevation
Degree in u = 3
in v = 3
v
u
Surfaces
Bezier Surface
Bicubic Patch
Surfaces
Bezier Surface
Shape Control
Surfaces
Bezier Surface
Derivatives
n
∂ m,n
∂ m

b (u,v ) = ∑  ∑ bij Bim (u ) B nj (v )
∂u

j =0  ∂u i =0
n m −1
= m ∑ ∑ Ä1,0 bij Bim −1(u )B nj (v )
j =0 i =0
Ä1,0 bij = bi +1 j − bij
Surfaces
Bezier Surface
Derivatives
m  ∂ n
 m
∂ m,n
n
b (u,v ) = ∑  ∑ bij B j (v ) Bi (u )
∂v
i =0  ∂v j =0

m n −1
= n ∑ ∑ Ä0,1bij B nj −1(v )Bim (u )
i =0 j = 0
Ä0,1bij = bij +1 − bij
Surfaces
Bezier Surface
Derivatives
Cross Boundary Derivatives
b03
b00
b33
Ä0,1b30
b30
Surfaces
Bezier Surface
Derivatives
Normal Vectors
∂ m ,n
∂ m ,n
b ( u, v ) × b ( u, v )
∂v
n( u, v ) = ∂u
∂ m ,n
∂ m ,n
b ( u, v ) × b ( u, v )
∂u
∂v
n
∂
∂v
∂
∂u
Surfaces
Bezier Surface
Derivatives:
m −1n −1
∂ 2 m,n
−1
n −1
b (u,v ) = mn ∑ ∑ Ä1,1bi , j J m
(
u
)
J
(v )
j
i
∂u∂v
i =0 j =0
Geometric Interpretation:
bi+1,j+1
Pi , j − bi + 1, j = bi , j +1 − bi , j
∆
∆11 bi , j = (bi +1 , j + 1 − bi +1 , j )
11
bi,j+1
− (bi , j + 1 − bi , j )
Pi,j
bi,j
bi+1,j
∆11 bi , j = (bi +1 , j + 1 − Pi , j )
Twist Vector
Surfaces
Bezier Surface
Composite Patches
Surfaces
Bezier Surface
Composite Patches
C0 Continuous
Surfaces
Bezier Surface
Composite Patches
C1 Continuous
Surfaces
Bezier Surface
Utah Teapot
32 Bicubic
Bezier patches
Surfaces
B-Spline Surface
B-Splines
Polynomial spline function of order k (degree k-1)
n +1
P (t ) = ∑ Bi Ni ,k (t )
i =1
Bi : Control point
Nik : Basis function
tmin ≤ t ≤ tmax
2 ≤ k ≤ n +1
Surfaces
B-Spline Surface
m +1 n +1
P (u,v ) = ∑ ∑ Bij Ni ,p (u ) N j ,q (v )
i =1 j =1
Bij : Control point
Nip, Njq : Basis functions
Surfaces
B-Spline Surface
Properties
• Affine invariance
• Convex hull (stronger)
• Change in control point position changes surface
shape (local control)
Surfaces
B-Spline Surface
Properties
• local control
Surfaces
B-Spline Surface
Examples
Surfaces
Polygonal Representation
For rendering often object is represented as collection of
polygons
Object
Surfaces
Polygonal
Mesh
Surfaces
Polygonal Representation
Polygonal mesh is a collection of edges, vertices and
polygons such that each edge is shared by at most two
polygons
Edge: Connects two vertices
Polygon: Closed sequence of edges
Surfaces
Polygonal Representation
Explicit Representation
Each polygon is represented by
P=((x1,y1,z1) (x2,y2,z2) … (xn,yn,zn))
i.e. vertices are stored in the order of traversal
Edges connect the successive vertices plus the last one
This representation has restrictive manipulation and has
multiple storage of points.
Surfaces
Polygonal Representation
Pointer to Vertex List
V2
Each vertex is stored once in a list V
V=((x1,y1,z1) (x2,y2,z2) … (xn,yn,zn))
V1
Each polygon is represented as
P=(V1,V2,V3)
e.g. P1=(1,2,4) and P2=(4,2,3)
V3
P2
P1
V4
In this representation it is difficult to find polygons that
share an edge.
Surfaces
Polygonal Representation
Pointer to Edge List
V2
Edge: E = (Vi,Vj,Pm,Pn)
Polygon=(Ep,Eq,Er)
E1
V1
V = (V1,V2,V3,V4)
E1 = (V1,V2,P1,null)
E2 = (V2,V3,P2,null)
E3 = (V3,V4,P2,null)
E4 = (V4,V2,P1,P2)
E5 = (V4,V1,P1,null)
E2
P1
E4
E5
P2
E3
V4
P1= (E1E4E5)
P2= (E2E3E4)
V3
Surfaces
Polygonal Representation
Winged Edge Data Structure
Edge
Vertices
Start
End
Faces
Left
Right
Left Traverse Pred
Succ
Right Traverse Pred
Succ
a
X
Y
1
2
b
d
e
c