Transcript Document 7168435

Surfaces
Chiew-Lan Tai
Reading
Required
 Hills Section 11.11
 Hearn & Baker, sections 8.11, 8.13
Recommended
 Sections 2.1.4, 3.4-3.5, 3D Computer
Graphics, Watt
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Mathematical surface
representations
 Explicit
z = f(x,y) (a.k.a. “height field”)
What if the surface isn’t a function?
 Implicit
g(x,y,z) = 0
x2 + y2 + z2 = 1
 Parametric
S(u,v) = (x(u,v), y(u,v), z(u,v))
x(u,v) = r cos 2v sinu
y(u,v) = r sin 2v sinu
z(u,v) = r cos u
0≤u≤1, 0≤v≤1
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Tensor product Bézier surfaces
 Given a grid of control point Vij, forming a control net, construct
a surface S(u,v) by:


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Each row in u-direction is control points of a curve Vi(u).
V0(u), …, Vn(u) at a specific u are the control points of a curve
parameterized by v (i.e. S(u,.v))
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Tensor product Bezier surfaces, cont.
 Let’s walk through the steps:
 Which control points are interpolated by the surface? Only the 4 corner
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points
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Tensor product Bezier surfaces,
cont.
 Writing it out explicitly:
 Linear Bezier surface:
S (u, v)  (1  v)[(1  u )V00  uV01 ]
 v[(1  u )V10  uV11 ]
 Quadratic Bezier surface?
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Matrix form
 For the case of cubic Bezier surface:
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Tensor product B-spline surfaces
 Like spline curves, we can piece together a sequence of Bezier
surfaces to make a spline surface. If we enforce C2 continuity
and local control, we get B-spline curves:
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Tensor product B-spline surfaces
u


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Use each row of B control points in u to generate Bezier control points in u.
Treat each row of Bezier control points in v direction as B-spline control points to
generate Bezier control points in v direction.
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Tangent plane and surface normal
S
S
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Partial Derivatives

S (u, v)  [v 3
u

S (u, v)  [3 v 2
v
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v2
v
3u 2 
 
2u 
T

1]M BezierV M Bezier
 1 
 
 0 
u 3 
 2
u 
T

2v 1 0]M BezierV M Bezier
u 
 
 1 
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Surface normal
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S
S
S
S
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Other construction methods
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Rotational surfaces
(surface of revolution)
 Rotate a 2D profile curve about an axis
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Constructing surfaces of revolution
 Given a curve C(u) in the yz-plane:
y
Profile curve, C(u)
z
x
 Let Ry() be a rotation about the y-axis
 Find: A surface S(u,v) obtained by applying Ry() on C(u)
S(u,v) = Ry(v) [0, Cy(u), Cz(u), 1]t
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y
Surface of revolution
 Profile curve on yz plane:
C(u) = ( 0, Cy(u), Cz(u) )
x
z
 The surface S(u,v) obtained by rotating C(u) about
the y-axis is
S(u,v) = ( -Cz(u) sin(v), Cy(u), Cz(u) cos(v))
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General sweep surface
 Given a planar profile curve C(u) and a
general space curve T(v) as the trajectory,
sweep C(u) along the trajectory
 How to orient C(u) as it moves along T(v)?
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Orientating C(u)
 Define a local coordinate frame at any point along the trajectory
 The Frenet frame (t,n,b) is the natural choice
 As we move along T(v), the Frenet frame (t,b,n) varies smoothly
(inflection points where curvature goes to zero needs special
treatment)
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Sweep Surfaces
 Orient the profile curve C(u) using the Frenet frame
of the trajectory T(v):
 Put C(u) in the normal plane.
 Place Oc at T(v).
 Align xc of C(u) with b.
 Align yc of C(u) with n.
 Sweep surface
S(u,v) = T(v) + n(v)Cx(u) + b(v) Cy(u)
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Variations
 Several variations are possible:


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Scale C(u) as it moves, possibly using length of T(v) as a
scale factor.
Morph C(u) into some other curve C’(u) as it moves along
T(v)
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Summary
What to take home:
 How to construct tensor product Bezier surfaces
 How to construct tensor product B-spline surfaces
 How to construct surfaces of revolution
 How to construct sweep surfaces from a profile and a
trajectory curve with a Frenet frame
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Subdivision Surfaces
What’s wrong with B-spline/NURBS surfaces?
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Subdivision curves
Idea:
 repeatedly refine the control polygon
P 0  P1  P 2  
C  lim P i
i 
 curve is the limit of an infinite process.
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Subdivision curves
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Chaikin’s algorithm
 In 1974, Chaikin introduced the following “corner-
cutting” scheme:


Start with a piecewise linear curve
Repeat
 Insert new vertices at the midpoints (the splitting

step)
Average two neighboring vertices (the average
step)
Averaging mask
(0.5, 0.5)
 With this averaging mask, in the limit,
the resulting curve is a quadratic Bspline curve.
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Local subdivision mask
 Subdivision mask: ¼ (1, 2, 1)
 Splitting and averaging:
 Applying the step recursively, each vertex converges to a specific
point
 With this mask, in the limit, the resulting curve is a cubic B-spline
curve.
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General subdivision process
 After each split-average step, we are closer to the limit
surface.
 Can we push a vertex to its limit position without infinite
subdivision? Yes!
 We can determine the final position of a vertex by
applying the evaluation mask. The evaluation mask for
cubic B-spline is 1/6 (1,4, 1)
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DLG interpolating scheme (1987)
 Algorithm:
 splitting step introduces midpoints, and averaging step
only changes these midpoints
 Dyn-Levin-Gregory scheme:
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Building Complex Models
 This simple idea can be extended to build
subdivision surfaces.
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Subdivision surfaces
 Iteratively refine a control polyhedron (or control mesh) to
produce the limit surface using splitting and averaging step:
 There are two types of splitting schemes:


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vertex schemes
face schemes
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Vertex schemes
 A vertex surrounded by n faces is split into n
subvertices, one for each face:
 Doo-Sabin subdivision:
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Face schemes
 Each quadrilateral face is split into four subfaces:
 Catmull-Clark subdivision:
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Face scheme, cont.
 Each triangular face is split into four subfaces:
 Loop subdivision:
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Averaging step
 Averaging masks:
 (n)Q  Q1    Qn
Q
 ( n)  n
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Adding creases
 Sometimes, a particular feature such as a crease should be
preserved.
 we just modify the subdivision mask.
 This gives rise to G0 continuous surfaces.
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Creases
 Here’s an example using Catmull-Clark
surfaces:
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Spatial Object
Deformation
Object Deformation
 Many objects are not rigid

jello, mud, gases, liquids
 In animation, to give characterization to rigid
objects

stretch and squash
 Two main classes of techniques


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Geometric-based
Physically-based
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Geometric Deformations
 Deform the object’s geometry directly

control point / vertex manipulation
 Deform the object’s geometry indirectly


Warp the space in which the object is
embedded (Spatial Deformation)
Main techniques


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Nonlinear Deformation (Barr)
Free Form Deformation (FFD)
Curve-based Deformation
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Spatial Deformation: General framework
 Warps the space that the
mesh is embedded in
 User inputs pairs of features
to guide the deformations

point pairs
local coordinate frames

continuous curves

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Spatial Deformation: General framework
 User inputs pairs of
features to guide the
deformations


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discrete point pairs
local coordinate frames
Continuous curves
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Spatial Deformation: General framework
 Spatial deformation defines a mapping on the
space that surrounds the model

It can be performed on any models that use
point-based data





Deformation is propagated along the space

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pixels in images
control points in spline surfaces
Vertices in mesh
point cloud data
May not be what the user expects
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Freeform Deformation (FFD)
[Sederberg and Parry 1986]
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Freeform Deformation (FFD)
[Sederberg and Parry 1986]
 Define a volume using
parallelepiped lattice
 Lattice defines a coordinate
system (S,T,U)
 Modify the lattice points
 Deformation of a point in the
space depends on its (s,t,u)
coordinates
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Freeform Deformation (FFD)
The lattice defines a Bezier volume:
P( s, t , u )   p ijk Bi ( s ) B j (t ) Bk (u )
ijk
For each object point v,
determine its lattice
coordinates (s, t, u)
Alter the lattice points p ijk
New position of v is evaluated
as P(s,t,u)
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Freeform Deformation (FFD)
[Sederberg and Parry 1986]
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Freeform Deformation (FFD)
[Sederberg and Parry 1986]
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Axial Deformation [Lazarus 1994]
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Axial Deformation [Lazarus 1994]
Source curve R(u)
target curve T(u)
P
P’
y’(u)
Y(u)
R(u)
x(u)
s(u )  (P  R (u ))  x(u)
t (u )  (P  R (u ))  y(u)
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T(u)
x’(u)
P'  T(u)  s(u) x' (u)  t (u) y' (u)
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Wires”
[Singh and Fiume 1998]
 Defined curves “bound” on the surface
 User deforms the mesh by editing the curves
 The original and modified curves are feature pairs
 Includes local rotation and scaling controls
 Provide smooth composite deformation for multiple
curve features
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Rendering Techniques
leading to geometry illusions
Texture Mapping
 Add surface texture, or colour to a computer-
generated graphic or 3D model


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texture map is applied (mapped) to the
surface of a shape
It is a rendering technique
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Bump Mapping (normal mapping)
 A perturbation to the surface normal at each pixel of
the object being rendered



look up in a height map and apply before the
illumination calculation is done
Normal map is the most commonly used technique
It is a rendering technique
w/o bump mapping
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Bump map
With bump mapping,
same geometry but
rendered with
detailed normals
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Displacement mapping
 An alternative technique in contrast
to bump mapping
 using a texture- or height map to
displace the actual geometric
position


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Displacements are often along the
local surface normal
Can be modeling OR rendering
technique
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Displacement mapping
Displacement maps
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Displaced Surfaces
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Alpha Mapping
 Specify transparency level at each pixel of the object
being rendered

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It is a rendering technique
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