Surface and Solid
Geometry
1
3D Polygons
• Once we know our plane equation:
Ax + By + Cz + D = 0,
we still need to manage the truncation which
leads to the polygon itself
Functionally, we will need to
do this to know if a point
lies in a polygon or not,
for example
2
1
3D Polygons
• To do this, we can project the 3D polygon into
2D and see if the point is in the 2d
using the inside test
z
(xi,yi,zi)
y
(xi’,yi’)
x
3
3D Polygons
• Project 3d to 2d based on largest of A,B,C
This example:
z
(xi,yi,zi)
Z (or C) is principal
component of N,
normal so project on
to xy-plane
y
x
(xi’,yi’)
4
2
Z-buffer
• For 3D with multiple polygons, must deal
with visibility
image
plane
Project vertices into image plane and with projected
vertex include depth in additional depth- or Z-buffer
5
Z-buffer
Scan convert each projected polygon:
For each polygon in scene
project verticies
for each pixel inside poly
calculate z
if z < closest
draw into frame buffer
update z buffer
6
3
Z-buffer
• How do we calculate z for buffer, quickly?
z = (-Ax -By -D)/C from plane eq.
Same scanline (x + 1):
z' = (-A(x + 1) - By - D)/C or
z' = z - A/C
z z'
7
Z-buffer
• How do we calculate z for buffer, quickly?
• For the next scanline, following edge
coherence:
xs+1 = xs + 1/m, ys+1 = ys +1
plus zs = (-Axs -Bys -D)/C,
zs+1 = zs - (A/m + B)/C
zs+1
z z'
8
4
Z-buffer
• Z-buffering is very common approach,
also often accelerated with hardware
• OpenGL embeds this approach
3D Polygon
GRAPHICS PIPELINE
Image Pixels
9
10
5
Surface Geometry
11
Surfaces
• Interpolating points for 2 parameters, u and v
Bi-cubic patch
12
6
Subdivision Surfaces
Refinement
Types of Subdivision
• Interpolating Schemes
- Limit Surfaces/Curve will pass through original set
of data points.
• Approximating Schemes
- Limit Surface will not necessarily pass through the
original set of data points.
7
Subdivision Surfaces
Interpolation example
• Approach Limit Curve Surface through an Iterative Refinement
Process.
Refinement 1
Refinement 2
Refinement ∞
Subdivision Surfaces
A Primer: Chaiken’s Algorithm (approximating surface)
Q2 Q3 P2
1
3
P1
Q0  P0  P1
Q4
4
4
Q1
3
1
Q5
Q1  P0  P1
Q0
4
4
P
3
P0
1
3
Apply
Iteratively
Q2 
P1  P2
4
4
3
1
Q3  P1  P2
4
4
1
3
P2  P3
4
4
3
1
Q5  P2  P3
4
4
Q4 
Limit ‘Curve’ Surface
8
Surface Example
Surface evolution with subdivision level
Limit surface
Catmull-Clark Subdivision (1978)
FACE
f 
1 n
 vi
n 1
EDGE
e
v1  v 2  f1  f 2
4
VERTEX
v i 1 
n2
1
vi  2
n
n
e
j
j

1
n2
f
j
j
9
Subdivision used in production
• Traditionally B-spline patches have
been used in production for character
animation.
• Difficult to control B-spline patch
density in character modelling.
Subdivision in Character Animation
Tony Derose, Michael Kass, Tien Troung
(SIGGRAPH ’98)
(Geri’s Game, Pixar 1998)
Solid Constructive
Geometry
10
Solid geometry
• Inherently 3D, geometric elements
describe sets of spaces enclosed by 2D
boundaries
• For example, a solid sphere is the
simplest solid element. Other simple
primitives include the cube, cylinder,
cone, and torus
21
Solid geometry
• Other objects are defined by
combinations of primitives.
• An entire math has been explored related
to the combination of solid primitives
called constructive solid geometry (CSG)
• With CSG, complex shapes may be
generated from operations formed on
primitives
22
11
Solid geometry
• Boolean operators are defined tools for
combining solid geometry for CSG
• These perform group operations on the
points included in the solid primitives
• They are:
- Union
- Subtraction
- Intersection
• These bool-op's are implemented in Maya
23
Solid geometry
• Boolean operator: Union combines two
elements into a single one
24
12
Solid geometry
• Boolean operator: Union combines two
elements into a single one
25
Solid geometry
• Boolean operator: Subtract take the
difference between two elements
26
13
Solid geometry
• Boolean operator: Subtract take the
difference between two elements
27
Solid geometry
• Boolean operator: Subtract take the
difference between two elements
28
14
Solid geometry
• Boolean operator: Subtract
29
Solid geometry
• Boolean operator: Intersection finds the
common points in the given primitives
30
15
31
Solid Constructive
Geometry
32
16
CSG
Tree
Solid geometry
Graph for
hierarchy
of Boolean
operations
Often used
for CAD and
Mech Eng
33
Solid geometry
Adams & Dutre 2003
34
17
Surfaces splitting
Split along the intersection curve
A
A
C A
A∩B
Label each part of the object (inside/outside)
A
A
C A
C
35
Example
36
18
Intersection curve
Compute approximate intersection of both mesh sets
37
Split and label operations
Interior
faces
Exterior
faces
38
19
Reconstruction
Depending on boolean operation: Merge
operation along the intersection curve
Intersection
Union
A-B
B-A
39
WHY?
Solid Constructive
Geometry
40
20
Solid Constructive
Geometry
41
Solid Constructive
Geometry
42
21