Part IV Geometric and Solid Modeling

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Part IV Geometric and Solid Modeling
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Modeling through Polygon Mesh

• Computer graphics modeling rendering
  animation
• A model is a computer representation of an
  object. Models may include molecules, human
  fingers, jet fighters, mountains, etc.
• Rendering is the process of creating an
  image/picture from a model.
• The most popular modeling technique in CG is
  polygon mesh where a surface is approximated by a
  mesh of polygons.
• The simplest polygon is a triangle. Then,
  triangular mesh.

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Triangular Mesh Representation I

• Each triangle is represented by 3 points.
• Problems
• storage waste
• no information on connectedness among polygons
• etc.

T1 ( (x1, y1, z1) (x2, y2, z2) (x4, y4, z4)
) T2 ( (x2, y2, z2) (x3, y3, z3) (x4, y4,
z4) )
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Triangular Mesh Representation II

• A triangle is represented by 3 vertex indices.
• A vertex is stored only once.
• Vertex coordinates can be easily changed.

V1 (x1, y1, z1) V2 (x2, y2, z2) V3 (x3, y3,
z3) V4 (x4, y4, z4) T1 (V1, V2, V4) T2 (V2,
V3, V4)
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Triangular Mesh Representation II -File Format
? Vertex Vertex
keyword integer coordinates of vertex
(floating point) ? Face Face
keyword integer vertex id of a
triangle ? Starting with are comments.
Vertex 331 0.370581
0.208696 0.190467 Vertex 335 0.36654
0.197382 0.118951 Vertex 338 0.353952
0.220184 0.144893 Vertex 343 0.420128
0.177147 0.132059 Vertex 348 0.382545
0.183089 0.185215
Face 12 335 351 296 Face 17 3824
3864 4009 Face 28 1507 1506 1578 Face 161
296 257 45 Face 171 335 296 246
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Triangular Mesh Representation II -Data Structure

• Extremely simplified versions of vertex/triangle
  structures.
• Linked list implementation?
• How about large-enough static array?
• How about dynamic array?
• Think about that!!

• struct Vertex
• unsigned short v_no
• double x
• double y
• double z
• / double normal3 /
• Vertex link

• struct Face
• / unsigned short f_no /
• Vertex v1
• Vertex v2
• Vertex v3
• / double normal3 /
• Face link

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Triangular Mesh Representation III

• How to construct a mesh data structure depends on
  applications.
• For example, for this class, only vertices and
  faces would be needed to be explicitly defined.
  However, some other applications would need
  explicit data structures for edges.
• We can represent a triangle by edge indices which
  are again represented by vertex indices.

V4
E1 (V1, V4) E2 (V1, V2) E3 (V2, V4) E4
(V2, V3) E5 (V3, V4) T1 (E1, E2, E3) T2
(E3, E4, E5)
E1
E5
V1 (x1, y1, z1) V2 (x2, y2, z2) V3 (x3, y3,
z3) V4 (x4, y4, z4))
E3
V1
T2
V3
T1
E4
E2
V2
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Geometric Transformation of a Mesh

• Mesh transformation
• Apply transformations to every vertex. Then,
  geometry is transformed.
• However, the topology(connectivities) is not
  changed.
• In the example,
• Mesh A is displayed by drawing all of its edges
  (wireframe display).
• As all vertices are rotated, scaled up and
  translated to make B, and then
  wireframe-displayed.

B
A
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Mesh Generation - Mathematical Procedure

• A lot of ways to generate a polygon mesh
• mathematical procedure
• 3D scanner
• etc.
• Mathematical procedure
• As an example, lets make a sphere of radius 1 by
  recursive subdivision.
• Consider the following coarse triangular mesh
  (diamond) for a sphere where the center is at
  origin and all 6 points are of distance 1 from
  the center.

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Mesh Generation - Mathematical Procedure (contd)

• Subdivide each triangle into 4 sub-triangles.
• The newly generated points are not of distance 1
  from the origin.
• So, elevate the points so that they are of
  distance 1 from the origin.

y
(0,1,0)
(0,1/ 2,1/ 2)
(1/ 2,1/ 2,0)
(1,0,0)
x
z
(1/ 2,0,1/ 2)
(0,0,1)
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Mesh Generation - Mathematical Procedure (contd)

• When we recursively do the subdivision for all
  triangles, we come closer to a sphere. Below is
  an example of Level Of Detail (LOD).
• Do recursive subdivision to all polygons.
  Otherwise, we have cracks.
• We can make meshes for other objects in a similar
  way.
• The algorithm is limited to simple objects such
  as quadric surfaces, but the concept has been
  extended to subdivision surfaces.

quadrilateral mesh for a donut
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Mesh Generation - 3D Scanner

• Use a laser ranger (or CCD video camera) and a
  motorized turntable.
• Its output is a set of discrete contours, which
  is to be converted into a mesh by a skinning
  algorithm.
• The skinning algorithm produces a mesh which is
  often unnecessarily detailed for many parts of
  the surface. So, mesh simplification is usually
  needed to reduce the number of polygons.
• The 3D scanning suffers from concave objects.

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What Else?

• There exist more ways to construct a polygon
  mesh.
• Interactively construct an object, e.g. using 3D
  Studio Max, which is then automatically converted
  into a mesh.
• There are good algorithms for constructing a mesh
  from CT data.
• Use digital elevation model (DEM) for terrain.
• etc.
• Polygon meshes are by far the most popular in CG.
  However, there are many other modeling techniques
  in Computer-Aided Design (CAD) and Computer-Aided
  Geometric Design (CAGD).
• Boundary Representation, which is a
  generalization of meshes
• Constructive Solid Geometry (CSG)
• space subdivision such as Octree
• free-form surfaces
• etc.

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CSG Primitives and Boolean Operations

• CSG is built upon primitives such as box and
  cylinder.
• Primitives are defined by dimensional parameters
  with default positions and orientations.
• Three Boolean operations union, intersection,
  and subtraction

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3
4
2.5
5
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box(x4,y3,z5)
cylinder(r1.5,h4)
box(x2.5,y4,z6)
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CSG Tree

• tree

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Quadtree

• Divide the 2D region into quadrants, and color
  each quadrant.
• Gray areas are recursively subdivided.
• Subdivision continues until all quadrants are
  homogeneous (either full or white) or until a
  pre-determined cutoff depth is reached.

1
2
full - Black partially full - Gray empty -
White
4
3
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Octree

• Octree is a 3D extension of Quadtree.
• The 3D region is divided into octants.
• Quadtree and octree are pretty useful for spatial
  directory, etc.

W
B
B
W
G
B
B
W
W
B
B
W
W
B
B
W
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Free-form Surfaces

• Curved quadrilaterals (often called bi-cubic
  parametric patches) such as Bézier surfaces,
  NURBS, etc.
• The shape and curvature of the surface can be
  changed by editing its control points. So,
  powerful interactivity!!
• Hard to directly render the free-form surfaces.
  So, usually converted into a mesh and then
  rendered.