Geometric Modeling

• Volumetrico Collection device obtains regular grid of measurement values

• Examples: CT, MRI, PET, Ultrasound

o Values are interpreted/visualized with color, transparency, texture• Example: Marching Cubes

Geometric Modeling

Geometric Modeling• Procedural

o Geometry is calculated by some equation or algorithm and rendered. The algorithms can be as creative as one can imagine, but we can divide them into two types: fractal (natural) and non-fractal (man-made)

o Examples:

Geometric Modeling• Implicit Surface

o Geometry is assumed at all points that satisfy equation f(x, y, z) = 0o Examples: A sphere centered at (a, b, c) of radius r is the set of all

points satisfying the equation (x-a)2 + (y-b)2 + (z-c)2 = r2

o Multiple implicit surfaces can be grouped together to create more complicated shapes.

Geometric Modeling• Parametric Patch

o Geometry is calculated using parametric equations and varying a parametric parameter over some range.

o Wavefront uses the term “free-form geometry.”

o Examples:

• Video describing how parametric equations are used with parametric patches(see 1:17-5:20)

• The equation p(u, v) = [ u v k1u2+k2v2 ] describes the parabola seen here as u and v vary over [0, 1]

Geometric Modeling• Modeling with Higher-Order Surfaces (Quadrics

and Cubics)o Quadrics: Supports common shapes such as circles, ellipses, and

parabolaso Examples:

o Cubics: General purpose patches allows for favorable continuity conditions

o Examples:

• Other Modeling Methodso Constructive Solid Geometry (CSG) builds intricate shapes from simple

ones using set operations.• This type of modeling is common to CAD (such as with the SolidWorks

modeler), because interiors may be defined easily• Example:

o Terrain Generators are a procedural modeling tool, but often customized for generating “terrain-like” models (i.e., usually controlled by some type of height map)

Geometric Modeling

Geometric Modeling• Modeling with Planer Polygons

o This is by far the most common case in (non-engineering) computer graphics today• Engineering models prefer CSG or bi-cubic patches for analysis and

manufacture• Commodity display hardware is optimized for planer polygons• Further optimized for triangles (and sometime quads).

o Types of Polygonal Models• Extrusions• Replications• Symmetric

o Bi-symmetryo Surface of Revolution

• Compositions of Simpler Surfaces

3D Modeling Terminology

• Coordinate System:o XYZ (RGB)o Right Handedo Handedness effects rotation

• Pos rotation follows finger curl• Reference Frames

o Modelo Worldo Eyeo View

ModelWorld

Eye

View

3D Modeling Terminology

• Vertex• Edge• Normal

o Computation Rule• Face

o Front Faceo Back Faceo CCW Traversal Rule

• Mesho Manifold

• Subdivisiono Triangulation

• Simplification/Decimation

• Transformationso Translateo Rotateo Scaleo Shearo Mirror

Rules for Oriented Faces

• CCW Rule: An in-order traversal of the vertices of a face will be counter-clockwise for a front-facing polygon.

• Normal Computation Rule: The normal of a face can be calculated by the cross product of any three, convex vertices of the face:o E.g., (v5 – v3) X (v1 – v3)o The normal is typically normalized (made unit length) by dividing by its

length, denoted || (v5 – v3) X (v1 – v3)||

v1

v2

v4

v3

v5v6

v7

Wavefront obj File Example

# Blender v2.74 (sub 0) OBJ File: 'cube.blend'# www.blender.orgmtllib cube.mtlo Cubev 1.000000 -1.000000 -1.000000v 1.000000 -1.000000 1.000000v -1.000000 -1.000000 1.000000v -1.000000 -1.000000 -1.000000v 1.000000 1.000000 -0.999999v 0.999999 1.000000 1.000001v -1.000000 1.000000 1.000000v -1.000000 1.000000 -1.000000vn 0.000000 -1.000000 0.000000vn 0.000000 1.000000 0.000000vn 1.000000 0.000000 0.000000vn -0.000000 -0.000000 1.000000vn -1.000000 -0.000000 -0.000000vn 0.000000 0.000000 -1.000000usemtl Materials offf 1//1 2//1 3//1 4//1f 5//2 8//2 7//2 6//2f 1//3 5//3 6//3 2//3f 2//4 6//4 7//4 3//4f 3//5 7//5 8//5 4//5f 5//6 1//6 4//6 8//6