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Implicit%20Surfaces

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What are Implicit Surfaces? 2D Geometric shapes that exist in 3D space ... A manifold surface is 'watertight' and contains no holes or dangling edges. Manifold ... – PowerPoint PPT presentation

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Title: Implicit%20Surfaces


1
Implicit Surfaces
  • Tom Ouyang
  • January 29, 2004

2
Outline
  • Properties of Implicit Surfaces
  • Polygonization
  • Ways of generating implicit surfaces
  • Applications

3
What are Implicit Surfaces?
  • 2D Geometric shapes that exist in 3D space
  • Surface representation through a function f(x, y,
    z) 0
  • Most methods of analysis assume f is continuous
    and not everywhere 0.

4
Example of an Implicit Surface
  • 3D Sphere centered at the origin
  • x2 y2 z2 r2
  • x2 y2 z2 r2 0

5
Point Classification
  • Inside Region f lt 0
  • Outside Region f gt 0
  • Or vice versa depending on the function

f 0
f gt 0
f lt 0
6
Manifold
  • A 2D Manifold separates space into a natural
    inner and natural outer region
  • A manifold surface is watertight and contains
    no holes or dangling edges

7
Manifold
  • It is difficult to determine enclosed region in
    non-manifold surfaces

8
Surface Normals
  • Usually gradient of the function
  • ??f(x,y,z) (df/dx, df/dy, df/dz)
  • Points at increasing f

9
Properties of Implicits
  • Easy to check if a point is inside the implicit
    surface
  • Evaluate f at that point
  • Fairly easy to check ray intersection
  • Substitute ray equation into f for simple
    functions
  • Binary search

10
Properties of Implicits
  • Simple set operations
  • Union min(f, g)
  • Intersection max(f, g)
  • Difference max(f,-g)
  • Complement -f

11
Polygonal Representation
  • Partition space into convex cells
  • Find cells that intersect the surfacetraverse
    cells
  • Compute surface vertices

12
Spatial Partitioning
  • Exhaustive Enumeration
  • Divide space into regular lattice of cells
  • Traverse cells polygonized

13
Spatial Partitioning
  • Subdivision
  • Start with root cell and subdivide
  • Continue subdividing traverse cells

14
Spatial Partitioning
  • Adaptive Polygonization

15
Determining Intersections
16
Implicit Surfaces vs Polygons
  • Advantages
  • Smoother and more precise
  • More compact
  • Easier to interpolate and deform
  • Disadvantages
  • More difficult to display in real time

17
Implicits vs Parametrics
  • Advantages
  • Implicits are easier to blend and morph
  • Interior/Exterior description
  • Ray-trace
  • Disadvantages
  • Rendering
  • Control

18
Types of Implicit Surfaces
  • Mathematic
  • Polynomial or Algebraic
  • Non polynomial or Transcendental
  • Exponential, trigonometric, etc.
  • Procedural
  • Black box function

19
Algebraic Surfaces
Degree 6
Cubic
Degree 4
20
Non-Algebraic Surfaces
21
Distance Functions
  • D(p) R
  • Sphere Distance to a point
  • Cylinder Distance to a line

22
Distance Functions
23
Interpolation
  • Interpolate corresponding algebraic equations

24
Solid Modeling
  • Solid model consists of a surface and its
    interior
  • Point classification
  • Constructive solid geometry (CSG)

25
Variational Implicit Surfaces
  • Specify boundary locations
  • boundary, interior, and exterior
  • Generate surfaces that interpolate boundary
    points

26
Compression
Mesh of 473,000 vertices and 871,000 facets
Implicit function of 32,000 terms
27
Procedural Methods
  • f as an arbitrary process or algorithm
  • Fractal (Julia Set)

28
Deformation
  • p D(p)
  • D maps each point in 3-space to some new location
  • Twist, bend, taper, and offset

29
Visualization
  • Contours

30
Visualization
  • Particle Display

31
Particle Display Demo
32
Visualization
  • Ray Tracing

33
Other Coordinate Systems
Cylindrical Coordinates
Spherical Coordinates
34
Summary
  • Surface defined implicitly by f(p) 0
  • Easy to test if point is on surface, inside, or
    outside
  • Easy to handle blending, interpolation, and
    deformation
  • Difficult to render
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