Implicit%20Surfaces

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Implicit Surfaces

• Tom Ouyang
• January 29, 2004

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Outline

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

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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.

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Example of an Implicit Surface

• 3D Sphere centered at the origin
• x2 y2 z2 r2
• x2 y2 z2 r2 0

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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
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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

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Manifold

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

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Surface Normals

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

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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

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Properties of Implicits

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

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Polygonal Representation

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

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Spatial Partitioning

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

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Spatial Partitioning

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

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Spatial Partitioning

• Adaptive Polygonization

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Determining Intersections
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Implicit Surfaces vs Polygons

• Advantages
• Smoother and more precise
• More compact
• Easier to interpolate and deform
• Disadvantages
• More difficult to display in real time

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Implicits vs Parametrics

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

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Types of Implicit Surfaces

• Mathematic
• Polynomial or Algebraic
• Non polynomial or Transcendental
• Exponential, trigonometric, etc.
• Procedural
• Black box function

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Algebraic Surfaces
Degree 6
Cubic
Degree 4
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Non-Algebraic Surfaces
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Distance Functions

• D(p) R
• Sphere Distance to a point
• Cylinder Distance to a line

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Distance Functions
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Interpolation

• Interpolate corresponding algebraic equations

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Solid Modeling

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

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Variational Implicit Surfaces

• Specify boundary locations
• boundary, interior, and exterior
• Generate surfaces that interpolate boundary
  points

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Compression
Mesh of 473,000 vertices and 871,000 facets
Implicit function of 32,000 terms
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Procedural Methods

• f as an arbitrary process or algorithm
• Fractal (Julia Set)

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Deformation

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

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Visualization

• Contours

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Visualization

• Particle Display

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Particle Display Demo
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Visualization

• Ray Tracing

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Other Coordinate Systems
Cylindrical Coordinates
Spherical Coordinates
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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