Implicit Surfaces

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 < 0 • Outside Region: f > 0 • Or vice versa depending on the function f = 0 f > 0 f < 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. f < 0 g < 0 f < 0 g < 0 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 Cubic Degree 4 Degree 6

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