Last Time

1. Last Time • B-splines • Recall a property of B-spline surfaces – the control point grid must be rectangular • Project 3 was made available

2. Today • Subdivision schemes • Homework 6 available

3. B-splines as Approximation • B-splines were developed as approximation functions • Given a set of points – the control points – the B-spline approximates them with a smooth curve or surface • Ideally, we would like to be able to provide a polygonal mesh, and then smooth it out with a B-spline surface • The 3D equivalent of smoothing a 2D poly-line with a B-spline • But we can’t do this in general, because B-spline control meshes must be rectangular • Can’t even to simple cases like a sphere

4. Subdivision Schemes • Basic idea: Start with something coarse, and refine it into smaller pieces, smoothing along the way • We have seen how subdivision may be used to render parametric curves and Bezier surfaces • We will see how it can be used for modeling specific objects, and as a modeling scheme in itself • In this lecture: • Subdivision for tessellating a sphere, and implementation details • Subdivision for fractal surfaces • Subdivision for B-spline patches • General subdivision surfaces

5. Tessellating a Sphere • Spheres are best parameterized in polar coordinates: • Note the singularity at the poles • Tessellation: The process of approximating a surface with a polygon mesh • One option for tessellating a sphere: • Step around and up the sphere in constant steps of  and  • Problem: Polygons are of wildly different sizes, and some vertices have very high degree

6. Subdivision Method • Begin with a course approximation to the sphere, that uses only triangles • Two good candidates are platonic solids with triangular faces: Octahedron, Isosahedron • They have uniformly sized faces and uniform vertex degree • Repeat the following process: • Insert a new vertex in the middle of each edge • Push the vertices out to the surface of the sphere • Break each triangular face into 4 triangles using the new vertices Octahedron Isosahedron

7. The First Stage Each new vertex is degree 6, original vertices are degree 4 Each face gets split into 4:

8. Sphere Subdivision Advantages • All the triangles at any given level are the same size • Relies on the initial mesh having equal sized faces, and properties of the sphere • The new vertices all have the same degree • Mesh is uniform in newly generated areas • This is a property we will see later in subdivision surfaces • Makes it easier to analyze what happens to the surface • The location and degree of existing vertices does not change • The only extraordinary points lie on the initial mesh • Extraordinary points are those with degree different to the uniform areas

9. Fractal Surfaces • Fractals are objects that show self similarity • The word is overloaded – it can also mean other things • Landscapes and coastlines are considered fractal in nature • Mountains have hills on them that have rocks on them and so on • Continents have gulfs that have harbors that have bays and so on • Subdivision is the natural way of building fractal surfaces • Start with coarse features, Subdivide to finer features • Different types of fractals come from different subdivision schemes and different parameters to those schemes

10. Fractal Terrain (1) • Start with a coarse mesh • Vertices on this mesh won’t move, so they can be used to set mountain peaks and valleys • Also defines the boundary • Mesh must not have dangling edges or vertices • Every edge and every vertex must be part of a face • Also define an “up” direction • Then repeatedly: • Add new vertices at the midpoint of each edge, and randomly push them up or down • Split each face into four, as for the sphere

11. Fractal Terrain Example A mountainside

12. Fractal Terrain Details • There are options for choosing where to move the new vertices • Uniform random offset • Normally distributed offset – small motions more likely • Procedural rule – eg Perlin noise • Scaling the offset of new points according to the subdivision level is essential • For the subdivision to converge to a smooth surface, the offset must be reduced for each level • Colors are frequently chosen based on “altitude”

13. Fractal Terrains http://members.aol.com/maksoy/vistfrac/sunset.htm

14. Terrain, clouds generated using procedural textures and Perlin noise http://www.planetside.co.uk/ -- tool is called Terragen

15. Terrain, clouds generated using procedural textures and Perlin noise http://www.planetside.co.uk/ -- tool is called Terragen

16. Fractal Terrain Algorithm • The hard part is keeping track of all the indices and other data • Same algorithm works for subdividing sphere Split_One_Level(struct Mesh terrain) Copy old vertices for all edges Create and store new vertex Create and store new edges for all faces Create new edges interior to face Create new faces Replace old vertices, edges and faces

17. Subdivision Operations • Split an edge, create a new vertex and two new edges • Each edge must be split exactly once • Need to know endpoints of edge to create new vertex • Split a face, creating new edges and new faces based on the old edges and the old and new vertices • Require knowledge of which new edges to use • Require knowledge of new vertex locations

18. Data Structure Issues • We must represent a polygon mesh so that the subdivision operations are easy to perform • Questions influencing the data structures: • What information about faces, edges and vertices must we have, and how do we get at it? • Should we store edges explicitly? • Should faces know about their edges?

19. Edge Data (iteration 1) • The mesh must store all the edges, because we need to loop over them to split them • The edge must store its endpoints, because we need them to split the edge • A flag to turn off vertex perturbation is useful for boundary edges struct Edge { int v_start; // Index of start vertex int v_end; // Index of end vertex bool no_perturb; }

20. Face Data • Faces are triangles • To split, a face must know its vertices or edges. Which? • To split, the face needs to know where the new vertices and new edges are stored • They were created when each edge was split • Where do we store them so that the face can find them?

21. Face Data (iteration 1) • When an edge is split, it makes sense to store the new vertex and new edges as part of the existing edge • Faces can then store their edges, and access the information for splitting from the edges • Each edge is stored once, so neighboring faces see the same information • Edges may point “forwards” or “backwards” around a face, and we must store this struct Face { int edges[3]; bool forward[3]; }

22. Face Data • When an edge is split, it makes sense to store the new vertex and new edges as part of the existing edge • Faces can then store their edges, and access the information for splitting from the edges • Each edge is stored once, so neighboring faces see the same information • Edges may point “forwards” or “backwards” around a face, and we must store this struct Face { int edges[3]; bool forward[3]; float n[3]; }

23. Edge Data (iteration 2) • Need to know endpoints, new vertex when split, and new edges when split • Store as indexes into a list struct Edge { int v_start; // Index of start vertex int v_end; // Index of end vertex int v_new; // Index of new vertex int e_start; // Index of one sub-edge int e_end; // Index of other sub-edge bool boundary; }

24. Rendering • To render, we must be able to find the vertices around a face • We have these, as part of the edges around the face • We also require vertex normals for smooth shading • These can be computed by first computing face normals, then averaging to find vertex normals • And we might require texture coordinates • When an edge is split to create a new vertex, average the endpoint texture coordinates to get the coordinates for the new vertex

25. Computing Normals • Can be done in two passes through the vertices, and one pass through the face • Loop over vertices: Set the vertex normal to the zero vector and set a counter to zero • Loop over faces: Compute the face normal, and add it to each vertex normal around the face, and increment the counter at each vertex • Face normal comes from taking cross product of two edge directions and normalizing • Loop over vertices: Divide the normal by the counter, and normalize

26. Vertex Data struct Vertex { float x[3]; float tex[2]; float norm[3]; int num_faces; }

27. Mesh Data Structure • Mesh stores: • Vertices, edges and faces • The up direction for offsetting vertices struct Mesh { int num_vertices; struct Vertex *vertices; int num_edges; struct Edge *edges; int num_faces; struct Face *faces; float up[3]; }

28. General Subdivision Schemes • Subdivision schemes can also be used where there is no “target” surface • They aim to replace a polygonal mesh with a smooth surface that approximates the coarse mesh • There are many schemes: • Butterfly scheme (for triangular meshes) • Catmull-Clark subdivision (for mostly rectangular meshes, converges to B-splines in uniform regions) • Loop’s scheme (for triangular meshes) • Modified butterfly scheme (for triangular meshes) • Many more…

29. Butterfly Scheme • Subdivides the same way we have been discussing • Each edge is split • Each face is split into four • Rules are defined for computing the splitting vertex of each edge • Basic rule for a uniform region • Splitting an edge with endpoints that have degree 6 • As before, all new interior vertices will have degree 6 • Take a weighted sum of the neighboring vertices • Weights define rules • http://www.gamasutra.com/features/20000411/sharp_01.htm

30. Butterfly Scheme (1) c b c a a d d c b c • Multiply each vertex by its weight and sum them up • w is a control parameter – determines how closely the shape conforms to the original mesh

31. Modified Butterfly Scheme • The butterfly scheme must be modified to deal with edges with an endpoint of degree  6 • In that case, compute new vertex based only the neighbors of the extraordinary vertex • If an edge has two extraordinary endpoints, average the results from each endpoint to get the new endpoint • The modified butterfly scheme is provably continuous about extraordinary vertices • Proof formulates subdivision as a matrix operator and does eigen-analysis of subdivision matrix

32. Modified Butterfly Scheme e3 e2 e1 e0 v eN-1 eN-3 eN-2

33. Modified Butterfly Example • Notes: • The mesh is uniform everywhere except the original vertices • It interpolates the original vertices • It has smoothed out the underlying mesh