Last TimeTodaySmooth versus GeneralSubdivision SchemesTessellating a SphereSubdivision MethodThe First StageSphere Subdivision AdvantagesFractal SurfacesFractal Terrain (1)Fractal Terrain ExampleFractal Terrain DetailsFractal Terrain AlgorithmSubdivision OperationsData Structure IssuesGeneral Subdivision SchemesModified Butterfly SchemeSlide 18Data StructuresSlide 20Slide 21BoundariesModified Butterfly ExampleImplicit FunctionsBlobs and MetaballsExample with BlobsBlob MathRendering Implicit SurfacesImplicit Surfaces Summary11/30/04 © University of Wisconsin, CS559 Fall 2004Last Time•More modeling:–Hierarchical modeling–Instancing and Parametric Instancing–Constructive Solid Geometry–Sweep Objects–Octrees11/30/04 © University of Wisconsin, CS559 Fall 2004Today•Subdivision schemes•Implicit surfaces•Homework 6 due in class•Project 3 ongoing11/30/04 © University of Wisconsin, CS559 Fall 2004Smooth versus General•Polygon meshes are very general, but hard to model with–In a production context (film, game), creating a dense, accurate mesh requires lots of work–Biggest problem is smoothness•We desire a way to “smooth out” a polygonal mesh–We can model at a coarse level, and automatically fill in the smooth parts•Subdivision surfaces are part of the answer11/30/04 © University of Wisconsin, CS559 Fall 2004Subdivision Schemes•Basic idea: Start with something coarse, and refine it into smaller pieces, smoothing along the way–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–Subdivision for fractal surfaces–General subdivision surfaces11/30/04 © University of Wisconsin, CS559 Fall 2004Tessellating a Sphere•Spheres are frequently 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 degree22 ,20sin ,cossin ,coscos zyx11/30/04 © University of Wisconsin, CS559 Fall 2004Subdivision 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 verticesOctahedronIsosahedron11/30/04 © University of Wisconsin, CS559 Fall 2004The First StageEach face gets split into 4:Each new vertex is degree 6, original vertices are degree 411/30/04 © University of Wisconsin, CS559 Fall 2004Sphere 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 regular (or 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 areas11/30/04 © University of Wisconsin, CS559 Fall 2004Fractal 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 schemes11/30/04 © University of Wisconsin, CS559 Fall 2004Fractal 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 sphere11/30/04 © University of Wisconsin, CS559 Fall 2004Fractal Terrain ExampleA mountainside11/30/04 © University of Wisconsin, CS559 Fall 2004Fractal 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•Reducing the offset of new points according to the subdivision level is essential–Define a scale, s, and a ratio, k, and at each level: si+1=ksi•Colors are frequently chosen based on “altitude”11/30/04 © University of Wisconsin, CS559 Fall 2004Fractal Terrain Algorithm•The hard part is keeping track of all the indices and other data•Same algorithm works for subdividing sphereSplit_One_Level(struct Mesh terrain)Copy old verticesfor all edgesCreate and store new vertexCreate and store new edgesfor all facesCreate new edges interior to faceCreate new facesReplace old vertices, edges and faces11/30/04 © University of Wisconsin, CS559 Fall 2004Subdivision 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 locations11/30/04 © University of Wisconsin, CS559 Fall 2004Data 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?11/30/04 © University of Wisconsin, CS559 Fall 2004General Subdivision Schemes•Subdivision schemes can also
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