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Computer Graphics SS 2014 Geometric Models - I

Learn about geometric models, polygon representation, polygon meshes, and their properties in computer graphics. Explore Euler's formula, approximation methods, and mesh representations in this comprehensive guide.

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Computer Graphics SS 2014 Geometric Models - I

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  1. Computer Graphics SS 2014 Geometric Models - I Rüdiger Westermann Lehrstuhl für Computer Graphik und Visualisierung

  2. Object representation • In computergraphicsweusually deal withpolygonal approximationsofcontinuousobjects

  3. Object representation • What is a polygon? • It consists of points, typically in 3D-space ℝ3 • A point is represented by its x,y,z-coordinates; it is called a vertex • Points areconnected via linesegments (edges) • Line segments are specified in terms of the vertices at their endpoints • A polygon is the interior of a closed planar connected series of line segments • The edges do not cross each other and exactly two edges meet at every vertex

  4. Object representation • What is a polygon? concave convex

  5. Object representation • What is a polygon mesh? • Objects are typically considered to be hollow • They are modelled by a (closed) surface composed of polygons (faces) – the boundary representation (B-Rep) or surface mesh • i.e., objects are modelled aspolyhedra

  6. Object representation • Types of polygon meshes • Triangle meshes and quad meshes

  7. Object representation • A polygon mesh must satisfy the following criteria • The intersection of two polygons is either empty, or a common vertex, or a common edge Not allowed Allowed Facesoverlap T-vertex

  8. Objectrepresentation • We only consider 2-manifoldmeshes • A mesh where every edge must be shared by exactly 2 faces, except the border edges

  9. 1 2 7 3 6 4 5 Polygon meshes • Polygon meshes and graphs • A graph is a 2-tupel <V,E> • V: the set of nodes • E: the set of edges connecting two nodes • A planargraph:Graph whose vertices and edgescan be embedded in ℝ2such that its edges do not intersect • A plane graph: A planargraphdrawn in a plane nonplanar planar

  10. Polygon meshes • Polygon meshes and graphs • A connected planar graph is a 2-tupel <V,E> • V: the set of nodes • E: the set of edges connecting two nodes • All nodes are connected via a path along the edges • F: the faces (Gebiete) are the connected parts of the plane, which emerge when cutting the plane along the edges • Triangulation: graph where every face is a triangle. • Euler’s Formula: V − E + F = 2 • Every convex polyhedron can be turned into a connected planar graph • Ie, Euler’s Formula holds as well

  11. Polygon meshes • Polygon meshes and graphs • n-ring of a node is the set of nodes for which a path to that node of length n exist • Valence of a node is the number of adjacent edges (= number of adjacent nodes) 2-ring 1-ring

  12. Polygon meshes • Polygon meshes and graphs • Seeing polygon meshes as graphs allows transforming problems on meshes into graph theoretical problems • Shortest path along edges from one vertex to another • Does a path exist along which every vertex is visited exactly once • How many edges E can a triangle mesh of V vertices have 3F = 2E⇒E≈ 3V, F≈ 2V • What is the average valence of the vertices in a triangle mesh ⇒ < 6 (< 4 in a mesh of quadrilaterals)

  13. Polygon meshes • Genus of a polyhedron • Genus: Maximal number of closed cutting curves that do not disconnect the graph into multiple components. • Ie. the number of holes or handles Genus 0 Genus 1 Genus 2 Genus 3

  14. Polygon meshes • Euler-Poincare Formula • For a closed polygonal mesh of genus g, the relation of the number V of vertices, E of edges, and F of faces is givenV - E + F = 2(1-g) • The term χ= 2(1-g) is called the Euler characteristic χ= 2 χ= 0 χ= -2 χ= -4

  15. Polygon meshes • Euler-Poincare FormulaV = 3890E = 11664F = 7776Genus 0, χ= 2

  16. Polygon meshes • Genus 1, χ= 0knot

  17. Example: Approximationsofa curve Different approximationsof a sphere Polygon meshes • A polygonal approximation is an approximation of a continuous surface by a polygon mesh

  18. Polygon meshes • A polygonal approximation typically consists of a discrete set of points of the continuous surface, which are connected via edges • Constructing the connectivity (edges) for a given set of points is called triangulation

  19. Polygon meshes • Approximation propertiesof polygonal approximations • Forpiecewise linear approximationstheerrorbetweentheapproximationandthesurfaceis Error goes down asweincreasethenumberofinterpolationpoints; errordecreasesasthesizeofthesquareoftheinterval

  20. Polygon meshes • Polygonal approximation are typically adaptive, i.e., more/smaller polygons are used in regions of high curvature (“high ”)

  21. Polygon meshes • Adaptive polygonal/polyhedral approximations

  22. Polygon meshrepresentation • Triangle mesh representation • Geometry:where are the vertices positioned • (Polygon-)topology: how are the vertices connected • (Mesh-)topology: how are the polygons connected • Explicitrepresentation (Face Set; STL files):store three vertices for each of the n triangles n*3*3 = 9n floats • High redundancy • 36 B/f = 72 B/v • no connectivity!

  23. Polygon meshrepresentation • Triangle mesh representation • Shared vertex („indexed face set“) representation (OBJ, OFF files) • Array of coordinates of all vertices (3n floats) • Array of triangles with indices into the vertex list (3n integers) • 12 B/v + 12 B/f = 36 B/v • Use, eg. Meshlab, to view .obj files http://meshlab.sourceforge.net/

  24. Polygon meshrepresentation • Indexed face set representation • Less redundancy than explicit representation • Separationof geometry and (polygon-)topology • More efficient computations on surfaces • Mesh operators still difficult to realize • Think about how to implement the operation to find all 1-ring neighbors

  25. Polygon meshrepresentation • Extendedshared vertex representation (Face-Based Connectivity): • Array of vertices, each with a reference to one adjacent triangle: 3n floats + 1n integers • Array of triangles, each with references to 3 vertices and face neighbours (adjacent triangles) (at the border: -1): 6n integers • 16 B/v + 24 B/f = 64 B/v

  26. VEND Polygon meshrepresentation ENCW EPCCW • Edge-Based Connectivity: • Winged Edge data structure (Baumgart94) • Arrays ofEdges(primary key): • start (VSTART) and end (VEND) vertices • two adjacent faces (FCW and FCCW) • preceding and next edges in one face (EPCW and ENCW) • preceding and next edges in other face (EPCCW and ENCCW) Vertices (key): position, one adjacent edgeFaces (key): one adjacent edge • 120 Byte/v FCCW FCW EPCW ENCCW VSTART

  27. Polygon meshrepresentation • On board example

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