Saab. Kyle McDonald. Polygon Meshes

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1 Saab Kyle McDonald Polygon Meshes Siddhartha Chaudhuri

2 What is a polygon mesh? Like a point cloud, it is a discrete sampling of a surface... but, it adds linear (flat) approximations of local regions of the actual underlying surface

3 What is a polygon mesh? Like a point cloud, it is a discrete sampling of a surface... but, it adds linear (flat) approximations of local regions of the actual underlying surface Points sampled from true surface

4 What is a polygon mesh? Like a point cloud, it is a discrete sampling of a surface... but, it adds linear (flat) approximations of local regions of the actual underlying surface Planar polygons: linear approximations to true surface

5 What is a polygon mesh? Like a point cloud, it is a discrete sampling of a surface... but, it adds linear (flat) approximations of local regions of the actual underlying surface The original samples become vertices of the polygons

6 What is a polygon mesh? Like a point cloud, it is based on a discrete sampling of a surface... but, it adds linear (flat) approximations of local regions of the actual underlying surface Like point clouds, meshes can have different resolutions gamedev.stackexchange.com

7 What is a polygon mesh? Like a point cloud, it is based on a discrete sampling of a surface... but, it adds linear (flat) approximations of local regions of the actual underlying surface Like point clouds, meshes can have different resolutions at different places ( adaptive meshing ) Leif Kobbelt

8 Elements of a mesh

9 Elements of a mesh Edges Vertices Faces

10 Elements of a mesh Non-Boundary Edge Boundary Edge NonBoundary Vertex Boundary Vertex Quadrilateral (Quad) Face Triangular Face (should be planar!)

11 A mesh is a graph This cannot be stressed strongly enough!

12 A mesh is an undirected graph

13 The vertex positions capture the geometry of the surface (1.0, 2.3, -4.5) (4.3, 1.9, 2.5) (0.2, -3.9, 1.2) (1.1, 0.1, 2.2) (3.3, -2.2, 1.0) (-1.3, 4.1, 3.1) (1.5, 3.1, 1.5) (-0.1, -1.1, -1.4)

14 The mesh connectivity captures the topology of the surface

by triangulation Many mesh formats allow non-planar faces, but most algorithms assume

15 Mesh Geometry: Planes and Normals Each polygon is (assumed to be) planar Triangular faces are always planar Quads and higher degree faces need not be Ambiguity revealed by triangulation Many mesh formats allow non-planar faces, but most algorithms assume planar faces. Caveat emptor. Same 4 noncoplanar vertices, different geometry! Always planar

16 Mesh Geometry: Planes and Normals The plane of each polygon has an associated normal vector

17 Mesh Geometry: Planes and Normals The plane of each polygon has an associated normal vector v2 Q: The computed normal may point into or out of the object. Which one to pick? e2 = v2 v0 v0 A: Either (typically outwards), but be consistent across the shape! Using our formula here, the normal is outwards if the vertices wind counter-clockwise around the face when seen from outside the shape. e1 e1 e2 n= ^ e1 e2 = v1 v0 v1

18 Mesh Geometry: Planes and Normals The plane of each polygon has an associated plane ^ equation: n (p v 0)=0 v2 n^ v0 v1

19 Mesh Geometry: Planes and Normals We can also associate vertices with normals Sometimes they come with the mesh (e.g. if they were estimated when the mesh was constructed from a point cloud) Sometimes we have to estimate them

20 Estimating vertex normals Simplest: Add up the normals of adjacent faces and unitize n^ 3 n^ 2 n^ 0 n^ 1

21 Estimating vertex normals Simplest: Add up the normals of adjacent faces and unitize Simple and usually a bit better: Add up the normals of adjacent faces, weighted by face areas Without area-weighting With area-weighting bytehazard.com/articles/vertnorm.html

22 Estimating vertex normals Simplest: Add up the normals of adjacent faces and unitize Simple and usually a bit better: Add up the normals of adjacent faces, weighted by face areas Complex: Detect sharp edges Without area-weighting With area-weighting bytehazard.com/articles/vertnorm.html

23 Mesh Topology Topology (loosely): The structure of a shape ignoring any measurements of distance, angle etc i.e. the properties invariant to bending, twisting, folding, stretching... (but not tearing) E.g. Genus: The number of holes in a shape Wikipedia

24 Mesh Topology Manifold: A topological space that is locally Euclidean (neighborhood has the topology of the unit ball) Some manifold shapes Not manifold Manifold structure of a surface is approximated by its mesh connectivity

Tutorial 3 Comparing Biological Shapes Patrice Koehl and Joel Hass

Tutorial 3 Comparing Biological Shapes Patrice Koehl and Joel Hass University of California, Davis, USA http://www.cs.ucdavis.edu/~koehl/ims2017/ What is a shape? A shape is a 2-manifold with a Riemannian