9. Meshes

Models

• Modeling: the mathematical definition of graphical objects for rendering
• Modeling objects is an art. Many different techniques have evolved for creating more realistic objects.
• CAD systems have been a major source of tools
• CAD systems concerned with precision, and typically model rigid geometries
• new interest in organic shapes employ new tools: fractals, metaclay
• Two interrelated aspects:
• 1. Tools: types of modeling tools, techniques for use, and interfaces to them
• success measured by their utility, practicality
• examples: polygon editing, lathes, extrusions, 3D scanners, ...
• 2. Mathematical modeling: mathematical means for defining objects
• success measured by the types of objects definable and editable, and their efficiency
• examples: polygonal models, 2D spline curves, 3D spline surfaces, fractals

Polygon models

• basic modeling approach
• given a solid object, we can define its geometry in terms of polygons
• ie. points in 3D space, with edges connecting them
• if we only draw the edges, we get a wireframe
• if we draw solid polygon faces, we get solid surfaces
• such objects have vertices, edges, and faces
• planar simple convex polygons are used to draw surfaces
• an advanced topic is the use of non-planar surfaces ie. surfaces with curves and bumps

Polygon meshes

• polygon mesh: a set of polygons with shared edges
• connected mesh: every faces shares at least one edge with some other, face
• can move from one face to any other by traversing edges
• simple polygon: no edges intersect except at endpoints
• planar polygon: all vertices lie on a plane
• triangles are only polygon guaranteed to be planar
• polyhedron: connected mesh of simple polygons such that every edge shared by exactly 2 faces --> closed
• convex polyhedron: a volume is convex iff for any 2 vertices, the line segment connecting them is within the volume

Examples of polygon meshes

• More examples: Example 1. Example 2.

Building meshes

• meshes are a fundamental way of constructing 3D volumes in graphics
• ways to build them:
• 1. Manually
• define points table, and then define the faces composing mesh
• polygons are often squares or triangles, but can be more complex
• time consuming, but a very common method esp. for games
• OpenGL: supports triangular meshes and quadrilateral strips

glBegin(GL_TRIANGLE_STRIP);
 ...vertices...
glEnd();

also: GL_TRIANGLE_FAN, GL_QUAD_STRIP

Building meshes

• 2. Automatically
• i) Patches: use computed grids of triangles/squares
• especially useful when surface can be defined mathematically

• ii) point sampling + triangulation: using lasers or other devices, get different (x,y,z) coordinates of random points on object
• then you apply a 2d or 3d hull/triangulation algorithm, which creates triangular mesh

Data structures for meshes

• Your application model can use a variety of data structures and their implementations (arrays, lists, trees, ...)
• the crudest way is to define a sequential list of vertice coordinates
• eg. square = { (0,0,0), (0,1,0), (1,1,0), (1,0,0)}
• disadvantages:
• coordinate values are duplicated, which wastes space
• during processing, identical coordinates may become different due to roundoff error
• difficult to match different polygon's shared vertices & edges
• better: table of points, then table of edges, faces, or both
• depending on application, you can have data structure tell you as much information as you need eg. edge list can indicate the polygons that use it (at least 1, max of 2)
• Composite objects important: defining hierarchical models
• can use different tools for subcomponents
• can copy/duplicate components
• animation: components can be assigned separate movement characteristics

Meshes

---

COSC 3P98 Computer Graphics
Brock University
Dept of Computer Science
Copyright © 2006 Brian J. Ross (Except noted figures).
http://www.cosc.brocku.ca/Offerings/3P98/course/lectures/modeling/mesh.html
Last updated: November 13, 2006