9. Meshes

Brock University
COSC 3P98 Computer Graphics
Instructor: Brian J. Ross

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
3D scanners: accurate digital modeling of real objects

Two interrelated aspects:

1. Tools: types of modeling tools, techniques for use, and their GUIs

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

Face list for the prism
Face number	Vertices
1	1, 3, 2
2	6, 2, 3, 4
3	1, 5, 4, 3
4	4, 5, 6
5	1, 2, 6, 5

: Note that we can split the quadrilateral (4-vertex polygons) into triangles. This simplifies the data structure definition, but then may result in duplicate vertex definitions.
More examples:

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);
   glVertex3fv(A);
   glVertex3fv(B);
   glVertex3fv(C);
   glVertex3fv(D);
   glVertex3fv(E);
...
glEnd();

also GL_TRIANGLE_FAN, GL_QUAD_STRIP
These are used to optimize the transferral of mesh vertices.
Results in fewer calls to OpenGL, as well as faster rendering (lighting).

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

Back to COSC 3P98 index

COSC 3P98 Computer Graphics
Brock University
Dept of Computer Science
Copyright © 2020 Brian J. Ross (Except noted figures).
http://www.cosc.brocku.ca/Offerings/3P98/course/lectures/modeling/mesh.html
Last updated: February 1, 2023