CS488 - Introduction to Computer Graphics - Lecture 12

Polygons

Ordered sets of vertices, joined sequentially (and cyclically) by straight lines.
Simple polygons
- convex or concave
- no line crossings
- no holes
Adding extra vertices removes the problems
- at the cost of adding degeneracies
Polygons cover the surface of an object
- each polygon has a normal vector,
- usually pointing out
- polygonal "skin" is often called a mesh

Clipping

Clipping a polygon reduces to clipping edges against

lines in 2D
planes in 3D

The clipping algorithm takes an ordered set of vertices [vi] and produces an ordered set of vertices [wi]. It relies on the following

If one edge leaves the clipping region, a later edge must enter it.

The algorithm is

for each edge of clipping region
   for each edge of polygon // must be in sequence
      clip edge against region
      switch (result of clip)
      case "all inside"
         output leading vertex of edge
      case "all outside"
         do nothing
      case "cross edge leaving region"
         output crossing point
      case "cross entering region"
         output crossing point
         output leading vertex of edge

Exercise. Hand execute this algorithm on several cases to make sure that you understand exactly how it works.

Projecting a Polygon

Lines project to lines so projecting the vertices projects a polygon in 3D to a polygon on the view plane.

Triangles

Sooner or later almost all polygons are converted to triangles

there is not a unique way of doing so
try to avoid long skinny triangles

Scan Conversion

Scan converting a polygon (in 2D)

sort vertices in direction perpendicular to the scan lines
for each scan line
   if scan line contains the next vertex
      update and sort edge list
   inside = false
   for each pixel
      if on next edge
         inside = !inside
      if (inside)
         paint pixel

Exercise. Expand "update and sort edge list" to make this a working algorithm.

Exercise. Hand execute your expanded algorithm on a triangle to make sure that you understand how it works.