CS488 - Introduction to Computer Graphics - Lecture 13

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
• Convention
  • from outside vertices go around the polygon clockwise
• Polygons cover the surface of an object
  • each polygon has a normal vector,
  • usually pointing out
  • polygonal "skin" is often called a mesh

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 triangle (in 2D)

1. Start at the scan line of the uppermost vertex
2. Until you get to the middle vertex
   • Intersect each scan line with two triangle edges
   • Paint pixels between the intersections
3. Until you get to the lowest vertex
   • Intersect each scan line with two triangle edges
   • Paint pixels between the intersections

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.

Exercise. Hand execute your expanded algorithm on a non-convex polygon.

Hidden Surface Removal

Every polygon, in 3D, has two sides

• the normal vector differentiates the sides
• different sides may have different properties

It is very convenient to assume that surfaces are closed

• Why?
• Naturally open surfaces, e.g., a piece of paper, can be closed by adding polygons

Then we can say that every polygon has only a single side with visual properties

• The normal vector points away from this surface

Backface Culling

Do not render any polygon with n.E < 0.

• On average this cuts the rendering effort in half.
• Remember that E is the ray from the polygon to the eye, not the view direction.

Does not handle occlusion, so we need to finish off with one of the following algorithms.

Painter's Algorithm

Sort polygons back to front, then render backmost first.

• device-independent
• slow if there's lots of occlusion
• sorting may not be possible, which makes things hard.
  • not easy to implement
• potentially O(n^2), but rarely is that bad. n: number of polygons.

Warnock's Algorithm

Divide and conquer

1. Start with whole window
2. Divide into four subwindows
3. For each window
   • If window has a single polygon
     • paint it clipped to the window
   • else if window is a single pixel
     • paint closest one pixel polygon
   • else
     • go to 1 with subwindow (recurse)
4. exit

Warnock's algorithm is

• easy to implement, device independent
• slow, often O(p.n). p: number of pixels.
• often used in quasi 2D applications where there is little occlusion.

Depth-Buffer Algorithm (also called zed-buffer)

Use a second frame buffer with depth values stored in it

1. Initalize depth-buffer to far plane distance
2. If a pixel has z less than the current depth buffer value
   • Write it, and
   • Update the depth buffer

The depth buffer algorithm is

1. easy to implement
2. fast, O(n)
3. no longer memory-intensive, though it used to be

Almost all graphics cards have hardware depth buffers.

Using the depth buffer may be hard the first time.

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