Lecture05 Rasterization 1 (Triangles)

Slide

Perspective Projection

• What’s near plane’s l,r,b,t then?
  • If explicitly specified, good
  • Sometimes people prefer: vertical field-of-view (fovY) and aspect ratio (assume symmetry i.e. l=-r, b = -t)
• How to convert from fovY and aspect to l, r, b, t?
  • Trivial
    $$\tan \frac{fovY}{2}=\frac{t}{|n|}$$
    $$aspect=\frac{r}{t}$$

Canonical Cube to Screen

• What is a screen?
  • An array of pixels
  • Size of the array: resolution
  • A typical kind of raster display
• Raster = screen in German
  • Rasterize = drawing onto screen
• Pixel (FYI, short of ‘picture element’)
  • For now: A pixel is a little square with uniform color
  • Color is a mixture of (red, green, blue)
• Defining the screen space
  • Slightly different from the ‘tiger book’
  • Irrelevant to z
  • Transform in xy plane: $[-1,1{]}^2$ to $[0,\text{width}]\times [0,\text{height}]$
  • Viewport transform matrix:
    $$M_{viewport}=\left(\begin{array}{cccc}\frac{width}{2}& 0& 0& \frac{width}{2}\\ 0& \frac{height}{2}& 0& \frac{height}{2}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

Rasterization: Drawing to Raster Displays

Triangles - Fundamental Shape Primitives

Why triangles?

• Most basic polygon
  • Break up other polygons
• Unique properties
  • Guaranteed to be planar
  • Well-defined interior
  • Well-defined method for interpolating values at vertices over triangle (barycentric interpolation)

A Simple Approach: Sampling

Evaluating a function at a point is sampling.

We can discretize a function by sampling.

for (int x = 0; x < xmax; ++x) {
  output[x] = f(x);
}

Sampling is a core idea in graphics.

We sample time (1D), area (2D), direction (2D), volume (3D)…

1. Rasterization = Sampling A 2D Indicator Function

for (int x = 0; x < xmax; ++x) {
  for (int y = 0; y < ymax; ++y) {
    image[x][y] = inside(tri, x + 0.5, y + 0.5); 
  }
}

2. Evaluating inside(tri, x, y)