光栅化(Rasterization)

What’s after MVP?

What’s after MVP?

• Model transformation (placing objects)
• View transformation (placing camera)
• Projection transformation
  • Orthographic projection
  • Perspective projection

Canonical Cube to Screen(规范到屏幕)

Canonical Cube to Screen(规范到屏幕)

• 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 the screen
• Pixel (FYI, short for “picture element”)
  • For now: A pixel is a little square with uniform color
  • Color is a mixture of (, , )
• Defining the screen space
  • Pixels’ indices are in the form of (x, y), where both x and y are integers
  • Pixels’ indices are from (0, 0) to (width - 1, height - 1)
  • Pixel (x, y) is centered at (x + 0.5, y + 0.5)
  • The screen covers range (0, 0) to (width, height)
  • Irrelevant to z (与Z轴无关)
  • Transform in xy plane: [-1, 1]² to [0, width] x [0, height]
  • Viewport transform matrix:
  $$M_{viewport}=\left(\begin{array}{cccc}\frac{width}{2}& 0& 0& \frac{width}{2}\\ \\ 0& \frac{height}{2}& 0& \frac{height}{2}\\ \\ 0& 0& 1& 0\\ \\ 0& 0& 0& 1\end{array}\right)$$

Rasterizing Triangles into Pixels

Rasterizing Triangles into Pixels

Polygon Meshes

Triangle Meshes

Triangles - Fundamental Shape Primitives

Why triangles?

• Most basic polygon
  • Break up other polygons
• Unique properties
  • Guaranteed to be planar[保证是平面的]
  • Well-defined interior[内部]
  • Well-defined method for interpolating values at vertices over triangle (barycentric interpolation)[三角形顶点插值方法(质心插值)]

What Pixel Values Approximate a Triangle?

A Simple Approach: Sampling (一个简单的方法:采样)

Sampling a Function

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) …

Rasterization As 2D Sampling

Sample If Each Pixel Center Is Inside Triangle

Define Binary Function: inside(tri, x, y)

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);

Inside? Recall: Three Cross Products!(三次叉乘)

Some problems

Edge Cases (Literally) 约定即可

Checking All Pixels on the Screen?

Incremental Triangle Traversal (Faster?)

Rasterization on Real Displays

So, If We Send the Display the Sampled Signal The Display Physically Emits This Signal Compare: The Continuous Triangle Function

Jaggies!

Aliasing (Jaggies)