Last Lectures

• Shading 1 & 2
  • Blinn-Phong reflectance model
  • Shading models / frequencies
  • Graphics Pipeline
  • Texture mapping
• Shading 3
  • Barycentric coordinates
  • Texture queries
  • Applications of textures

Interpolation Across Triangles: Barycentric coordinates 重心坐标

Interpolation Across Triangles

• Why do we want to interpolate?
  • Specify values at vertices
  • Obtain smoothly varying values across triangles
• What do we want to interpolate?
  • Texture coordinates, colors, normal vectors, ...
• How do we interpolate?
  • Barycentric coordinates

Barycentric Coordinates

• A coordinate system for triangles $(\alpha,\beta,\gamma)$

  • $(x, y) = \alpha A+\beta B+\gamma C\\\alpha+\beta+\gamma=1$
  • Inside the triangle if all three coordinates ($\alpha \beta \gamma$) are non-negative

• What’s the barycentric coordinate of $A$?

  $$ \begin{align*}(\alpha,\beta,\gamma)&= (1,0,0)\\(x,y)&= \alpha A+\beta B+\gamma C\\&= A\end{align*} $$

• Geometric viewpoint - proportional areas

  $$ \alpha = \frac{A_A}{A_A+A_B+A_C}\\\beta = \frac{A_B}{A_A+A_B+A_C}\\\gamma = \frac{A_C}{A_A+A_B+A_C} $$

• What’s the barycentric coordinate of the centroid?

  $$ \begin{align*}(\alpha,\beta,\gamma)&= (\frac{1}{3},\frac{1}{3},\frac{1}{3})\\(x,y)&= \frac{1}{3}A+\frac{1}{3}B+\frac{1}{3}C\end{align*} $$

• Barycentric Coordinates: Formulas

  $$ \begin{align*}\alpha&= \frac{-(x-x_B)(y_C-y_B)+(y-y_B)(x_C-x_B)}{-(x_A-x_B)(y_C-yB)+(y_A-y_B)(x_C-x_B)}\\\beta&= \frac{-(x-x_C)(y_A-y_C)+(y-y_C)(x_A-x_C)}{-(x_B-x_C)(y_A-y_C)+(y_B-y_C)(x_A-x_C)}\\\gamma&= 1-\alpha-\beta\end{align*} $$

Using Barycentric Coordinates

• Linearly interpolate values at vertices
  • $V = \alpha V_A+\beta V_B+\gamma V_C$
  • $V_A$, $V_B$, $V_C$ can be positions, texture coordinates, color, normal, depth, material attributes...
  • However, barycentric coordinates, are not invariant under projection!

Applying Textures

Simple Texture Mapping: Diffuse Col

for each rasterized screen sample (x, y): // usually a pixel's center
	(u, v) = evaluate texture coordinate at (x, y) // using barycentric coordinates!
	texcolor = texture.sample(u, v);
	set sample's color to texcolor; // Usually the diffuse albedo Kd (recall the Blinn-Phong reflectance model)

Texture Magnification - Easy Case (too small)