• A triangle is an affine combination of three points \[ \vec{X}\of{\alpha, \beta, \gamma} \;=\; \alpha\vec{A} + \beta\vec{B} + \gamma\vec{C}\]

  with \(\alpha+\beta+\gamma=1\).

• Affine transformation preserve affine combinations

  • Triangles are transformed to triangles

• Planar projections map triangles to triangles

  • Similar derivation as for lines

• A point \(\vec{x} = \transpose{(x,y,z,1)}\) lies on its tangent plane, specified by \(\vec{n} = \transpose{(n_x, n_y, n_z, d)}\): \[ n_x x + n_y y + n_z z + d = 0 \quad\Leftrightarrow\quad \transpose{\vec{n}} \vec{x} = 0 \]

• The same equation should be satisfied after an affine transformation \(\mat{M}\) maps \(\vec{x}\) to \(\vec{x}'\) and \(\vec{n}\) to \(\vec{n}'\) \[ 0 \;=\; \transpose{\vec{n}'} \vec{x}' \;=\; \transpose{\vec{n}'} \mat{M} \vec{x} \;=\; \transpose{\left(\transpose{\mat{M}} \vec{n}' \right)} \vec{x} \]

• Comparing the two equations yields \(\vec{n}' = \mat{M}^{\mathsf{-T}} \vec{n}\)

  • In practice use upper-left \(3 \times 3\) block of \(\mat{M}\)
  • Don’t forgot to re-normalize \(\vec{n}'\)

• So far: color/material varies per model or per vertex

• Textures add visual detail without raising geometric complexity: Paste 2D images onto 3D geometry

• So far: color/material varies per model or per vertex

• Textures add visual detail without raising geometric complexity: Paste 2D images onto 3D geometry

• Textures allow us to change many surface properties:
  • reflectance (diffuse + specular colors/coefficients)
  • normal vector (normal mapping, bump mapping)
  • geometry (displacement mapping)
  • opacity (alpha mapping)
  • reflection/illumination (environment mapping)
  • …

• Assign 2D texture coordinate \(\vec{u}=(u,v)\) to each vertex
• Interpolate texture coordinate by barycentric coordinates \[ \vec{u}\of{\vec{X}} = \alpha \vec{u}\of{\vec{A}} + \beta \vec{u}\of{\vec{B}} + \gamma \vec{u}\of{\vec{C}} \]
• Fetch color value from texture: \(\vec{c}\of{\vec{x}} = \mathrm{texture}\of{\vec{u}\of{\vec{x}}}\)

• Linear interpolation in world coordinates yields nonlinear interpolation in screen coordinates!
• Choose screen-space (noperspective) or perspective (smooth) interpolation for vertex shader outputs (smooth is default)

• Assign 2D texture coordinate \(\vec{u}=(u,v)\) to each vertex
• Interpolate texture coordinate by barycentric coordinates in 3D object space
• Fetch color value from texture

• Many ways to parameterize a sphere!
• Some parameterizations have special properties:
  • preserve angles (conformal, 2nd image)
  • preserve areas (equi-areal, 4th image)

• Point sampling is the wrong model
  • Texture minification leads to aliasing
• Integrate over image pixel’s area in texture space
  • Approximated by an ellipse
  • Elliptically weighted averaging (EWA filtering)

• Point sampling is the wrong model
  • Texture minification leads to aliasing
• Integrate over image pixel’s area in texture space
  • Approximated by an ellipse
  • Elliptically weighted averaging (EWA filtering)
  • Computationally too expensive
• Approximate EWA filtering by
  • mip-mapping
  • anisotropic texture filtering (not discussed)

• Store texture at multiple levels-of-detail
  • MIP comes from the Latin “multum in parvo”: a multitude in a small space.
  • Precompute down-scaled versions of texture image
• Use lower-resolution versions when far from camera
  • OpenGL picks the most suitable image resolution for each per-pixel texture lookup based on pixel’s depth value

• Discard transparent texture pixels (alpha=0) in fragment shader. Frequently used for real-time rendering of plants.

• Discard transparent texture pixels (alpha=0) in fragment shader. Frequently used for real-time rendering of plants.

• Approximate reflections of environment at surface.
• Environment is assumed to be far away from object.

• Cube maps are the preferred representation.

• Add surface detail without increasing geometric complexity
  • Perturb surface normal before lighting
• Assume bumps are small compared to geometry
  • Bump pattern is taken from a texture

• Shadows are important for 3D depth perception

• Shadows are important for 3D depth perception

• Shadows are important for 3D depth perception

• Shadows are important for 3D depth perception

Shadow computation is very similar to visibility determination

• Visibility
  • Which objects can be seen from the viewpoint?
• Shadows
  • Which objects can be seen from the light source?

• If a light source is occluded, simply skip its diffuse and specular contribution in the Phong lighting computation

• Visibility
  • Which objects can be seen from the viewpoint?
• Shadows
  • Which objects can be seen from the light source?

1. Render scene as seen from light source
   • Store z-buffer (holds distance to light)
   • Light’s z-buffer is called shadow map

1. Render scene as seen from light source
   • Store z-buffer (holds distance to light)
   • Light’s z-buffer is called shadow map

1. Render scene without light contribution
2. For each light source
   1. Render scene from light souce
      • Store z-buffer in shadow map
   2. Render scene with light contribution (accumulate)
      • Shadow map look-up for each pixel
      • Pixels in shadow are discarded
      • Other pixels are lit and rendered

• Avoid the perspective aliasing!
• Acquire shadow map in post-perspective space
  • Normalized Device Coordinates
  • Parallel viewing rays
• Optimal configurations
  • Directional light parallel to image plane
  • Point light in the camera plane

• Don’t filter shadow map pixels!
  • Corresponds to shadow map test with filtered version of the geometry
• Filter boolean shadow map results
  • Sample in current pixels’s vicinity
  • How many shadow map tests yield light/shadow?
  • Percentage closer filtering

• Works for everything that can be rasterized
  • Self-shadowing, alpha-textured objects
• Soft shadows by percentage closer filtering
• Fixed resolution shadow map
  • Aliasing issues
  • Resolved by perspective shadow maps
• Omni-direction lights?
  • Needs “shadow cube maps”

• McGuire et al: Fast, Practical, and Robust Shadow Volumen, Nvidia Tech Report, 2002.
• Everitt et al: Hardware Shadow Mapping, Nvidia Tech Report, 2005.
• Stamminer & Drettakis: Perspective Shadow Maps, Siggraph 2002.