Figure 1: Eye rays reflected from an object. The reflected rays are shown as
blue arrows. Red arrows are the normal vectors. Given the normal vectors, one
can compute the reflected rays using the formulas we used to do Phong shading.
Note that environment mapping will not produce correct results for rays which
have to bounce from the object several times before escaping to the environment
(like one of the rays shown above).
Environment Mapping
1 Geometry of reflection
Consider an ideally reflective object and a ray through a pixel. What color
should one use for that pixel? The same as the color incoming to the reflection
point from the direction of the reflected ray (see Figure 1). Therefore, if one is
able to precompute and somehow store the colors incoming towards the reflective
object along different rays, rendering would be really simple.
In environment mapping, one stores the colors arriving at a 3D object from
different directions in one or more textures. Typically, one assumes that colors
incoming to different points along parallel rays are the same. This means that
we need to keep a color for each incoming ray direction rather than for every
possible incoming ray. Below we discuss two specific examples.
2 Spherical environment map
Imagine a photograph (or a synthetic image, e.g. rendered using ray tracing)
of a mirror sphere. This image stores colors incoming along rays of all possible
directions (Figure 2). It is directly used as texture for spherical environment
mapping. Such a texture might look like the one shown in Figure 3. To simplify
things, let’s say that the sphere is exactly inscribed into the (square) photograph.
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