Advanced Global Illumination, 2nd Edition

Appendix B

Hemispherical Coordinates

B.1 Hemispherical Coordinates

In photorealistic rendering, one often wants to work with functions defined over a hemisphere (one-half of a sphere), centered around a surface point. A hemisphere consists of all the directions in which one can look when standing at the surface point: one can look from the horizon all the way up to the zenith and all around. A hemisphere is therefore a two-dimensional space, in which each point on the hemisphere defines a direction. Spherical coordinates are a useful way of parameterizing the hemisphere.

In the spherical coordinate system, each direction is characterized by two angles (Figure B.1). The first angle, φ, represents the azimuth and is measured with regard to an arbitrary ...