# mathr / blog / #

## Prismatic

Around this time last year I was playing around with a physics-based ray-tracer for spherically curved space. In spherical space the ray geodesics eventually wrap around, meeting at the opposite pole to the observer. To compound the sphericity I used a projection that wraps the whole sphere-of-view from a point into a long strip.

It's been so long I've forgotten the details of how it works, but I embedded the 3D spherical space in 4D Euclidean (flat) space. I represented ray directions by points on the "equator" around the ray source, and a lot of trigonometry was involved to transform these ray directions appropriately when tracing the rays through curved space. Eventually I optimized the code to use simpler functions like sqrt and arithmetic instead of costly sin and cos calls.

The materials are all physically based, with refractive index varying with simulated light wavelength, which gives a rainbow effect when different colours are refracted by different angles. To get the final image requires tracing a monochrome image at many different wavelengths, which are then combined into the XYZ colour space using tristimulus response curves. I collected some Wikipedia articles together into a little A5 booklet: colour.pdf (and a version with the pages rearranged for printing using bookletimposer: colour-booklet.pdf).

Here are some miscellaneous links related to how Prismatic works:

Projection
transformation projection (section cubic to/from spherical map)
Lambert equal-area projection
spherical coordinates
Ray-surface intersection distance
solve a = b cos x + c sin x
double angle formulae
Reflection and refraction
Lambertian reflectance
Fresnel equations
Snell's law
reflect()
refract()
Kramers-Kronig relations
Absorption and emission
Beer-Lambert law
complex refractive index
absorption spectroscopy
emission spectrum
Colour, wavelength, CIE XYZ, sRGB
colour matching functions (section database / CMFs)
sRGB specification
illuminant D65
CMYK colour model
Materials
refractiveindex.info
corundum
sapphire
ruby and sapphire
ruby
chemistry webbook
chemistry webbook UV/visibile wavelengths
crown glass
flint glass

The prismatic code itself is online at code.mathr.co.uk/prismatic.