2005F


Fall 2005

ARTS 102 Aesthetics of the Algorithmic Image


Emanuel Garcia



Caustics
Definition
   Caustic is a method of deriving a new curve based on a given curve and a point.
Literal Meaning
   Caustics are complex patterns of shimmering light that can be seen on surfaces in presesnce of reflective or refractive objects such as those formed on the floor of a swimming pool in sunlight. Caustics occur when light rays from a source, such as the sun, get refracted, or reflected, and converge at a single point on a non-shiny surface, which creates the non-uniform distribution of bright and dark areas. The algorithm has the simplistic nature of shadow mapping, yet produces impressive results comparable to those created using off-line rendering.
Description
   Caustic is a method of developing a new curve based on a given curve and a point. A curve produced this way may also be called caustic. Given a curve x and a fixed point y, which is the light source, catacaustic is the envelope of light rays coming from the y and reflected from the curve. Diacaustic is the envelope of refracted rays. Light rays may also be parallel as when the light source is at infinity.
History
   Caustics were first introduced and studied by Tschirnhausen in 1682. Other contributors were Huygens, Quetelet, Langrange, and Cayley.
Caustics
  

Shown below are some example of curve relations by caustics, formulas, and caustics.

Eitrochoid

The catacaustic of a cardioid

Catacaustic of sinusoid

Catacaustic of an ellipse
The Curve Relations by Caustics
  

Base Curve Light Source Catacaustic
circle on curve cardioid
circle not on curve limacon of Pascal
circle Infinity nephroid
parabola rays perpendicular to directrix Tschirnhausen's cubic
Tschirnhausen's cubic focus semicubic parabola
cissoid of Diocles focus cardioid
cardioid cusp nephroid
quadrifolium center astroid
deltoid Infinity astroid
equiangular spiral pole equiangular spiral
cycloid rays perpendicular to line through cusps cycloid 1/2
y==E^x rays perpendicular y-axis catenary

Links
  

http://mathworld.wolfram.com/TschirnhausenCubic.html
http://www.mrs.umn.edu/~sungurea/introstat/history/w98/Quetelet.html
www.xahlee.org/SpecialPlaneCurves_dir/Caustics_dir/caustics.html http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html http://www.xahlee.org/