On Caustics of Plane Curves

Using Rolling Circles to Generate Caustic Envelopes Resulting from Reflected Light

“…Light from any radiant on the outer solid circle of diameter 5π 4 will focus at a point on the smaller solid circle of diameter 5π 6 . In [3], geometrical methods are used to study the location of the focus F as a result of reflection at P = α(s 0 ) from the source at S. It is shown that there is a unique conic that has at least three point contact with α(s) at P such that S is at one of the foci of the conic. The point F will then be located at the other focus of the conic.…”

“…Let C d be the circle of diameter d tangent to α at P and let r be the radius of curvature at P. The circle C r/2 is called the discriminant circle (see [3]). There are a number of simple consequences of Theorem 2 that we now list.…”

“…A. Cayley [6] wrote a memoir in 1857 describing caustics both from reflections and refractions, giving calculations for all cases involving a circular mirror (see [22] for equations). See also [3,7,11,14]. Lockwood [13] "A Book of Curves" discusses caustics, gives a table of common curves and some of their associated caustics, and provides a useful glossary for plane curves.…”

A novel non-central catadioptric auto-calibration approach

“…In [4], they used the Jacobian method. In [3], the method is based on a local conic approximation. They use the fact that the light rays emanating from one focal point will meet the second focal point in a conic section.…”

“…Section (3) illustrates what is polarization imaging and how it is used as a necessary pre-step to our auto-calibration method. Section (4) explains caustics, its importance, and how to compute.…”