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CAUSTIC


Notion studied and named by Tschirnhausen
in 1681 and then by Jacques Bernoulli in 1691, and La Hire in 1703.
From the Latin causticus, copy of the Greek kaustikos: burning. 
The word caustic refers, in a general fashion, to the envelope of light rays emitted from a finite distance (the source is then called the radiant) or an infinite one after they are modified by an optical instrument. Every modified ray is considered as a whole, and includes the virtual ray.
A) CAUSTIC BY REFLECTION
In a plane, the caustic by reflection (or catacaustic) of a curve with respect to a light source S is the envelope of the rays emitted by S after reflection on a mirror with profile .
I) Case where S is a radiant.
In this case, the caustic by reflection of the reflecting
curve
is the evolute of the orthotomic,
who is, then, rather called anticaustic or secondary caustic.
Recall that this curve is itself the homothetic image with centre S
and radio 2 of the pedal of
with respect to S.


Construction of the characteristic point M on the reflected ray: the centre of curvature I at M_{0} is projected on J on the incident ray, then on K on the normal at M_{0}. S, K and M are aligned.  The red caustic is the evolute of the green orthotomic of the blue ellipse.  The red caustic is the evolute of the green orthotomic of the blue astroid. 
Application of this result: the caustic by reflection of the negative pedal of a cycloidal curve is a similar cycloidal curve.
Examples:
reflecting curve  light source  caustic by reflection 
circle  on the circle and at the top of the cardioid  cardioid 
circle  any point
pole of the limaçon 
caustic of a circle 
parabola  focus of the parabola  reduces to a point at infinity in the direction of the parabola 
bifocal conic  focus of the conic  reduces to the other focus 
logarithmic spiral  asymptotic point of the spiral  logarithmic spiral 
Tschirnhausen cubic  focus  semicubical parabola 
cissoid of Diocles  point (4a, 0)  cardioid 
cardioid  cuspidal point  nephroid 
inverse caustic of a circle  centre  circle 
II) Case where S is at infinity.
In this case (the incident rays are parallel), the caustic can also be defined as an evolute. Given a line D orthogonal to the rays, the caustic is the evolute of the anticaustic associated to the line D, locus of the symmetrical image of the projection of M_{0} on D about the tangent at M_{0}. Note that the various anticaustics associated with the lines D are parallel and therefore have the same evolute.
Cartesian parametrization for rays parallel to Ox: 
Examples:
reflecting curve or
inverse caustic 
direction of the rays  caustic 
circle  any direction  nephroid (curve of the coffee cup) 
parabola  parallel to the axis  focus 
parabola  any other direction  Tschirnhausen cubic 
cubic parabola ay^{2}=x^{3}  parallel to Oy  Tschirnhausen cubic 
generalised parabola
(; the case k=1/2 corresponds to the previous case) 
parallel to Oy  pursuit curve
(k = speed of the master / speed of the dog) 
logarithmic: y = a ln (x/a)  parallel to Oy  pursuit curve
(speed of the master = speed of the dog) 
arch of a cycloid  perpendicular to the axis of the rolling motion  two arcs of a cycloid reduced by half. 
deltoid  any direction  astroid 
exponential
y = a e^{x/a} 
parallel to Oy  catenary 
B) CAUSTIC BY REFRACTION (generalises the previous case)
The caustic by refraction (or diacaustic) of a curve with respect to a light source S is the envelope of the rays emitted by S after refraction on a dioptre with profile
If M_{0} is a point on the curve and n a constant (that can be negative), the refracted ray associated to the incident ray (SM_{0}) is the line (D) such that r is the angle formed between (D) and the normal (N) to at M_{0}, with , where i is the angle .
Only the case n > 0 corresponds to the physical refraction (n is then the ratio of the refractive indices in the side to which S does not belong and the one to which it does); the case n = –1 corresponds to the reflection.
The reunion of the caustics for the constants n > 0 and –n is referred to as complete caustic by refraction for the constant n. It is the evolute of the anticaustic of with respect to S associated to the constant n.
Examples:
 the complete caustics by refraction
of the straight line are the evolutes of conics and the complete caustics
by refraction of the circle are the evolutes of complete Cartesian
ovals.
 the curves for which the caustic
by refraction reduces to a point are the conics for parallel incident rays,
and the Cartesian ovals for
a radiant.
See also on this page the example of the caustics by refraction of circles, for a light source at infinity.
C) ORTHOCAUSTIC.
The orthocaustic of
with respect to a light source S is the envelope of the lines perpendicular
to the rays emitted by S at their point of impact on .
The orthocaustic is therefore none other than the negative
pedal.
Examples: the orthocaustic of a straight line (D) is a parabola with focus S tangent to (D); the orthocaustic of a circle is an ellipse or a hyperbola depending on whether S is inside or outside the circle.
D) OTHER CAUSTICS.
For a pool table delimited by a convex curve, the envelope
of the consecutive trajectories of a pool ball (in the case where this
trajectory is not periodic) is also referred to as "caustic". For example,
for an elliptic pool table, the caustic is an homofocal ellipse or hyperbola
(see this
site).
See also, in the field of curves defined by optical means,
the anamorphoses.
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© Robert FERRÉOL 2017