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CAUSTIC

Caustique par réflexion au flambeau
Caustique par réflexion au soleil
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 M0 is projected on J on the incident ray, then on K on the normal at M0. 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 M0 on D about the tangent at M0. Note that the various anticaustics associated with the lines D are parallel and therefore have the same evolute.


Construction of the caustic from an anticaustic; the characteristic point on the ray reflected at M0 is determined as the projection on this ray of the middle of the segment line joining M0 to the centre of curvature of (G0) at M0.


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 ay2=x3 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 ex/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 M0 is a point on the curve  and n a constant (that can be negative), the refracted ray associated to the incident ray (SM0) is the line (D) such that r is the angle formed between (D) and the normal (N) to  at M0, 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