In differential geometry and geometric optics, a caustic is the envelope of rays either reflected or refracted by a manifold. It is related to the concept of caustics in optics. The ray's source may be a point (called the radiant) or parallel rays from a point at infinity, in which case a direction vector of the rays must be specified.
More generally, especially as applied to symplectic geometry and singularity theory, a caustic is the critical value set of a Lagrangian mapping (π ○ i) : L ↪ M ↠ B; where i : L ↪ M is a Lagrangian immersion of a Lagrangian submanifold L into a symplectic manifold M, and π : M ↠ B is a Lagrangian fibration of the symplectic manifold M. The caustic is a subset of the Lagrangian fibration's base space B.
A catacaustic is the reflective case.
With a radiant, it is the evolute of the orthotomic of the radiant.
The planar, parallel-source-rays case: suppose the direction vector is
(
a
,
b
)
and the mirror curve is parametrised as
(
u
(
t
)
,
v
(
t
)
)
. The normal vector at a point is
(
−
v
′
(
t
)
,
u
′
(
t
)
)
; the reflection of the direction vector is (normal needs special normalization)
2
proj
n
d
−
d
=
2
n
n
⋅
n
n
⋅
d
n
⋅
n
−
d
=
2
n
n
⋅
d
n
⋅
n
−
d
=
(
a
v
′
2
−
2
b
u
′
v
′
−
a
u
′
2
,
b
u
′
2
−
2
a
u
′
v
′
−
b
v
′
2
)
v
′
2
+
u
′
2
Having components of found reflected vector treat it as a tangent
(
x
−
u
)
(
b
u
′
2
−
2
a
u
′
v
′
−
b
v
′
2
)
=
(
y
−
v
)
(
a
v
′
2
−
2
b
u
′
v
′
−
a
u
′
2
)
.
Using the simplest envelope form
F
(
x
,
y
,
t
)
=
(
x
−
u
)
(
b
u
′
2
−
2
a
u
′
v
′
−
b
v
′
2
)
−
(
y
−
v
)
(
a
v
′
2
−
2
b
u
′
v
′
−
a
u
′
2
)
=
x
(
b
u
′
2
−
2
a
u
′
v
′
−
b
v
′
2
)
−
y
(
a
v
′
2
−
2
b
u
′
v
′
−
a
u
′
2
)
+
b
(
u
v
′
2
−
u
u
′
2
−
2
v
u
′
v
′
)
+
a
(
−
v
u
′
2
+
v
v
′
2
+
2
u
u
′
v
′
)
F
t
(
x
,
y
,
t
)
=
2
x
(
b
u
′
u
″
−
a
(
u
′
v
″
+
u
″
v
′
)
−
b
v
′
v
″
)
−
2
y
(
a
v
′
v
″
−
b
(
u
″
v
′
+
u
′
v
″
)
−
a
u
′
u
″
)
+
b
(
u
′
v
′
2
+
2
u
v
′
v
″
−
u
′
3
−
2
u
u
′
u
″
−
2
u
′
v
′
2
−
2
u
″
v
v
′
−
2
u
′
v
v
″
)
+
a
(
−
v
′
u
′
2
−
2
v
u
′
u
″
+
v
′
3
+
2
v
v
′
v
″
+
2
v
′
u
′
2
+
2
v
″
u
u
′
+
2
v
′
u
u
″
)
which may be unaesthetic, but
F
=
F
t
=
0
gives a linear system in
(
x
,
y
)
and so it is elementary to obtain a parametrisation of the catacaustic. Cramer's rule would serve.
Let the direction vector be (0,1) and the mirror be
(
t
,
t
2
)
.
Then
u
′
=
1
u
″
=
0
v
′
=
2
t
v
″
=
2
a
=
0
b
=
1
F
(
x
,
y
,
t
)
=
(
x
−
t
)
(
1
−
4
t
2
)
+
4
t
(
y
−
t
2
)
=
x
(
1
−
4
t
2
)
+
4
t
y
−
t
F
t
(
x
,
y
,
t
)
=
−
8
t
x
+
4
y
−
1
and
F
=
F
t
=
0
has solution
(
0
,
1
/
4
)
; i.e., light entering a parabolic mirror parallel to its axis is reflected through the focus.
