3D ANAMORPHOSIS
| From the Greek ana "going up, going back", and
morphe "shape". |
The word anamorphosis refers, in a general fashion,
to the transformation that maps an object onto the object of which it is
the virtual image by an optical system, for a given viewer at finite or
infinite distance.
In space, define the anamorphosis associated to
a surface 
(the mirror) and a point 
(the observer) as the relation that maps any point M onto its symmetrical
image(s) with respect to the mirror starting from ,
i.e. any point M' that is the symmetrical image of M with
respect to the tangent plane to 
at H, where H is an intersection point between the line (M)
and the mirror  .
This way, a light ray emitted by M' goes through the eyes of the
observer after reflection at H, and M' is a virtual image
of M. To put it simply, the observer thinks they are seeing M
when they are seeing M'. |
 |
The relation 
where 
is the normal vector to
at H provides the coordinates of M'. |
This relation transforms a curve 
into a curve 
,
said to be an anamorphosis of the former.
Examples:
- a plane anamorphosis (
= plane) is none other than a reflection.
- spherical anamorphosis:
Here,
is the sphere with center O and radius 1,
the observer is at infinity in the direction of Oz,
and M(x,
y, z), M'(x',
y',
z'):
we have 
where 
and u = x, v =y
(therefore,  ,
e
= 1 for the half-sphere z > 1,
e = -1
for the other half); we get, in cylindrical coordinates given by  :
 ,
i.e.  .
Approximate formula (see opposite) when  :  .
|
 |
 
|
View of a spherical anamorphosis for an observer located
at infinity in the direction of Oz, along with the transformation
of a grid pattern and a line.
The dome-like grid pattern is the real grid pattern the
virtual image of which is, for an observer located on Oz at infinity,
the planar grid pattern. |
|
Image of a grid pattern placed on the plane z
= 0 reflected by the half-sphere z < 0. |
Real result:
See also:
melusine.eu.org/syracuse/mluque/BouleMiroir/boulemiroir.html
- cylindrical anamorphosis: see the
page dedicated to the
planar case; the transformation formulas for the cylinder 
and an observer at infinity in the direction of Oy are 
| Image of vertical lines and a sinusoid placed on the
plane y = - 1 reflected by the cylinder . |
 |
- conical anamorphosis:
 |
View of a conical anamorphosis for an observer located
at infinity in the direction of Oz, along with the transformation
of a grid pattern and a curve.
The image of the grid pattern is composed of portions
of conchoids
of Nicomedes, and the image of the black circle is a portion of the
conchoid of a circle, or limaçon
of Pascal.
|
View of the reflection of a lattice of conchoid of Nicomedes
(made
by Alain Esculier using povray)
|
Inverse transformation of an exterior grid pattern.
The reflections are, again, portions of conchoids
of Nicomedes. |
For some authors, the word anamorphosis refers more simply
to the transformation that maps an object onto its symmetrical image with
respect to a curved mirror.
| In space, the anamorphosis (in the second sense)
associated to a surface
(the mirror) is the relation that maps any point M onto its symmetrical
image(s) with respect to the mirror, i.e. any point M' symmetrical
image of M with respect to an orthogonal projection H of
M
on . |
 |
As opposed to the previous anamorphosis, this relation
is symmetrical.
Some images made by Alain Esculier:
The distorting mirrors in the Jardin d'acclimatation, Paris.
© Robert FERRÉOL
2018
,
the grid pattern is reflected by the part of the sphere with colatitude
between 0 and 
/4.
,
and 
.
:
if the anamorphosis of a curve is located on a plane perpendicular to the
axis, then it is a 
La Géode, Cité
des Sciences, Paris