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 obtained by using the approximate formula above Image of a grid pattern placed on the plane z = 1, reflected by the sphere, for an observer located at infinity in the direction of Oz. The radius of the internal ball is equal to 1/ , the grid pattern is reflected by the part of the sphere with colatitude between 0 and /4. Image made by Alain Esculier, with the software Povray; a cylinder was placed behind the sphere. Image of a grid pattern placed on the plane z = 0 reflected by the half-sphere z < 0.

Real result:  - 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:

 When the observer is at infinity in the direction of the axis of the cone, the anamorphosis is the transformation that amounts, in any plane passing by the axis of the cone, to the symmetry with respect to the corresponding generatrix (see figure). If is the cone with axis Oz and half-angle at the vertex f , then the anamorphosis relation can be written in cylindrical coordinates: , and . We derive from this : if the anamorphosis of a curve is located on a plane perpendicular to the axis, then it is a conchoid of the view from above of the initial curve, and vice versa.  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.

See a pantograph for the conical anamorphosis at www.museo.unimo.it/theatrum/macchine/095aogg.htm

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: Convex spherical anamorphosis Conical anamorphosis Concave cylindrical anamorphosis; this concave mirror switches right and left, so the writing is not reversed on the mirror, contrary to what happens with a planar mirror! Concave spherical anamorphosis: left and right as well as up and down are switched. M.C. Escher Hand with Reflecting Sphere 1935 M.C. Escher Three Spheres II 1946 La Géode, Cité des Sciences, Paris   The distorting mirrors in the Jardin d'acclimatation, Paris.