DESCRIPTIVE GEOMETRY AND ANAMORPHOSIS

Jean DRABBE
emeritus professor
University of Brussels

jdrabbe@skynet.be		 Chantal GABRIEL-RANDOUR
mathematics teacher
Athénée Gatti de Gamond
maître de stage ULB
ch.randour@brutele.be
These notes are part of a talk given by Chantal Gabriel-Randour at the IMECT3 Conference (University of Cambridge, UK - July 2003).
Some geometric constructions developed by Jean Drabbe are used.


1. DESCRIPTIVE GEOMETRY

Some basic ideas of Descriptive Geometry were already known to Albert Dürer (Underweysung - 1525) and Piero della Francesca (De Prospectiva Pingendi - written between 1474 and 1482) but their work had little effect on the history of geometry.

Gaspard MONGE is universally looked on as the father of the whole subject.

We wish to represent three-dimensional figures in a plane. We choose two mutually perpendicular planes: one is called the groud (or horizontal ) plane while the other is called the vertical plane. Their intersection is the ground line. We project a point A in space orthogonally on those two planes. This gives Av (in the vertical plane) and Ah (in the horizontal plane).

Now we rotate around the ground line till both planes coincide.






(Av , Ah) represents A 		The following figures represent a cone and a cylinder


2. CONICAL ANAMORPHOSES

Imagine a right circular cone mirror standing on the ground plane and an eye point O directly above the tip of the the cone. Let P be a point in the ground plane. It is required to construct au point P' in the ground plane so that reflected in the mirror - seen from O - it appears to be P.

In other words, P' satisfies the following condition :
Let OP intersect the cone at U. A ray from P' striking the cone at U is being reflected along UO.

Let p be the plane through the cone's axis parallel to the vertical plane. In case P is in p, the problem is trivial. We are going to make use of this special case to solve the general problem.
Rotate the ground plane around the (vertical) axis of the cone till P coincides with some point Q in p and solve the problem for Q. This gives R. Apply the inverse rotation to R to find P'.


In 1973 Michel Parré constructs a device drawing mechanically conical anamorphoses to be seen from infinity. The tool is based on the pantograph of Scheiner (1605) made to construct dilatations.
We show a reproduction of the device by Jean-Jacques Gabriel.


The secret of Parré's device lies in some simple mathematics.
Let C be the center of the cone's base circle. R marks the point where the line from C to P cuts the circle. When the eye point is at infinity, the ration PR / RP' does not depend on P. Its value only depends on the cone in use.



Conical anamorphose of a dog
(to be seen for infinity)



3. CYLINDRICAL ANAMORPHOSES

An eye point O and a cylindrical mirror standing on the horizontal (ground) plane are given.
Let p be the polar plane of O with respect to the cylinder (i.e. the plane determined by the lines of contact of the tangent planes through O) .
Because of the eye does not distinguish between points on the same visual ray, we assume that the light seen from O is coming from points in p.
So we need to solve the problem : Let P be a point in p and let C be the point where the segment OP cuts the cylinder. Determine the point P' in the ground plane so that the ray P'C reflects along CO.

We let Q be the point of intersection of OP and the ground plane. The laws of reflexion and some elementary geometry show that P'hCh and ChOh make the same angle with the tangent line at Ch to the circular base and that ChP'h = ChQh.
This solves the problem.



An anamorphose constructed by means of descriptive geometry





and another one



We are indebted to Diane Van Bockstal for allowing us to reproduce two beautiful paintings by her father Raymond Van Bockstal.



REFERENCES

Andersen, K. The mathematical treatment of anamorphoses from Piero della Francesca to Niceron in History of Mathematics: States of the Art, Academic Press, 1996.

Hunt, J. et al. Anamorphic images, American Journal of Physics, 68 (2000), pp. 232-237.

Monge, G. Géométrie Descriptive, augmentée d'une théorie des ombres et de la perspective extraite des papiers de l'auteur par B. Brisson, Gauthier-Villars, Paris (1922).
Available at http://gallica.bnf.fr (Bibliothèque Nationale de France).

Niceron, J. La perspective curieuse (1652).
Available at http://gallica.bnf.fr (Bibliothèque Nationale de France).

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