Last update, Dec.10, 1998

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Publications on Projective Geometry :

• The Cross Ratio, by Jim Blinn, 1998.

Publications on Perspective Geometry :

• La Place de J.-H. Lambert (1728-1777) dans l'Histoire de la Perspective, by R. Laurent, 1987.
• Brook Taylor's work on Linear Perspective, by K. Andersen, 1992.

BibTeX references .

Explains the concept of cross ratio in projective geometry and proves it (easy explanation).

Restitution en géométral d'une perspective ou des règles inverses de la perspective

• Notes tirées (pp.167-192) de "La perspective affranchie de l'embaras du plan géometral", par Lambert, Johann Heinrich; Zurich: Heidegger et Cie., 1759, 192 p. Réimprimé en fac-similé, chez Alain Brieux, Paris, 1977.

Problème: étant donné une perspective, où en général ne figure pas la position de l'observateur par rapport au tableau, retrouver les dimensions géométrales du dessin ainsi représenté, à une échelle donnée. (Peut-on aussi reconstruire un bâtiment ainsi?)

La résolution de ce problème pré-suppose que le tableau soit perpendiculaire au plan géométral et que l'on puisse déduire les 4 données suivantes (qui caractérisent une perspective):

1. la ligne d'horizon;
2. le point de l'oeil sur le tableau;
3. la distance de l'oeil au tableau;
4. la hauteur de l'oeil au-dessus du plan géométral.

Problem 21

"Find the Horizon Line, the Eye Station and the Principal Distance, from the 2PtP of a square."

Solution:

Let abcd be the perspective image of the square ABCD in Ground Plane G. The parallel sides of the square, ab & cd define the VP m, while the parallels ad & bc define the VP M. These 2 VPs thus define the Horizon Line (HL).

Furthermore, mbM must be a right angle (in space). The same is true of the diagonals acN and bdn. Hence, the VPs m & M and the diagonal intercepts with the HL, n and N, determine two circles upon which the Eye Station Q must lie. This determines Q as the intercept of both (half-)circles. N.B.: These 2 half-circles are called "arcs capables d'angles droits" in French.

Finally, the orthogonal line from the Eye Station, Q, to the Horizon Line determines the Pincipal Point , P. The (scaled) length QP is then the Principal Distance, f (or focal distance).

Problem 22

"Find the Horizon Line, the Eye Station and the Principal Distance, from the 2PtP of a rectangle, knowing the ratio of its two sides."

Solution:

Assume the ratio AD/AB is known for the rectangle ABCD. Then, the internal angle alpha = DBA is also known (since DAB is a right angle): alpha = arctg(AD/AB). This is also the angle associated to dab, the perspective projection of DAB.

Consider now the VPs m & n associated to the directions BA & BD (or ba & bd in perspective), respectively. These 2 VPs determine the Horizon Line, as usual.

Furthermore, Q, the Eye Station, must be on the half-circle through m and M as usual, where M is the VP for the other side of the rectangle. i.e., BC (bc).

Then, alpha = mQn, since Qm is parallel to BA (ba) and Qn is parallel to BD (bd). Thus, we have the following geometrical construction to retrieve the locus of Q on the "cercle capable":

1. On the circle of diameter mM, find the point H, s.t. the angle HMm = alpha (=arctg(mH/MH)).
2. Then, construct the line through Hn, which intercepts the circle at Q.

Introduces a particular form of "cross-ratios", where one of the four points is a vanishing point (p.28).

Uses an angle-scale on a vanishing line and the semi-circle including the eye to find how to draw 2D figures making an angle with respect to the viewer (pp.30-33).

Considers certain Inverse problems of perspective.

VPs IKH & Station Point (O) generate the visual tetrahedron. S is the ortho. projection of O & orthocenter of triangle IKH .	Resolving the visual tetrahedron, given a cube in 3PtP.

Mention of John Hamilton's "Stereography or a Compleat body of Perspective" (London: 1738, 1749), where the curve of foreshortening is studied (an hyperbola):

• Let the distance between a point in the ground plane and the ground line be x and the distance between the perspective image of the point and the ground line be y. It is shown that the points (x,y) lie on a hyperbola.

Page created & maintained by Frederic Leymarie, 1998-2001
Comments, suggestions, etc., mail to: leymarie@lems.brown.edu