Michel Chasles (born November 15, 1793, Épernon, France—died December 18, 1880, Paris) was a French mathematician who, independently of the Swiss German mathematician Jakob Steiner, elaborated on the theory of modern projective geometry, the study of the properties of a geometric line or other plane figure that remain unchanged when the figure is projected onto a plane from a point not on either the plane or the figure.
Chasles was born near Chartres and entered the École Polytechnique in 1812. He was eventually made professor of geodesy and mechanics there in 1841. His Aperçu historique sur l’origine et le développement des méthodes en géométrie (1837; “Historical Survey of the Origin and Development of Geometric Methods”) is still a standard historical reference. Its account of projective geometry, including the new theory of duality, which allows geometers to produce new figures from old ones, won the prize of the Academy of Sciences in Brussels in 1829. For its eventual publication Chasles added many valuable historical appendices on Greek and modern geometry.
In 1846 he became professor of higher geometry at the Sorbonne (now one of the Universities of Paris). In that year he solved the problem of determining the gravitational attraction of an ellipsoidal mass to an external point. In 1864 he began publishing in Comptes rendus, the journal of the French Academy of Sciences, the solutions to an enormous number of problems based on his “method of characteristics” and his “principle of correspondence.” The basis of enumerative geometry is contained in the method of characteristics.
Chasles was a prolific writer and published many of his original memoirs in the Journal de l’École Polytechnique. He wrote two textbooks, Traité de géométrie supérieure (1852; “Treatise on Higher Geometry”) and Traité des sections coniques (1865; “Treatise on Conic Sections”). His Rapport sur le progrès de la géométrie (1870; “Report on the Progress of Geometry”) continues the study in his Aperçu historique.
Chasles is also remembered as the victim of a celebrated fraud perpetrated by Denis Vrain-Lucas. He is known to have paid nearly 200,000 francs (approximately $36,000) between 1861 and 1869 for more than 27,000 forged documents—many purported to be from famous men of science, one allegedly a letter from Mary Magdalene to Lazarus, and another a letter from Cleopatra to Julius Caesar—all written in French.
Projective drawingThe sight lines drawn from the image in the reality plane (RP) to the artist's eye intersect the picture plane (PP) to form a projective, or perspective, drawing. The horizontal line drawn parallel to PP corresponds to the horizon. Early perspective experimenters sometimes used translucent paper or glass for the picture plane, which they drew on while looking through a small hole to keep their focus steady.
projective geometry, branch of mathematics that deals with the relationships between geometric figures and the images, or mappings, that result from projecting them onto another surface. Common examples of projections are the shadows cast by opaque objects and motion pictures displayed on a screen.
Projective geometry has its origins in the early Italian Renaissance, particularly in the architectural drawings of Filippo Brunelleschi (1377–1446) and Leon Battista Alberti (1404–72), who invented the method of perspective drawing. By this method, as shown in the figure, the eye of the painter is connected to points on the landscape (the horizontal reality plane, RP) by so-called sight lines. The intersection of these sight lines with the vertical picture plane (PP) generates the drawing. Thus, the reality plane is projected onto the picture plane, hence the name projectivegeometry. See alsogeometry: Linear perspective.
Although some isolated properties concerning projections were known in antiquity, particularly in the study of optics, it was not until the 17th century that mathematicians returned to the subject. The French mathematicians Girard Desargues (1591–1661) and Blaise Pascal (1623–62) took the first significant steps by examining what properties of figures were preserved (or invariant) under perspective mappings. The subject’s real importance, however, became clear only after 1800 in the works of several other French mathematicians, notably Jean-Victor Poncelet (1788–1867). In general, by ignoring geometric measurements such as distances and angles, projective geometry enables a clearer understanding of some more generic properties of geometric objects. Such insights have since been incorporated in many more advanced areas of mathematics.
Fundamental theorem of similarityThe formula in this diagram reads k is to l as m is to n if and only if line DE is parallel to line AB. This theorem then enables one to show that the small and large triangles are similar.
A theorem from Euclid’s Elements (c. 300 bc) states that if a line is drawn through a triangle such that it is parallel to one side (see the figure), then the line will divide the other two sides proportionately; that is, the ratio of segments on each side will be equal. This is known as the proportional segments theorem, or the fundamental theorem of similarity, and for triangle ABC, shown in the diagram, with line segment DE parallel to side AB, the theorem corresponds to the mathematical expression CD/DA = CE/EB.
Now consider the effect produced by projecting these line segments onto another plane as shown in the figure. The first thing to note is that the projected line segments A′B′ and D′E′ are not parallel; i.e., angles are not preserved. From the point of view of the projection, the parallel lines AB and DE appear to converge at the horizon, or at infinity, whose projection in the picture plane is labeled Ω. (It was Desargues who first introduced a single point at infinity to represent the projected intersection of parallel lines. Furthermore, he collected all the points along the horizon in one line at infinity.) With the introduction of Ω, the projected figure corresponds to a theorem discovered by Menelaus of Alexandria in the 1st century ad:
C′D′/D′A′ = C′E′/E′B′ ∙ ΩB′/ΩA′.
Since the factor ΩB′/ΩA′ corrects for the projective distortion in lengths, Menelaus’s theorem can be seen as a projective variant of the proportional segments theorem.
With Desargues’s provision of infinitely distant points for parallels, the reality plane and the projective plane are essentially interchangeable—that is, ignoring distances and directions (angles), which are not preserved in the projection. Other properties are preserved, however. For instance, two different points have a unique connecting line, and two different lines have a unique point of intersection. Although almost nothing else seems to be invariant under projective mappings, one should note that lines are mapped onto lines. This means that if three points are collinear (share a common line), then the same will be true for their projections. Thus, collinearity is another invariant property. Similarly, if three lines meet in a common point, so will their projections.
The following theorem is of fundamental importance for projective geometry. In its first variant, by Pappus of Alexandria (fl. ad 320) as shown in the figure, it only uses collinearity:
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Let the distinct points A, B, C and D, E, F be on two different lines. Then the three intersection points—x of AE and BD, y of AF and CD, and z of BF and CE—are collinear.
There is one more important invariant under projective mappings, known as the cross ratio (see the figure). Given four distinct collinear points A, B, C, and D, the cross ratio is defined as
CRat(A, B, C, D) = AC/BC ∙ BD/AD.
It may also be written as the quotient of two ratios:
CRat(A, B, C, D) = AC/BC : AD/BD.
The latter formulation reveals the cross ratio as a ratio of ratios of distances. And while neither distance nor the ratio of distance is preserved under projection, Pappus first proved the startling fact that the cross ratio was invariant—that is,
CRat(A, B, C, D) = CRat(A′, B′, C′, D′).
However, this result remained a mere curiosity until its real significance became gradually clear in the 19th century as mappings became more and more important for transforming problems from one mathematical domain to another.
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