Lecture 26. Early Stage of ProjectiveGeometry

Figure 26.1 The woodcut book The Designer of the Lute illustrates how

one uses projection to represent a solid object on a two dimensional canvas.

Projective geometry was first systematically developed by Desargues 1 in the 17th centurybased upon the principles of perspective art. As a mathematical field, however, projectivegeometry was established by the work of Poncelet 2 and others.

Projective geometry is a branch of mathematics which deals with the properties andinvariants of geometric figures under projection. One source for projective geometry wasindeed the theory of perspective. One difference from elementary geometry is the way inwhich parallel lines can be said to meet in a point at infinity.

1Girard Desargues (1591-1661) was a French mathematician and engineer, one of the founders of projec-tive geometry.

2Jean-Victor Poncelet (1788 - 1867) was a French engineer and mathematician who served most notablyas the commandant general of the Ecole Polytechnique. He is considered a reviver of projective geometry.

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The theory of perspective The theory of perspective describes how to project a three-dimensional object onto a two-dimensional surface, i.e., perspective may be simply describedas the realistic representation of real scenes on a plane. This has been an interesting problemfor most painters since ancient times. Even though some Roman artists seem to haveachieved correct perspective about 100 B.C. However, it was simply an individual geniusrather than the success of a theory. The vast majority of ancient paintings, in fact, showincorrect perspective.

Medieval artists made some charming attempts at perspective but always got it wrong,and errors persisted well into the fifteenth century.

Figure 26.2 False perspective.

During the Renaissance, scientists and scholars began engaging in different kinds ofexperiments. Some artists conducted careful observations of nature and even anatomical

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dissections to try to better understand the world around them. But it wasn’t until theearly 15th Century that a Florentine architect and engineer named Filippo Brunelleschi(1377-1446) developed a mathematical theory of perspective through a series of opticalexperiments. Brunelleschi was able to understand the science behind perspective.

The basic principles Renaissance artists used were the following:

∙ A straight line in perspective remains straight.

∙ Parallel lines either remain parallel or converge to a single point (vanishing point).

These principles suffices to solve a problem artists frequently encountered: the perspec-tive depiction of a square-tiled floor.

Figure 26.3 Parallel lines converge to a single point (vanishing point).

Desargues’ Theorem Mathematical setting on perspective is the family of lines (“lightrays”) through a point (the “eye”). In this setting, the problems of perspective becamerelatively easy, but the concepts were a challenge to traditional geometric thought. Differentfrom Euclid, one had the following:

(i) Points at infinity (“vanishing point”) where parallels met.(ii) Transformations that changed lengths and angles (projective).

Projective geometry originated through the efforts of a French artist and mathematician,Gerard Desargues (1591-1661), as an alternative way of constructing perspective drawings,although the idea of points at infinity had already been used by Kepler(1604).

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Desargues published a book in 1639, but only one copy of this book is now known tosurvive, which was rediscovered in 1951. His two most important theorems, the so-calledDesargues’ theorem and the invariance of the cross-ratio, were published in a book aboutperspective by Bosse (1648).

Figure 26.4 Desargues’ theorem

Desargues’ theorem is a property of triangles in perspective illustrated by Figure 26.4.The theorem states that the points 𝐺,𝐹 and 𝐸 at the intersections of corresponding sideslie in a line. This is obvious if the triangles are in space, since the line is the intersection ofthe planes containing them. The theorem in the plane is subtly but fundamentally differentand requires a separate proof, as Desargues realized. In fact, Desargues’ theorem was shownto play a key role in the foundations of projective geometry by Hilbert (1899).

Pascal Theorem Blaise Pascal (1623-1662) was a very influential French mathematicianand philosopher who contributed to many areas of mathematics. He worked on conic sectionsand projective geometry and in correspondence with Fermat he laid the foundations for thetheory of probability.

When he was 12 years old, Pascal, gaving up his play-time to this new study, began toread a geometry book. In a few weeks, he had discovered for himself many properties offigures, including that the sum of the angles of a triangle is equal to 𝜋.

His father, struck by this display of ability, gave him a copy of Euclid’s Elements. BeforePascal turned 13 he had proved the 32nd proposition of Euclid and discovered an error inDescartes’ Geometry.

At 16, Pascal began preparing to study entire field of mathematics. Desargues’s studyon conic sections drew his attention and helped him formulate Pascal’s theorem. Pascal’s

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Essay on Conics was written in late 1639, who probably had heard about projective geometryfrom his father, who was a friend of Desargues. The Essay contained the first statement ofa famous result that became known as Pascal’s theorem, which is the dual of Brianchon’stheorem. It states that, given a (not necessarily regular, or even convex) hexagon inscribedin a conic section, the three pairs of the continuations of opposite sides meet on a straightline, called the Pascal line.

Figure 26.5 Pascal and Pascal’s theorem

Projective geometry was further developed in 18th century (Gaspard Monge, Jean-VictorPoncelet), 19th century (Julius Placker, Steiner, Clebsch, Riemann, Max Noether, Enriques,Segre, Severi, Schubert), and etc.

Riemann sphere The projective geometry has been continuously developing. One of thebasic notions is “Riemann sphere” —– one dimensional complex projective space.

Let 𝑆 be the unit sphere

𝑆 = {(𝑇1, 𝑇2, 𝑇3) ∈ ℝ3 ∣ (𝑇1)2 + (𝑇2)2 + (𝑇3)2 = 1}.

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If we regard 𝑆 as the earth, then the point {(0, 0, 1)} can be regarded as the north pole ofthe sphere. We define a map

𝑆 − {(0, 0, 1)} → ℂ𝛼 7→ 𝐴

where 𝐴 is the intersection point of the plane ℂ and the the straight line passing throughthe north pole (0, 0, 1) and the point 𝛼.

Figure 26.6 Riemann sphere

This map is one-to-one and onto. We call 𝑆 the Riemann sphere, and the map Stereo-graphic projection.

Notice that as 𝐴 moves to ∞, the corresponding 𝛼 moves to the north pole. Then wemay write the north pole 𝑁 = {∞}. We can denote 𝑆 = (𝑆 − {𝑁}) ∪ {𝑁} as

𝑆 = ℂ ∪ {∞}. (1)

In the Riemann sphere, the “infinite” is just a point in 𝑆. We can treat ∞, as any otherpoint in the complex space ℂ, as an ordinary point in 𝑆.

For example, in Calculus, the definitions of lim𝑛→∞ 𝑥𝑛 = 𝑎 ∈ ℝ and lim𝑛→∞ 𝑥𝑛 = +∞are quite different. Passing everything in the Riemann sphere 𝑆, lim𝑛→∞ 𝑧𝑛 = 𝑎 ∈ ℂ andlim𝑛→∞ 𝑧𝑛 = ∞ should have the same definition.

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