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projective geometry: Quick Facts

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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.
Fundamental theorem of similarity
Projective version of the fundamental theorem of similarityIn RP, Euclid's fundamental theorem of similarity states that CD/DA = CE/EB. By introducing a scaling factor, the theorem can be saved in RP as Cā€²Dā€²/Dā€²Aā€² = Cā€²Eā€²/Eā€²Bā€² āˆ™ Ī©Bā€²/Ī©Aā€². Note that while lines AB and DE are parallel in RP, their projections onto PP intersect at the infinitely distant horizon (Ī©).
Pappus's projective theoremPappus of Alexandria (fl. ad 320) proved that the three points (x, y, z) formed by intersecting the six lines that connect two sets of three collinear points (A, B, C; and D, E, F) are also collinear.
Pascal's projective theoremThe 17th-century French mathematician Blaise Pascal proved that the three points (x, y, z) formed by intersecting the six lines that connect any six distinct points (A, B, C, D, E, F) on a circle are collinear.
Cross ratioAlthough distances and ratios of distances are not preserved under projection, the cross ratio, defined as AC/BC āˆ™ BD/AD, is preserved. That is, AC/BC āˆ™ BD/AD = Aā€²Cā€²/Bā€²Cā€² āˆ™ Bā€²Dā€²/Aā€²Dā€².
Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated.
A diagram of a double cone intersected by two green planes labeled PP (picture plane) and RP (reality plane), illustrating projected conic sections in projective geometry, with the apex and a point Omega marked.
Figure 9: Use of auxiliary view to show true size and shape of an inclined surface (ABCD), which is not correctly represented in the front, top, or side view (see text).
Figure 1: Two techniques of representing an object. (A) Perspective drawing, suggesting that the object is cubical. (B) Orthographic top and front views, revealing that the object is not cubical.
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Key People

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Girard Desargues
French mathematician
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Oswald Veblen
American mathematician
Steiner surface. It was during a trip to Rome in 1844 that Jakob Steiner first discovered the fourth-degree surface that today bears his name; for this reason it is sometimes referred to as the Roman surface. Each of its tangent planes has the characteristic property that it intersects the surface in a pair of conics. The Steiner surface also contains three double lines that meet one another in a triple point. Steiner never published these and other findings concerning the surface. A colleague, Karl Weierstrass, first published a paper on the surface and Steiner's results in 1863, the year of Steiner's death.
Jakob Steiner
Swiss mathematician
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Michel Chasles
French mathematician
August Ferdinand Mƶbius, detail from an engraving by an unknown artist.
August Ferdinand Mƶbius
German mathematician and astronomer
Jean-Victor Poncelet, detail of a lithograph by Patout, 1849.
Jean-Victor Poncelet
French mathematician
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Karl Georg Christian von Staudt
German mathematician

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