The Casey angle. A Different Angle on Perspective

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A Different Angle on Perspective Marc Frantz Marc Frantz (mfrantz@indiana.edu) majored in painting at the Herron School of Art, where he received his.f.a. in 1975.

1 A Different Angle on Perspective Marc Frantz Marc Frantz majored in painting at the Herron School of Art, where he received his.f.a. in After a thirteen-year career as a painter and picture framer, he entered graduate school in mathematics at Indiana University Purdue University Indianapolis (IUPUI), where he received his M.S. in Since then he has taught mathematics at IUPUI and Indiana University, where he is currently a research associate. He loves the visual approach to mathematics, especially links between mathematics and art. In Figure 1, the angle between a fence post and the fence rail is a right angle. On the other hand, in a projective image, such as a perspective drawing or the photograph in Figure 2, the measure of the image of a right angle may well not be 90. ut should all angles be disregarded in projective geometry? We answer No, and describe a sense in which certain angles are preserved by projective transformations, although generally projective geometry disregards all considerations of distance and angle [3, p. xii]. The Casey angle Consider the four fence posts in Figures 1 and 2. It turns out that from the intersection of the two semicircles, the angle subtended by the inner semicircle endpoints is 60. This is easy to prove in Figure 1, and easy to check empirically in Figure 2 by measuring with a protractor. We go on to discuss some applications of the invariance of these angles to projective geometry, perspective drawing, and photography in hopes 60 α = 90 Figure 1. Two angles associated with a fence. MSC: 51N THE MATHEMATICAL ASSOCIATION OF AMERICA

60 α < 90 Figure 2. of convincing the reader that they deserve to be better known. These are examples of a concept we dub the Casey angle, after a theorem by the Irish geometer John Casey (1820 1891).

2 60 α < 90 Figure 2. of convincing the reader that they deserve to be better known. These are examples of a concept we dub the Casey angle, after a theorem by the Irish geometer John Casey ( ). Consider any four distinct points on a line, denoted in order by A, C,, D, as in Figure 3. Let O be an intersection point of the circles with diameters A and CD. The angle θ = CO is the Casey angle of the four points. O θ A C D Figure 3. The Casey angle of four collinear points. Observe that O lies on the circle with diameter A in Figure 3, hence m( AO) = 90, where m denotes the measure of an angle. Consequently, the Casey angle satisfies 0 < θ < 90. Observe also that unlike the angles α and α, the Casey angle in Figure 2 is not the projected image of that in Figure 1. For instance, if the semicircles in Figure 1 were physically in the plane of the fence, their projected images in Figure 2 would be elliptical arcs not semicircles and the corresponding angle would not be 60. Instead, we should think of the semicircles and the Casey angles as compass and straightedge constructions drawn on the figures. This is the sense in which the Casey angle is preserved: if we take a photograph of the fence, or a photograph of the photograph, et cetera, the corresponding semicircles may vary but the measure of the Casey angle remains the same. In fact, this construction serves as a partial check on a perspective drawing of a fence: if the Casey angle corresponding to the four fence post tops isn t 60, then there is a mistake in the drawing. VOL. 43, NO. 5, NOVEMER 2012 THE COLLEGE MATHEMATICS JOURNAL 355

3 Casey s theorem Casey s theorem is based on the cross ratio. The cross ratio (A, CD) of four collinear points A,, C, D is given by (A, CD) = AC D C DA, where each quantity e.g., AC, is a directed distance. This means that a positive direction is arbitrarily assigned to the line, so that AC and CA have the same magnitude but opposite sign. Since there are an even number of directed distances in the expression, the number of negative directed distances remains even or odd if the positive direction is reversed, so the choice of the positive direction does not affect the cross ratio. A fundamental result of projective geometry is that the cross ratio is projectively invariant: if A,, C, D are four collinear points and A,, C, D are their images under a projective transformation, then A,, C, D are collinear and (A, CD) = (A, C D ). The invariance holds when the labels of the points are permuted, so there is more than one invariant numerical quantity associated with a given quadruple of collinear points. Although there are 4! = 24 ways to apply the labels A,, C, D to four points, it is well known that permuting the labels leads to at most six distinct values of the cross ratio. Specifically, if one labeling gives λ for the cross ratio, the set of all cross ratio values of the four points is = { λ, 1 λ, 1 λ, 1 1 λ, λ 1 λ, λ λ 1 }. (1) Casey s theorem expresses the six cross ratio values in (1) in terms of the six trigonometric functions and the Casey angle: Casey s Theorem. The set of the cross ratio values of four distinct collinear points can be written = (θ) = { sin 2 θ, cos 2 θ, csc 2 θ, sec 2 θ, tan 2 θ, cot 2 θ }, (2) where θ is the Casey angle of the four points. In textbooks the proof of Casey s theorem is typically left as an exercise (see for example [4, p. 93], [8, p. 194], or [6, p. 155]). The statement of the theorem may also include the fact that 2θ is the angle at O between (the tangents of) the two semicircles in Figure 3. We also leave the proof for the reader, with the following hint. The fact that (A, CD) = cot 2 θ follows using the formula for the cross ratio and the law of sines. The other elements of (θ) are generated using the formulas in (1), and basic trigonometric identities. It s also helpful to use the fact that the inscribed angles AO and COD are right angles. (Readers familiar with the cross ratio of four concurrent lines can take a shortcut to this proof.) As suggested by Figures 1 and 2, the Casey angle is invariant in perspective drawing and photography. To be more precise, we first state without proof that if A, C,, D are collinear points appearing in that order, then their respective images A, C,, D in a photograph or perspective drawing are collinear points appearing in the corresponding order. Thus, as mentioned earlier, we have both (A, CD) = cot 2 θ and (A, C D ) = cot 2 φ, where θ and φ are the respective Casey angles. The invariance of the cross ratio then gives cot 2 θ = cot 2 φ. It is easy to check that the function β cot 2 β is one-to-one on (0, π/2), and hence φ = θ. 356 THE MATHEMATICAL ASSOCIATION OF AMERICA

Photographs and drawings Photogrammetry is the science of extracting three-dimensional information from twodimensional photographic images.

4 Photographs and drawings Photogrammetry is the science of extracting three-dimensional information from twodimensional photographic images. Many techniques of photogrammetry also apply to perspective drawings and paintings. Here we present a simple application of the cross ratio to photogrammetry, with the Casey angle in a central role. Our starting point is the perspective image of two books lying on a table as in Figure 4. The goal is to determine what the indicated angle is between the actual physical books on the table, omitting the trivial cases in which the angle is 0 or 90. Calling this the actual angle, we phrase the question as follows: What is the actual angle between the books depicted in Figure 4?? Figure 4. What is the actual angle between the books? Simply measuring the image of the actual angle with a protractor will not give the correct answer, because of the distortion caused by the oblique point of view. This is not an angle preserved by projective transformations. To frame the correct answer, we show the same books in Figure 5, with their sides extended to pairs A, and C, D of vanishing points on the horizon line. ecause the actual angle is not 0 or θ A C D Figure 5. It is the Casey angle of the vanishing points! VOL. 43, NO. 5, NOVEMER 2012 THE COLLEGE MATHEMATICS JOURNAL 357

5 90, the vanishing points are distinct. Therefore the four points have a Casey angle θ. This angle is the solution we seek. To explain, we briefly review some concepts from the theory of perspective. Figure 6 is a bird s-eye view of an observer looking with one eye from the viewpoint E through a picture plane at a rectangle lying in the ground plane (or a plane parallel to the ground plane). In the balloon we have the observer s view of the image in the picture plane, including the vanishing points A, on the horizon line. ground plane rectangle in ground plane what the viewer sees parallel A picture plane parallel A E viewing semicircle parallel to ground plane Figure 6. Observer looking through a picture plane at a rectangle lying in the ground plane. The key idea is the relationship between the viewpoint, the vanishing points, and the rectangle in the ground plane. It is a basic fact of linear perspective that every perspective drawing or photograph has a unique viewpoint, from which the image is meant to be viewed with one eye [5]. This point is the center of projection when the real world is projected onto the picture plane. From this viewpoint, the line of sight to a vanishing point is parallel in space to the lines in the real world whose images converge to that point. Therefore, since adjacent sides of the rectangle in the ground plane (or a book on the table) meet at right angles, the lines of sight EA and E are mutually perpendicular and parallel to the ground plane. It follows from plane geometry that the viewpoint E lies on the semicircle A parallel to the ground plane, indicated in Figure 6. We call this semicircle the viewing semicircle. In Figure 7 we look down on an observer viewing a wall-sized version of the picture of the two books. As in Figure 5, the points A, are the vanishing points of the book on the right, and the points C, D are the vanishing points of the book on the left. The corresponding viewing semicircles, both parallel to the ground plane, determine the viewpoint E at their intersection. Two edges of the image of the left-hand book have been extended to meet C, and two edges of the image of the right-hand book have been extended to meet. Since the lines of sight EC and E are parallel to the 358 THE MATHEMATICAL ASSOCIATION OF AMERICA

Conclusion In the Renaissance, art lovers had their first experience of seeing realistic twodimensional representations of the real world, in the form of paintings and frescoes.

6 A C? D θ E Figure 7. The viewpoint E is the vertex of the Casey angle of the vanishing points A, C,, D. corresponding edges of the actual books, the actual angle between the books is the Casey angle θ = m( CE), as we claimed. Conclusion In the Renaissance, art lovers had their first experience of seeing realistic twodimensional representations of the real world, in the form of paintings and frescoes. Today our experience is flooded with the counterparts of these paintings: photographic images from newspapers, magazines, and the Internet, and we frequently make our own such images with digital cameras and cell phones. Embedded in these many images are interesting geometric relationships, including examples of projective invariants. One of the easiest invariants to visualize is the Casey angle, which can be represented as a single measurement with a protractor. In many cases we can use it to extract information about the real world despite the distortion of perspective and better understand the relationship of the viewpoint and vanishing points in perspective drawings. An appropriate selection of photographs, together with straightedges, compasses, and protractors, can serve as materials for interesting classroom experiments with the Casey angle. Such experiments reveal surprising mathematical relationships right in front of our eyes all the time and make an interesting counterpoint to the notion that angles have no place in projective geometry. On a historical note, John Casey is perhaps best known for another Casey s theorem, which is sometimes regarded as a generalization of Ptolemy s theorem (see for example [7]). The first prominent mathematician to investigate perspective with multiple vanishing points as used in this article was rook Taylor, who wrote two versions of a treatise on the subject for artists [9, 10]. For an interesting account of this work, see the excellent books by Kirsti Andersen [1, 2]. VOL. 43, NO. 5, NOVEMER 2012 THE COLLEGE MATHEMATICS JOURNAL 359

7 Summary. When a plane figure is photographed from different viewpoints, lengths and angles appear distorted. Hence it is often assumed that lengths, angles, protractors, and compasses have no place in projective geometry. Here we describe a sense in which certain angles are preserved by projective transformations. These angles can be constructed with compass and straightedge on existing projective images, giving insights into photography and perspective drawing. References 1. K. Andersen, rook Taylor s Work on Linear Perspective: A Study of Taylor s Role in the History of Perspective Geometry. Including Facsimiles of Taylor s Two ooks on Perspective, Springer-Verlag, New York, , The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge, Springer, New York, H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited, Mathematical Association of America, Washington DC, H. Eves, A Survey of Geometry, Allyn and acon, oston MA, M. Frantz and A. Crannell, Viewpoints: Mathematical Perspective and Fractal Geometry in Art, Princeton University Press, Princeton NJ, A. F. Horadam, A Guide to Undergraduate Projective Geometry, Pergamon Press, Australia, R. Johnson, Advanced Euclidean Geometry, Dover, Mineola NY, D. C. Kay, College Geometry, Holt, Rinehart and Winston, New York, Taylor, Linear Perspective, R. Knaplock, London, , New Principles of Linear Perspective, R. Knaplock, London, THE MATHEMATICAL ASSOCIATION OF AMERICA

Chapter 5. Drawing a cube. 5.1 One and two-point perspective. Math 4520, Spring 2015

Chapter 5. Drawing a cube. 5.1 One and two-point perspective. Math 4520, Spring 2015 Chapter 5 Drawing a cube Math 4520, Spring 2015 5.1 One and two-point perspective In Chapter 5 we saw how to calculate the center of vision and the viewing distance for a square in one or two-point perspective.