The Fundamental Theorem of Projective Geometry and the Desargues Theorem in Projective Geometry
By Luotong He
Published July 23, 2023
Abstract
This research project will briefly present projective geometry and discuss two main theorems in this geometry. The two theorems that will be discussed are the fundamental theorem of projective geometry and the Desargues theorem. Beyond presenting the theorems themselves, this article will also showcase the proofs of these two theorems.
Projective Geometry
Projective geometry can be found in many unexpected situations and contexts, such as the converging parallel lines of train tracks, the beautiful view of a coast, or even a flat piece of paper sitting on a table. People like Pappus of Alexandria have been studying this topic since the fourth century [1]. A comparison with Euclidean geometry reveals many differences and similarities between the two. Similarities are to be expected, as many theorems in projective geometry are extensions of the idea of points and lines that are defined in Euclidean geometry [2]. Projective geometry mainly focuses on the relationships between points and lines, whereas Euclidean geometry emphasizes other relationships between lines and points, such as perpendicularity [3]. A specific example of a difference between the two is the existence of parallel lines. Projective geometry states that lines will eventually converge to each other, and parallel lines do not exist, whereas Euclidean geometry defines their existence within axioms [4].
The Fundamental Theorem of Projective Geometry
The fundamental theorem of projective geometry was introduced by German mathematician Karl Georg Christian von Staudt in the 19th century. He contributed to many studies of geometry, such as the theory of conics and the development of the theory of algebraic curves. Beyond that, the fundamental theorem of projective geometry is also known as “Staudt’s theorem”, and this theorem helped many people understand more about the basics of projective geometry [7]. The theorem states that projectivity is determined when given three collinear points and the corresponding three collinear points [5]. Collinearity is one of the basic concepts of geometry in general. Three or more points are said to be collinear if they all lie on the same straight line.
Projectivity is also a basic concept within projective geometry, but before knowing what projectivity is, other concepts are important for understanding this theorem and projective geometry in general. The term pencil can be seen in many theorems of projective geometry; a pencil is a set of lines that pass through the same point. The term perspectivity is another basic concept that can be defined as a transformation/mapping between points and lines, whereas projectivity is a sequence of perspectivities [10]. Imagine there is a paper sitting on the table; any action performed on the paper is considered a kind of transformation. In this scenario, projectivity is a specific way of manipulating the paper that forms a certain relationship with the original paper. Now that we understand some of the basic concepts, it would be helpful to fully understand the theorem itself. As mentioned, perspectivity is a transformation/mapping; therefore, the difference between the two sets of collinear points stated in the theorem is the perspectivity that each collinear point forms. Let L and L’ be two distinct lines with L having four collinear points A, B, C, and D, and L’ having three collinear points A’, B’, and C’ (see Figure 1 step 1). By connecting point A to point A’, B’, and C’, perspectivity has been formed, since there is a mapping from point A to the points on line L’ (see Figure 1 step 2). Now repeat the same for point A’ by connecting point A’ to points on line L, which are points A, B, C, and D (see Figure 1 step 3). Let G and H be the intersecting points between these lines that A and A’ formed. Now extend GH to find two other intersection points (see Figure 1 step 4). Now a new line L” has been formed and has four collinear points. (see Figure 1 step 5). Since this new line was formed, two perspectivities have been found: A’ is perspective to L” and the points on L” are perspective to the points on L. A trivial case is shown by A’ perspective of F and F being a perspectivity of A. As stated previously, projectivity is simply a sequence of perspectivities; therefore A is projective to A’. Beyond that, connecting points A and I and extending them to line L’ will result in another point D’ (see Figure 1 step 6), which then shows projectivity as well, since every connection line has three collinear points [6].
Desargues Theorem
French mathematician Girard Desargues is considered one of the people that helped the development of projective geometry, and one of his many discoveries was the Desargues theorem [8]. With the use of this theorem, people were not just able to see the relationship between two triangles under the same perspective, but the theorem also revealed a new mathematical perspective on objects. The theorem states: In the real projective plane, two triangles’ perspectives from a point are the perspectivity from a line [9]. An important concept regarding this theorem is the idea of perspective or perspectivity. Imagine there is a pair of eyes, called the centre point. If any other points are seen by the center point, then this means perspectivity has been formed. There are many ways of proving this theorem, including using other theorems such as the Pappus Hexagon Theorem, or applying linear algebra knowledge, but it can also be shown directly. Given two triangles ABC and A’B’C’(see Figure 2 step 1), assume that they are being observed by a point O, or saying that point O is perspective to both triangles, as AA’, BB’, and CC’ are intersecting to point O (see Figure 2 step 2). When extending both triangles, there will be some intersection points being formed. Let R be the intersection of AB and A’B’, let S be the intersection of AC and A’C’, and let T be the intersection of BC and B’C’ (see Figure 2 step 3). They are also collinear by connecting the intersection points since they can form a straight line (see Figure 2 step 4). As mentioned, perspectivity can be interpreted as a pair of eyes. Now there are three pair of eyes, with the eyes being the intersection points R, S, and T. Since these eyes are on the same line, all these eyes can see both triangles because they are at the intersection points of the extension of the side of the triangle, therefore both triangles are perspective from the line, which means that the theorem holds.
Conclusion
Projective geometry is a topic that has been studied for many years. From Pappus of Alexandria in the fourth century, to Girard Desargues during the 17th century, and Karl Georg Christian von Staudt during the 19th century, the study of projective geometry never ends. Projective geometry provides a different view of geometry than Euclidean geometry and challenges many of its theorems and properties. Even though projective geometry requires a much stronger understanding of its theorem and properties, it is still a very interesting topic to study, as it has so many real-life applications.
Reference
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[2] Sauer, T., Sch¨utz, T. Einstein on involutions in projective geometry. Arch. Hist. Exact Sci. 75, 523–555 (2021). https://doi.org/10.1007/s00407-020-00270-z
[3] Weisstein, Eric W. ”Euclid’s Postulates.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/EuclidsPostulates.html
[4] Weisstein, Eric W.”Projective Geometry.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/ProjectiveGeometry.html
[5] Beauville, Arnaud. ”Fundamental Theorem of Projective Geometry.” From MathWorld–A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FundamentalTheoremofProjectiveGeometry.html
[6] C. H. S. M., Introduction to geometry, second edition. New York: Wiley, 1989. Pp228 -241
[7] E. A. Marchisotto, “The projective geometry of Mario Pieri: A legacy of Georg Karl Christian von Staudt,” Historia Mathematica, vol. 33, no. 3, pp. 277–314, 2006.
[8] B. A. Swinden, “Geometry and Girard Desargues,” The Mathematical Gazette, vol. 34, no. 310, pp. 253–260, 1950.
[9] Weisstein, Eric W. ”Desargues’ Theorem.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/DesarguesTheorem.html