Exploring the Theorems of Pappus and Desargues in Projective Geometry
By Bonnie Yam
Published March 14th, 2022
Projective Geometry
The popularity of projective geometry throughout the centuries was like a rollercoaster. Projective geometry started in the 4th century, became a prominent tool in Renaissance art, and its theorems are now heavily studied in mathematics. In contrast to Euclidean geometry that focuses on distances and angles, projective geometry does not study these quantities because distances and angles are not preserved by projection [1]. Projective geometry emphasizes on geometric images that are preserved by projection, such as lines and points [1].
Menelaus’ Theorem
Let A and B be distinct points on the line L. The signed ratio of the point C on L is defined as:
Menelaus’ Theorem: Suppose ∆𝐴𝐵𝐶 is a triangle. Let point X be on AB, point Y be on BC, and point Z on CA. The points X, Y, and Z are collinear if and only if
The Menelaus’ theorem will be used to prove the theorems of Pappus and Desargues, which are significant in projective geometry.
Pappus’ Theorem
The Greek geometer Pappus of Alexandra wrote his famous theorem in 320 AD, and it became a starting point for projective geometry.
Pappus’ Theorem: Suppose the points A, B, and C are on the line L. Suppose the points A’, B’, and C’ are on the line L’. If AB’ meet A’B at point X, AC’ meet A’C at point Y, and BC’ meet B’C at point Z, then the points X, Y, and Z are collinear.
Proof:
Suppose the lines AB’, BC’, and A’C form a triangle ∆𝑁𝑃𝑀. By the Menelaus’ theorem, the following holds true:
Divide the product of the first three expressions by the product of the last two expressions
Then by the Menelaus’ theorem, the points X, Y, and Z are collinear.
In the above proof, what happens if the lines AB’ and BC’ are parallel? In Euclidean geometry, we would think these lines will not be able to intersect and form the triangle. However, projective geometry is not concerned with the existence of parallelism and states that all pairs of lines meet at a point (including the point at infinity), thus it introduces the duality between lines and points [4]. The dual diagrams of Pappus have a total of nine lines and nine points with three points on each line and three lines intersecting each point [4]. Since the property of three collinear points is satisfied in the configuration depicted in the proof above, it will remain true for its dual diagrams.
Desargues’ Theorem
The French architect Girard Desargues triggered the development of projective geometry with his great discovery in 1600s about perspective triangles.
Desargues’ Theorem: Suppose the triangles ∆𝐴𝐵𝐶 and ∆𝐴′𝐵′𝐶′ are perspective from point P. If the corresponding sides of the triangles intersect (i.e. AC meet A’C’ at point X, BC meet B’C’ at point Y, and BA meet B’A’ at point Z), then the points of intersection X, Y, and Z are collinear.
The dual of the Desargues’ theorem is its converse: if the
triangles ∆𝐴𝐵𝐶 and ∆𝐴′𝐵′𝐶′are perspective from a line and each
pair of corresponding vertices lie on a line (i.e. A and A’ lie on L, B and B’ lie on L’, C and C’ lie on L’’), then the two triangles are perspective from the point of intersection of these three lines [3]. The dual diagrams of Desargues have a total of ten lines and ten points with three points on each line and three lines intersecting each point [4].
Although Pappus and Desargues lived in different generations, there exists a relation between their findings: the Pappus’ theorem implies the Desargues’ theorem, but not vice-versa [5].
Conclusion
Projective geometry may be difficult to grasp because it challenges the traditional properties of geometry. Nonetheless, it is an interesting mathematical topic to study, and it is found in a variety of applications: architecture, computer vision, and perspective art.
References
[1] J. Stillwell, Mathematics and Its History, 3rd ed. New York: Springer, 2010. [E-book] Available: SpringerLink. doi: 10.1007/978-1-4419-6053-5.
[2] Z. Shahbazi. MATD02. “Fall 2021 Assignment #1.” University of Toronto Scarborough, Scarborough, ON, n.d.
[3] H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited. Washington, D.C: Mathematical Association of America, 1967.
[4] J. Barnes, Gems of Geometry, 1st ed. Heidelberg: Springer, 2009. [E-book] Available: SpringerLink. doi: 10.1007/978-3-642-05092-3.
[5] E.A. Lord, Symmetry and Pattern in Projective Geometry, 1st ed. London: Springer, 2013. [E- book] Available: SpringerLink. doi: 10.1007/978-1-4471-4631-5