We begin our lecture series by describing the technique of perspective drawing which was responsible for popularizing projective geometry.
We use an image of railway tracks that appear to converge upon the horizon to introduce the idea of perspective drawing. We then describe the classic drawing technique used in the 1400's and investigate its consequences logically. In particular we show that parallel lines which are perpendicular to the picture plane will appear to meet at a special place within the horizon of the image (the principle vanishing point). We also give a proof of how parallel lines always meet somewhere upon the horizon within pictures produced via projective drawing. It is important to have a proof of this foundational fact about perspective drawing.
We also use Cartesian geometry to derive an expression for where the images of particular objects within the scene will get projected to within the picture plane. We use the formula we derive to examine the image a parabola in the scene. Remarkably we find the this parabola is projected onto a circle within the picture plane. By examining the underlying mathematics we obtain a rational parametrization of the circle which is rather similar to the one described within the video:
We are initiating our lecture series by explaining how the subject of projective geometry got popularized. Later we shall discuss how the study of projective geometry related to many other subjects including ancient Greek mathematics (e.g., Pappus's Theorem), conic sections, linear algebra, topology, and more specific types of geometry such as Euclidean, spherical and hyperbolic geometry.
Protective geometry is deeper and more fundamental than standard euclidean geometry, and has many applications in fundamental physics, biology and perspective drawing. We shall introduce it visually, without relying upon equations. The hope is make this beautiful subject accessible to anybody, without requiring prior knowledge of mathematics. At the same time, there are some very deep, rarely discussed ideas in this subject which could also benefit experts.