Projective geometry
Introduction
Dear friend, before starting I just want you to think for a second and answer the question: how do we perceive our three-dimensional world with a two-dimensional view? Are you sure you fully understand the “nature” of perspective? Why is the horizon inaccessible? Where do parallel lines intersect? What is infinity in geometry? In this article, I will try to answer these questions, without using scientific axioms. My goal is to teach you to love science and its beauty
For what it was invented?
2.1. Intro
In this short chapter, I would like to define the problem: why did mathematicians come up with projective geometry, and what is it all about? As I have mentioned earlier — we are living in a 3D space and perceive the information with a 2D view. What you are seeing now is just a projection of our world on the retina of your eye. Any object is two-dimensional for us and only by rotating this object, we can initialize its size (volume) of it. Oddly enough, our eyes are a great example of a projective plane.
2.2. Requirements
With the development of technology requirement of interaction with our world through the camera became using algorithms based on geometry. The answer wasn’t coming for long and mathematicians care about us long ago and invented the projective geometry which is universally used in telephones, computers, computer vision, and 3D graphics.
For example, when working with 3D graphics, the visible by us image on the screen — is a mathematical projection of the object. If you think about it, this is just a code and numbers mapped into human-readable form. The computer saves perspective by using projective mapping, in other words, the projection of the main cube on the screen depends on the position of the main 3D cube since the vertices of the cube are “linked” to the relative projection point Q’ and the monitor is the projection plane. Conventionally, by moving away or reducing the distance of the projection point from the monitor, you can adjust the scale of the object. (fig.2)
What about telephones? Nowadays are popular mobile applications that scan documents regardless of camera angle view. What about robot orientation equipped with cameras in space? Shortly speaking, projective geometry is used wherever analysis from the image is present.
2.3. Where did it come from?
Projective space based on affine space. If it doesn’t matter to you, affine space is what learned in school. Affine geometry includes dots and lines. Agree, it’s pretty limited. Therefore, the projective geometry was supplemented with points and lines at infinity and became a powerful instrument. About infinity, we will talk later, but worth to highlight — thanks to infinity, now any lines can intersect — even parallel lines.
Projection among us
3.1. This is not a repetition of the original, but its transformation or what is a perspective
In total, the projective geometry is mapped to the projection on the plane. Worth to highlight that projection — is not a repetition of the original, but its transformation. For a clear understanding, let’s consider the railways — an example from our life. Rails are parallel, but they converge at a horizon according to the position of the observer. Amazing, isn’t it?
Parallel lines intersect! Your geometry teacher says you’re crazy. Nonetheless, it’s true. How to describe it with geometry? Usual (Euclidian or affine) geometry doesn’t work with this, therefore projective geometry is to the rescue. Consider the plane (the surface of the earth). Now place point Q in space outside found the plane (is the observer). Let’s project rails onto a projective plane. Each point (imagine that rails are infinity) projects onto the projective plane according to the position of the observer. (fig.3)
Parallel lines at the projective plane α’ become not parallel. The point of intersection is the ideal point 𝐼′. Therefore, any lines intersect at infinity. This leads us to the conclusion that angles aren’t conserved on the projective plane, e.g., on the plane α rails (lines) are parallel (angle between lines is 0°), but in the projective plane angle > 0°.
Perspective — is a part of our life without which it’s impossible to perceive the world. It is fair to say that projection is only a synonym for projection.
Let’s meet at infinity or where do parallel lines intersect?
4.1. Proof of the intersection of parallel lines
In the projective plane absolutely, all lines intersect — even parallels. Now we are proof of it. We need to go up one dimension — from 2D into 3D. We already did it for example with rails.
Consider two parallel planes 𝑝 and 𝑝′. Parallel lines 𝑚 and 𝑛 belong to the plane 𝑝, and point 𝑄′ belongs to the plane 𝑝′ is the origin of the spherical coordinate system with surface 𝛼′. (fig.4)
The projection of lines is the projection of all points belonging to 𝑚 and 𝑛 onto the plane of the sphere, that is, onto the plane 𝛼! relative to the circle center 𝑄′. The farther we move away from the ball, the closer we approach infinity, in other words, the points are infinitely receding — this is very important to understand. The line connecting of infinity point with the center of the sphere will approach the equator (great circle). Each subsequent line will approach the equator even closer. At some point, they will be so close to the equator that it can be argued that they lie in the a plane, and therefore the point is at infinity — is at infinity. The projection of the (blue) line will also converge at this point. In the figure, this convergence is marked with a red line.
4.2. Ideal points 𝑰’ and 𝑰’’
The point at infinity is just a conventional description of infinity. When the projection points are so close to each other, they begin to resemble convergence at one point, which is more convenient to designate as ideal.
We’ve proofed the intersection of parallel lines at infinity into the projective plane at one ideal point, which belongs to the equator (great circle). Note: there are positive and negative infinity — 𝐼’ and 𝐼’’ respectively.
4.3. Horizon
Any point in a great circle (equator) is an ideal point 𝐼’. A set of points at infinity creates the line at
infinity (horizon). You guessed it, the horizon is the equator. That can be perfectly visualized with a panoramic camera (360°). I’d like to say that horizon it’s just a formal definition of all points at infinity, thus it’s not material and is inaccessible.
All parallel lines intersect in infinity, like several roadways extending towards the horizon. It’s worth noting that only parallel lines can intersect in infinity, in other words, not parallel lines have a different point of intersection in infinity.
Test
- What makes Projective geometry different from Euclidian?
a. Projective geometry is infinite
b. Projective geometry includes only points and lines at infinity
c. Projective geometry additionally includes lines and points at infinity - What is another name for perspective?
a. Perspective — the projection of 3D space onto a 2D plane
b. Perspective — is a formal definition of the convergence of all objects at one point. - Where is projective geometry used?
a. Only in scientific books
b. Only in 3D
c. Where you need to analyze/display objects based on photos (videos/code) - Where do parallel lines intersect?
a. In the affine plane
b. In (ideal) point at infinity
c. In (ideal) point at infinity in the projective plane - Is an ideal point (point of intersection of parallel lines) in the projective plane a projection of some point on the affine plane?
a. Yes, it is
b. No, it isn’t, because the parallel lines don’t intersect in the reality
Key: 1 — С ; 2 — A; 3 — C; 4 — C; 5 — B; 6 — A.
Bibliography
[1]. Basic ideas of projective geometry, O.A. Volberg, Edition 3-e. State Educational and Pedagogical Publishing House of the Ministry of Education of the RSFSR, Moscow, 1949, Leningrad.
[2]. Foundations of projective geometry, Robin Hartshorne, W.A. Benjamin, INC., New York, 1967.
