Where do Two Parallel Lines Meet?

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Geometry is one of the ancient branches of mathematics, yet it is the most advanced branch today. The perceptive growth of geometry provided the foundation to many scientific theories and discoveries. Euclid in Alexandria wrote Elements, a mathematical exposition (comprised on 13 volumes) in 300 BC. Euclid’s Elements has been referred as the most successful and influential books ever written. Euclid gave a systematic way to study planar geometry, prescribing five postulates of Euclidean geometry. The study of Euclidean spaces is the generalization of the concept to Euclidean planar geometry, based on the description of the shortest distance between the two points through the straight line passing through these two points. Mathematicians believed that the Euclidean geometry is the ultimate geometry for many centuries, based on the following five simple postulates:

  1. A straight line segment can be drawn joining any two points.
  2. Any straight line segment can be extended indefinitely in a straight line.
  3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.
  4. All right angles are congruent.
  5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.
5th postulate of the Euclidean Geometry

Later, Sir Issac Newton (1642–1727) built the whole calculus and laws of physics describing the dynamics of a moving particle in the Euclidean space, using the Euclidean geometry. He recorded all his work on describing the laws of physics to his logical mathematical framework in his book, Philosophiae  Naturalis Principia Mathematica (three volumes) also known as, Newton’s Principia Mathematica. The Newton’s Principia Mathematica took the shape of calculus, well acknowledged today.

The fifth postulate of the Euclidean geometry presented itself as a nuisance for many years. However, the fifth postulate later appeared as the motive behind the edifice of an entirely new kind of geometry, namely the non-Euclidean Geometry. Bernard Riemann (1826–1866) explained the limitations of these postulates (in particular the fifth postulate) and introduced the concept of non-Euclidean geometry. He developed the geometry of surfaces that are not essentially flat, for instance the surface of sphere. The lines on the surface of sphere behaved differently as compared with the lines on a flat surface. In particular, they do not follow the fifth postulate as the longitude lines are parallel at equator, yet they intersect at poles. Later on, Albert Einstein (1879 –1955) used the non-Euclidean geometry to describe his theory of general relativity.

Parallel lines on sphere

Abstract study of geometry through algebraic and topological characterization led to many important discoveries in mathematics and theoretical physics of today. The proof of Fermat’s Last Theorem by Sir Andrew Wiles (an English mathematician born in 1953) was the most significant development. The proof was based on many advanced techniques of abstract algebraic geometry. Algebraic geometry is proven to be the key area for many important developments of the recent times. The solutions’ space of linear systems of polynomial equations (studied in linear algebra) generalizes the character of Euclidean spaces, namely, the affine spaces. The linear algebra holds the key to understanding many important aspects of Euclidean spaces as well as affine spaces. 

One significant category of abstract spaces is the projective spaces. At the moment, I confine myself with the real projective spaces, only.  These spaces are formulated through the real Euclidean spaces in a systematic way. A real projective space of dimension ‘n’ is denoted by \mathbb{R}P^n and obtained through the Euclidean space of dimension \mathbb{R}^{n+1} excluding its origin. Each element of real projective space \mathbb{R}P^n represents a line in \mathbb{R}^{n+1}/{origin} and denoted as [x_{0}:x_{1}:...:x_{n}]. Now we extend the study of linear algebra in real projective spaces. The ratio between the coordinates implies that if (c_{0},c_{1},...,c_{n}) is a zero of a polynomial then so is any multiple of it, that is P(c_{0},c_{1},...,c_{n})=P(\alpha c_{0},\alpha c_{1},...,\alpha c_{n})=0 . It is concluded that we may only view the solutions of linear homogeneous polynomials in \mathbb{R}P^n, and these solutions correspond to linear subspaces of \mathbb{R}^{n+1}, called affine cones. One strange behavior of a projective space is that it contains all the infinite points of the Euclidean spaces. In particular, \mathbb{R}P^{2} represents all straight lines in \mathbb{R}^{2} and its points at infinities. In order to understand this concept mathematically, consider the equation of a line ‘L’ in \mathbb{R}^{2}.

1+ax_{1}+bx_{2}=0

The associated homogeneous polynomial

This process is known as homogenization. The line ‘L’ corresponds to the projective line (I) in \mathbb{R}P^2 . The projective line (I) intersects the line  x_{0}=0 in \mathbb{R}P^2 at the point [0:-b:a]. If b\neq 0, then it is easy to see that

[0:-b:a]=[0:1:\frac{-a}{b}]

Note that \frac{-a}{b} is the slope of line L. Hence the point at which any line intersects x_{0}=0 depends only on the slope of the line. This implies that any two parallel lines in \mathbb{R}^{2} will meet at one point in \mathbb{R}P^{2}. This point is known as the point at infinity. That’s why we can think of \mathbb{R}P^{2} as \mathbb{R}^{2} added infinity at each direction in \mathbb{R}^{2}. Hyperbolic geometry came out by the assumption that parallel line will meet at infinity.

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