Can two parallel lines meet?

As against the common notion in euclidean space, where it is generally known and believe that two parallel lines cannot meet, since the angle between them is zero (0°) unless one parallel line is skew against the other, in that case, the angle between them will definitely be greater than zero (0°). In projective space, it is not so, two parallel lines can intersect by meeting at the point called infinity.

_{Image from: commons.wikimedia}

To develop a real life athletics gaming application, like a car race, a developer must definitely have to assume that two parallel lines will meet at a point called ‘infinity’.

In order to get a similar idea of what am talking about, if you are playing a car race game for instance, at far road sides, both sides must be programmed in such away that it makes both sides narrower when it moves far away and as if the roads will collide with one and other, similar occurrence of this in real life is the train railroad in the picture above, it becomes narrower while it tends or moves far away from human eyes and finally the two parallel railroads later meet at the horizon, which is a point at infinity.

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Why Projective Space

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No doubt, euclidean space or Cartesian space describe our two dimensions (2D) and three dimensions (3D) geometry very well, however, they are not sufficient enough to handle the projective space since euclidean geometry is always a subset of projective geometry. Also, the fact that when dealing with athletics gaming, an object always goes far away from eyes and to the point called infinity (∞,∞), this becomes meaningless in Euclidean space.

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Homogeneous Coordinates

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In order to make calculations of computer graphics and geometry possible in projective space, mathematicians have discovered a way to solve this issue and this is what is led to 'Homogeneous Coordinates'

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August Ferdinand Möbius was the first mathematician to introduce homogeneous coordinates into projective geometry. He proposed that in order to make calculations of graphics and geometry possible in projective space, then homogeneous coordinates must be represented with (N+1) numbers.

For instance, to make 2D Homogeneous coordinates, one will only add an additional variable, say w, into amn already existing coordinates. This results in making a point in cartesian coordinates (x,y) to becomes (x, y, w) in Homogeneous coordinates and X and Y in Cartesian are represented with x, y and w in Homogeneous as;
X = x/w
Y = y/w

For example, a point in cartesian coordinate with (1, 2) represents (1, 2, 1) in Homogeneous coordinate, whenever a point say (1, 2), tends towards infinity, it becomes (∞,∞) in cartesian coordinates and (1, 2, 0) in Homogeneous coordinates, since X = x/w, Y = y/w

thereby of (1/0, 2/0) = (∞,∞)

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Mathematical proof: Two parallel lines can meet

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_{Image from: texample.net}

Proof: Two parallel lines can meet

putting into consideration, the following linear algebra system in Euclidean space;
Ax+By+C = 0
Ax+By+D = 0
since there is no solution for above equations because of C ≠ D, else If C = D, then two lines are identical (overlapped).
By replacing x and y to x/w, y/w in a projective space respectively, it becomes
A(x/w)+B(y/w)+C = 0
A(x/w)+B(y/w)+D = 0
multiplying through by w, we have
Ax+By+Cw = 0
Ax+By+Dw = 0

Applying elimination method, we have
Cw-Dw = 0
w(C-D) = 0
w = 0 & C-D = 0
Thereby, we have a solution, (x, y, 0) since (C - D)w = 0, ∴ w = 0, C = D, Therefore, two parallel lines meet at (x, y, 0), which is the point at infinity.

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Conclussion

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In conclusion, with the poof, explanation and the references quoted, there are no reasons to think that parallel lines cannot intersect, especially in projective space where Homogeneous coordinates are very useful and fundamental concept in computer graphics, such as projecting a 3D scene onto a 2D plane.

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References
_{1.Programmer’s guide to homogeneous coordinates}.
<sub>2. Two parallel lines can intersec.
3. Do Parallel Lines Meet At Infinity?
4. August Ferdinand Möbius
5. Weisstein, Eric W. "Homogeneous Coordinates." From MathWorld--A Wolfram Web Resource.</sub>

_{*All images are from free source websites}

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