Two Lines Meeting at Infinity

If two lines separated by a finite distance meet at infinity, then what is the angle between them?

In the image example, the lines are separated by a distance of 17 squares, but why should this even have an effect?

Obviously the angle can't be more than \(180\) degrees, and since the \(3\) lines form a triangle the sum of all \(3\) angles can't be more than \(180\) degrees. Is the angle \(0\) or just infinitesimally small?

Are we even dealing with Euclidean geometry since the lines are meeting at infinity... or do we have to throw the regular rules out the window?

Or perhaps is the angle simply impossible to define?

Share your thoughts please :)

Note by Alex Delhumeau
2 years, 11 months ago

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Visually, it's a matter of perspective, (literally), but mathematically it might be best to analyze this using limits.

First suppose the two red lines meet at a finite distance from the foreground, in effect forming an isosceles triangle with "base" length \(17\) and side length \(x.\) Then by the Cosine rule we have that

\(17^{2} = x^{2} + x^{2} - 2x^{2}\cos(\theta) \Longrightarrow \cos(\theta) = 1 - \dfrac{289}{2x^{2}}.\)

Now as \(x \rightarrow \infty\) we see that \(\cos(\theta) \rightarrow 1,\) implying that \(\theta \rightarrow 0.\)

Brian Charlesworth - 2 years, 11 months ago

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We're not dealing with Euclidean geometry. This is about the projective geometry

Agnishom Chattopadhyay Staff - 2 years, 10 months ago

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