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 180180 degrees, and since the 33 lines form a triangle the sum of all 33 angles can't be more than 180180 degrees. Is the angle 00 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 :)

#Geometry #Distance #Angles #Infinity #Note

Note by Alex Delhumeau
5 years, 10 months ago

No vote yet
1 vote

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Comments

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 1717 and side length x.x. Then by the Cosine rule we have that

172=x2+x2−2x2cos⁡(θ)⟹cos⁡(θ)=1−2892x2.17^{2} = x^{2} + x^{2} - 2x^{2}\cos(\theta) \Longrightarrow \cos(\theta) = 1 - \dfrac{289}{2x^{2}}.

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

Brian Charlesworth - 5 years, 10 months ago

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

Agnishom Chattopadhyay - 5 years, 10 months ago

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?

In affine and Euclidean geometry, it is postulated that two parallel lines DO NOT MEET. You now have to extend your framework to projective geometry, where you can have two lines meeting at a "point in infinity"... because of this, everything you just wrote is meaningless.

A Former Brilliant Member - 2 years, 10 months ago
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