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From what I understand, the hyperbolic distance between two points in hyperbolic space is the length of the line(semicircle) that connects those two points.

The hyperbolic length then would be the sum of all the little tangents on any curve connecting those two points.

I am having trouble differentiating the two ideas and don't even know if they're right to be honest.

user338365
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    Think about the corresponding Euclidean concepts (Euclidean lines are straight lines instead of circles intersecting the boundary, of course). Distance is a thing associated to two points, and is the length of the shortest curve connecting them. The length of a curve is a property of the curve, and is given by an integral of arc-length. – Chappers Apr 05 '17 at 19:41

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While the line may appear to be a semi-circle in your model of the hyperbolic plane, it is nonetheless straight. If you were an ant walking on the hyperbolic surface you would not be aware that the line is not straight. So, there are no tangents to speak of. All the "tangents" lie on the same line.

Suppose you lived on a spherical surface. Oh, wait! you do!

And you followed the shortest distance from Los Angeles to London. You would take a route over Canada, and Greenland on the way to London. And if you looked at it on an Atlas, the line would appear to be curved. But, if you were the pilot on the route, it would feel like a straight line.

Doug M
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