Linguofreak
Well-known member
Minimalistic statement of problem to be solved:
Given a straight line (geodesic) A in hyperbolic space, a hypercycle H with A as its axis, and a second straight line B tangent to H, what is the separation between B and H as a function of distance from their point of tangency?
Actual problem to be solved:
We have a fantasy world set in a hyperbolic 3D space, with a "planet" consisting of a wall of rock of thickness T dividing the space in half, with continents and oceans and an atmosphere at the surface of the wall on either side. The center of the wall is a 2D hyperbolic plane with the same curvature as the 3D space, its surfaces on either side are 2D hyperbolic planes of a lower curvature than the overall space. Lines on these surfaces deviate from geodesics in the overall space, so the surfaces will appear to their inhabitants to curve downward, just like the Earth's surface does for us. To be precise, lines on the surfaces will be hypercycles in the overall space, and the surfaces themselves will be the 2D equivalent of a hypercycle. Hovering at some distance above the surfaces we have "stars" that provide light. The questions we can now ask are:
1) "What is the horizon distance for an observer O standing at height X above one surface of the planet?"
2) "What is the furthest point on the surface that a star S at height Y will illuminate?" (Essentially the same question with a different height)
3) "What is the furthest distance at which O can see S?" (should be the sum of 1) and 2) )
1) and 2) are basically two instances of the minimalistic problem statement above with different parameters.
Given a straight line (geodesic) A in hyperbolic space, a hypercycle H with A as its axis, and a second straight line B tangent to H, what is the separation between B and H as a function of distance from their point of tangency?
Actual problem to be solved:
We have a fantasy world set in a hyperbolic 3D space, with a "planet" consisting of a wall of rock of thickness T dividing the space in half, with continents and oceans and an atmosphere at the surface of the wall on either side. The center of the wall is a 2D hyperbolic plane with the same curvature as the 3D space, its surfaces on either side are 2D hyperbolic planes of a lower curvature than the overall space. Lines on these surfaces deviate from geodesics in the overall space, so the surfaces will appear to their inhabitants to curve downward, just like the Earth's surface does for us. To be precise, lines on the surfaces will be hypercycles in the overall space, and the surfaces themselves will be the 2D equivalent of a hypercycle. Hovering at some distance above the surfaces we have "stars" that provide light. The questions we can now ask are:
1) "What is the horizon distance for an observer O standing at height X above one surface of the planet?"
2) "What is the furthest point on the surface that a star S at height Y will illuminate?" (Essentially the same question with a different height)
3) "What is the furthest distance at which O can see S?" (should be the sum of 1) and 2) )
1) and 2) are basically two instances of the minimalistic problem statement above with different parameters.
