Recent content by nixed

I forgot to post the attached picture of all great circles from a point which cross a given latitude.

Here is another helpful picture of great circles tangent to a given latitude

The analysis is shown in the attached sheet for those interested in following the details.

Given two points A(θA,øA) and B(ΘB,øB) and a latitude restriction Θrim how do we tell if the great circle between A and B will rise to higher latitudes than Θrim? The parametric form of the great circle passing through two points was derived in the post "great circle problem". The z(ψ)/R...

Consider the attached diagram If point B is on the sphere surface outside the shaded spherical triangle, then the shortest route B to A is the shorter arc of the the great circle passing through A and B. (N.B. there is always one and only one such great circle for a given A and B unless they...

The general solution for the shortest path from A to H case 3 is a path A-C-P-C'-H via point P(∏/2) on the rim as discussed above. The critical point C where the great circle from A just touches the rim tangentially occurs at an angle ψc around the great circle from A and at an angle øc around...

What then is the shortest path between any two points A and B on a sphere, if the shorter arc of the great circle connecting them is interrupted by a restriction that we cannot go above a certain latitude? The insight from Case 3a above may suggest the answer.

OK thanks ! By the way I discovered that my assumed shortest path is only a close approximation to the true shortest path so the proof I was attempting based on the inequality is invalid! This is the shortest path when the great circle from A to H is disrupted by a latitude restriction imposed...

For case 3a the Ant in the northern hemisphere I think I understand. The shortest route always passes through the P(∏/2) point on the rim. If we imagine the initial ant position as the pole of a sphere radiating 'longitude lines' (all the great circles from A) we are only interested in those...

I am trying to solve the 'ant and honey problem on a spherical bowl' to find the shortest route between two points on a sphere when the path is constrained by not being allowed to pass higher than a certain latitude (so interrupting some great circles connecting the two points). I intuitively...

NOT True! The route on the great circle going straight up to the rim then tracking down through the bowl is not the shortest route between A and H! The shortest route over the rim and to H must leave the plane of AOH- the question is with the constraint of not being able to walk paths that go...

By depth of bowl ,D, I mean the vertical distance from the rim to the bottom of the bowl so that 0≤ D ≤ 2R cover all the possible depths. The bowl is always a part of a sphere of radius R, just that case 1 is a sort of dish, case2 a hemisphere and case 3 a larger bowl that is more than a...

In considering the shortest paths between two points on a sphere I came across the following interesting problem: An ant sits on the outside of a glass bowl of spherical curvature (radius R), at a distance d from the lip of the bowl. It sees a drop of honey on the inside of the bowl directly...

Yes, I noticed the age of the thread, but I posted because I have been working on the 'ant and honey' problem on a spherical bowl and needed to work out the great circles going through given points on a sphere and couldn't find the information easily on-line so had to derive it. Having done so...

So the answer to the original problem: A(2,10,11) B(14,5,2) both are points on sphere radius 15. So a= (2/15,10/15, 11/15) and b=(14/15,5/15,2/15) are unit vectors which give us the components ax, ay etc. ∴Cosδ = (28+ 50+22)/152 ∴ δ = 1.1102 rad = 63.61 deg the shortest arc length along...