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I have 2 focal points of ellipse defined on a sphere: $F_1 = (q_1, p_1)$ and $F_2 = (q_2, p_2)$ and length of major axis $2a$. $R$ of a sphere is 1.

$q_1, q_2$ are latitudes

$p_1, p_2$ are longitudes

I need to find 4 points:

  • point on ellipse with max latitude
  • point on ellipse with min latitude
  • point on ellipse with max longitude
  • point on ellipse with min longitude

I tried to convert points to vectors and use a property that sum of angles $$\sphericalangle F_1OP + \sphericalangle F_2OP$$ is constant for each $P$ on ellipse, but it lead me nowhere. Any idea how it can be solved?

KReiser
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kosmo16
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    What is an ellipse on a sphere? – H. Gutsche Jul 20 '18 at 16:52
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    Define an ellipse as the locus of points which is a constant summed distance away from two focal points, whenever the distance is calculated as the shortest path along the surface of the sphere. I believe this turns out to be the intersection of a cylinder and a sphere. – Narlin Jul 20 '18 at 17:43
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    Alas @Narlin, I do not think it is that simple. I believe that the problem is soluble with spherical trig but I will need a bigger block of time to formulate an answer. – Oscar Lanzi Jul 20 '18 at 21:06
  • As @Narlin mentioned ellipse is the locus of points which is a constant summed distance away from two focal points. – kosmo16 Jul 24 '18 at 08:23
  • @OscarLanzi Can you give me a clue how to approach this problem, maybe I will be able to find the answer by myself. – kosmo16 Aug 01 '18 at 13:09

1 Answers1

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A partial answer based on a comment:

Let $A, B$ be two points on the sphere (Earth) with radius taken as unity. Define $p(A), q(A)$ as the latitude and longitude of $A$ on the Earth, analogously for other points. Define $N$ as the North Pole with $p(N)=+\pi/2, q(N)$ arbitrary.

Draw $\triangle ABN$. Then $|AN|=\pi/2-p(A), |BN|=\pi/2-p(B), |\angle N|=\pm (q(A)-q(B))$. Now from the Law of Cosines

$\cos(|AB|)=\sin(p(A))\sin(p(B))+\cos(p(A))\cos(p(B))\cos(q(A)-q(B))$.

(As an exercise you may want to derive the same formula replacing $N$ with $S$ where$p(S)=-\pi/2$. Use the identity $\sin(\pi/2+\theta)=\sin(\pi/2-\theta)$.) Use this formula for the distance from any point on the ellipse to the foci, adding up the inverse cosines to equal the major axis.

Good luck!

Oscar Lanzi
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