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distance

In any direction of the sphere, What is the distance from a point to a circle?Is there a vertical relationship between the planes of its circle?

There is a point $A$ and a circle $C$ on the sphere. Draw a big circle $D$ from point $A$. There are countless small circles $c$ that pass through point $A$ and tangent to $D$. Some of them will intersect with $C$. The intersection point is $B$. What is the shortest small arc from point $A$ to $B$? Its plane is perpendicular to the $C$ plane, regardless of the direction of $D$?

I guess they are always perpendicular to each other.

On the sphere, my work requires me to find the distance from the outer point of the circle to the circle, and the distance must be in a certain direction.

Conclusion:

Mr. Aretino has proved that when AB is the shortest, AB and C are not vertical. (2) What is the mathematical significance of this conclusion? Are there different shortest directions on the sphere? What is the relationship between this and the shortest distance from a point on the plane to a line in different directions?

enter image description here

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  • $\begingroup$ Could you please introduce some regulation on naming? $A$ is a point, $B$ represents a family of points, $C,D$ are circles, but also $c$ are circles... If would be a lot easier if, for example, big letters are points and small letters are circles. If you can then also give them some sort of intuitive name, like for example $B_c$ the point on $c$, that would help a lot with understanding your question. $\endgroup$ – Dirk Apr 16 '19 at 6:46
  • $\begingroup$ Is $D$ any great circle through $A$? Or is it perpendicular to the great circle through $A$ perpendicular to $C$? $\endgroup$ – Intelligenti pauca Apr 16 '19 at 13:09
  • $\begingroup$ Some experimentation with GeoGebra suggests the shortest path is NOT, in general, perpendicular to $C$. $\endgroup$ – Intelligenti pauca Apr 16 '19 at 13:27
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    $\begingroup$ I'm afraid a full answer would involve a lot of tedious computations, which I'm not going to do right now. However I created a file with GeoGebra which allows one to get a feeling of the solution, for various positions of $A$: I'll upload it in a few minutes. $\endgroup$ – Intelligenti pauca Apr 18 '19 at 13:45
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    $\begingroup$ Here's the link to experiment with GeoGebra: ggbm.at/pnxcjy96. You must move point B until you get the minimum length for arc AB. $\endgroup$ – Intelligenti pauca Apr 18 '19 at 14:00
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@Aretino gave the following interesting answer, but he did not give the shortest AB plane and C plane is vertical, hope he or others can give this conclusion.

Pictorial shortest

Aretino: I thought it was quite evident that the plane of arc $AB$ is not perpendicular to the plane of circle $BE$. In any case I changed again my construction, to show the dihedral angle between those two planes.

Aretino: If circles $C$ and $D$ are perpendicular then of course arc $AB$ lies along circle $D$. You can experiment by yourself with the interactive construction to see if there are other cases.

Aretino:You didn't answer my question: If $D$ is parallel to $C$, is $AB$ the shortest when $AB$ is perpendicular to $C$?

Mr. @Aretino has proved that when AB is the shortest, AB and C are not vertical.

D is parallel to C

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  • $\begingroup$ I'm glad you appreciate my construction. As I already mentioned in a comment, the shortest path does NOT correspond, in general, to perpendicular planes. Later on I'll add the measure of $\angle ABE$, so you can verify that by yourself. $\endgroup$ – Intelligenti pauca Apr 22 '19 at 22:09
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    $\begingroup$ If circles $C$ and $D$ are perpendicular then of course arc $AB$ lies along circle $D$. You can experiment by yourself with the interactive construction to see if there are other cases. $\endgroup$ – Intelligenti pauca Apr 23 '19 at 22:39
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    $\begingroup$ If D is parallel to C, there is no arc AB tangent to D whose plane is perpendicular to C. $\endgroup$ – user856 Apr 25 '19 at 5:03
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    $\begingroup$ @E.wei Your picture refers to a different problem: point $A$ in your question lies on the great circle, while in your last figure it lies on the other circle- $\endgroup$ – Intelligenti pauca Apr 25 '19 at 7:20
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    $\begingroup$ @E.wei Anyway, even in this case, a check with GeoGebra shows that the shortest arc is NOT perpendicular to circle $C$. $\endgroup$ – Intelligenti pauca Apr 25 '19 at 7:32

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