In any direction of the sphere, What is the distance from a point to a circle?

Question

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?

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. — Dirk, Apr 16 '19 at 06:46
Is $D$ any great circle through $A$? Or is it perpendicular to the great circle through $A$ perpendicular to $C$? — Intelligenti pauca, Apr 16 '19 at 13:09
Some experimentation with GeoGebra suggests the shortest path is NOT, in general, perpendicular to $C$. — Intelligenti pauca, Apr 16 '19 at 13:27
@Dirk Capital C is a circle, lowercase c is a small circle, capital D is a big circle, capital B is the intersection of C and c. Different c will have different B. — E.wei, Apr 16 '19 at 23:04
@Aretino The great circle D through A is in any direction, and D is not perpendicular to a given C. Verticality refers to the angle between a plane and a plane. — E.wei, Apr 17 '19 at 00:09
I suspect that $B$ will be the near intersection of circle $C$ with the great circle through $A$ perpendicular to $D$. The circle joining $A$ to $B$ will be tangent to $D$ and $C$. — Ethan Bolker, Apr 18 '19 at 13:29
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. — Intelligenti pauca, Apr 18 '19 at 13:45
Here's the link to experiment with GeoGebra: https://ggbm.at/pnxcjy96. You must move point B until you get the minimum length for arc AB. — Intelligenti pauca, Apr 18 '19 at 14:00
Does anyone know the solution for the planar case? $A$ is a point on the plane, $C$ is a circle, $D$ is a line though $A$. What is the shortest circular arc from $A$ to $C$ tangent to $D$? — , Apr 18 '19 at 15:02
@EthanBolker The AB you gave is not tangent to D, so it is not required by the title. — E.wei, Apr 19 '19 at 23:48
@Aretino The picture you gave is very good and interesting. So what's your conclusion? When the small circular plane is perpendicular to the $C$ plane, the arc $AB$ is the shortest? — E.wei, Apr 20 '19 at 00:04
@Aretino It is recommended that your research be included in the answers to questions. This will enable more people to notice your answer. — E.wei, Apr 20 '19 at 01:32
Your title is misleading. What you are after is not the shortest distance between a point and a circle. — , Apr 25 '19 at 07:18

E.wei · Answer 1 · 2019-04-27T00:13:03.423

-1

@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.

edited Apr 27 '19 at 00:13

answered Apr 21 '19 at 23:03

E.wei

91

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. – Intelligenti pauca Apr 22 '19 at 22:09
@Aretino Notice that I'm talking about spherical angles, not tangent angles of curves. – E.wei Apr 22 '19 at 22:32
@Aretino I mean, when the plane of the AB arc is perpendicular to the plane of the circle C, the length of the AB arc is the shortest. Do you think this is wrong? – E.wei Apr 23 '19 at 05:41
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. – Intelligenti pauca Apr 23 '19 at 10:42
@Aretino Wouldn't it be vertical under any conditions? – E.wei Apr 23 '19 at 22:18
1

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. – Intelligenti pauca Apr 23 '19 at 22:39
@Aretino Your conclusion is very good and helpful to me. If D is parallel to C, then AB is perpendicular to C, right? My network speed is low, so interaction is difficult. – E.wei Apr 23 '19 at 22:50
If D is PERPENDICULAR to C, then the shortest arc $AB$ is obtained when $B$ is one of the intersections between D and C and arc AB is a part of circle D. – Intelligenti pauca Apr 24 '19 at 08:04
If you have a low speed connection, you can download the GeoGebra file to experiment off-line: choose "Open in App" from the icon with three vertical dots (upper right). In the new window, choose "Download as... -> ggb" form the three bars icon (upper right). Of course you must also download (for free) GeoGebra 5 classic, if you haven't already. – Intelligenti pauca Apr 24 '19 at 08:11
@Aretino If $D$ is parallel to $C$, is $AB$ the shortest when perpendicular to $C$? – E.wei Apr 24 '19 at 16:05
If $D$ is parallel to $C$ then a plane perpendicular to $C$ is also perpendicular to $D$. Hence your supposition is false. – Intelligenti pauca Apr 24 '19 at 16:15
@Aretino：You didn't answer my question: If $D$ is parallel to $C$, is $AB$ the shortest when $AB$ is perpendicular to $C$? – E.wei Apr 24 '19 at 22:35
1

If D is parallel to C, there is no arc AB tangent to D whose plane is perpendicular to C. – Apr 25 '19 at 05:03
@rahul Notice the other picture I added. https://i.stack.imgur.com/KzJl1.jpg – E.wei Apr 25 '19 at 06:30
1

@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- – Intelligenti pauca Apr 25 '19 at 07:20
1

@E.wei Anyway, even in this case, a check with GeoGebra shows that the shortest arc is NOT perpendicular to circle $C$. – Intelligenti pauca Apr 25 '19 at 07:32
@Aretino: Thank you very much for your answer. You have taught me more knowledge. I will continue to study your interaction map. I changed $D$, but even so, there must be a great circle in the tangent circle that passes through $A$. – E.wei Apr 25 '19 at 09:35

In any direction of the sphere, What is the distance from a point to a circle?

1 Answers1