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The following proposition is from 'Spherical Geometry and Its Applications' by Marshall A. Whittlesey:

Proposition 5.6 If two distinct points on a sphere are not antipodal then there exists a unique great circle passing through them [1]

Let there be two non-antipodal distinct points on a small circle of a sphere. I cannot imagine a great circle that passes through both of them. (Think of this small circle just below the 'equator' of the sphere and imagine the equator is on a horizontal plane) Could you help me understand this? Or do I misunderstand some concepts here?

[1] Whittlesey, Marshall A.(2020). Spherical Geometry and Its Applications. CRC Press Taylor & Francis Group.

Ali Kıral
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    If you draw the shortest path on the sphere between your two points, that path is a segment of the great circle you're looking for. (Maybe you've misunderstood the definition of a great circle?) – Karl Dec 21 '22 at 18:41
  • @Karl Any circle that has all its points on a sphere and whose diameter is equal to that of the sphere is a great circle. But, if I draw a path between my two points, won't that segment be an arc of the small circle I began with? – Ali Kıral Dec 21 '22 at 18:47
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    The arc of your small circle won't be the shortest path on the sphere. – Karl Dec 21 '22 at 19:00
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    Can you visualize a plane that contains your two points and the sphere's center? This plane intersects the sphere along a great circle. (This is ice1000's answer.) – Karl Dec 21 '22 at 19:07
  • Ok, I think I have just managed to visualize it in my head. Everything became full circle:D – Ali Kıral Dec 21 '22 at 19:21

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Here's a simple construction, using the fact that three points uniquely determine a plane if not collinear. Take the following three points:

  • The given two points.
  • The center of the sphere.

The plane determined by these three points should intersect with the sphere with a great circle.

  1. The plane should pass through the center of the sphere (so it intersects with the sphere with a great circle) and through the two points provided.
  2. The given two points are not antipodal so they are not collinear with the center.
ice1000
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    That's right. And if points are antipodal then we cam draw infinitely many great circles. – Vasili Dec 21 '22 at 18:10
  • @ice1000 I am really trying but I still cannot visualize it. Could you provide me with a visual with the great circle? I hope I am not demanding much. – Ali Kıral Dec 21 '22 at 18:39
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    @AliKıral: Form a triangle using the center and the two points. This triangle defines a plane. The intersection of this plane and the sphere will be a great circle and it will contain both points. – Vasili Dec 21 '22 at 19:08