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reading a book of spherical astronomy I've read this:

Three great circles pass through three points on a sphere. If for each great circle we consider only one of the two parts in which it is divided by the two points that determine it, we will have a spherical triangle. Three points on the sphere thus define eight spherical triangles, one of which is entirely situated in a hemisphere, i.e. such that the three arches that make it up are all smaller than a semicircle.

Now, the "smallest" triangle is obviously clear, but I can't understand exactly (... probably an image could help...) how the other triangles are built, and how an arch - the side of the triangle - can be greater than a semicircle.

Perhaps a mistake in the book or an imprecise description?

Thanks in advance Carlo

Arthur
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C.Baroni
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  • Is there a word missing? Seems like it should say "each one of which is entirely situated in a hemisphere". – Don Hatch Dec 18 '18 at 08:23
  • No. the text says "one of which...". Probably the text refers to "inner" and "outer" spherical triangle, and if the "inner" is entirely situated in an hemisphere, the "outer" is larger and extends beyond a hemisphere. – C.Baroni Dec 18 '18 at 22:31
  • Are you sure? It could be that I'm confused or misinterpreting something, but it looks to me like each of the 8 triangles is in fact the intersection of three hemispheres, no? – Don Hatch Dec 19 '18 at 03:01
  • Furthermore, it looks to me like each of the 8 spherical triangles is a central reflection of the opposite triangle, so it has the same size and shape as its opposite (except mirror-reversed). That alone makes it impossible for there to be only one out of the 8 triangles that is "situated in a hemisphere". The author of this passage seems quite confused. – Don Hatch Dec 19 '18 at 03:14
  • Or, positing the more charitable interpretation-- I think the passage is simply missing the necessary word "each", as I originally suggested. Every arch of every spherical triangle in the picture (see @Cristoph's picture) is smaller than a semicircle, as you observed when you wrote the question. – Don Hatch Dec 19 '18 at 03:24

2 Answers2

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Suppose the three points are $A$, $B$, $C$, and let $P$ be the center of the sphere. Then we have three planes that pass through $P$, namely $ABP$, $ACP$, and $BCP$.

Consider a given point on the sphere; it can be on one side or the other of each of the three planes, so it can lie in one of $2 \times 2 \times 2$ regions of the sphere. These 8 regions are the 8 spherical triangles.

Another explanation: each of the three planes divides the sphere into two pieces. So, the three planes together divide the sphere into $2 \times 2 \times 2$ pieces.

bubba
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In addition to bubbas answer that (as long as the three points do not lie on one great circle) the three points define three planes that split the sphere into 8 parts, let me provide a picture of the situation:

8 triangles on a sphere

Christoph
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  • Actually, there are MORE than 8 triangles. The above picture can be easily used to visualize one more that hasn't been discussed. You see the beige shaded triangle above. Now consider ALL the rest of the sphere except for the shaded triangle. Notice that this region has 3 arcs as its borders, the same 3 arcs as the small triangle. It also is a triangle, it just isn't what we are used to visualizing. So each of the 8 small triangles has a large (rest of the sphere) triangle associated with it. So there are 16 spherical triangles defined by the 3 planes. But only 2 with corners at A,B,C. – Mark T Sep 13 '21 at 20:25
  • @MarkT I take as definition of a spherical triangle an intersection of three half-spheres. By this definition, the complement you are describing is not a spherical-triangle. – Christoph Sep 15 '21 at 05:26
  • Christoph: Assume that the shaded area in the diagram is slightly larger, so it has three 90 degree angles. It thus takes up one eighth of the sphere. On a unit radius sphere, the sum of the three angles of a spherical triangle (in radians) minus Pi is the area of the triangle. (That fact is amazing to me.) So each angle is Pi/2 so 3Pi/2-Pi = Pi/2. Eight times that is 4Pi, the area of a unit sphere. The 3 angles on the "rest" are 270 or 3Pi/2. Then 33Pi/2-Pi = 7Pi/2 exactly the known area of the rest of the sphere. So that proves it IS a triangle. It just isn't what we expect to see. – Mark T Oct 03 '21 at 03:18
  • @MarkT I just gave you my definition of a spherical triangle and it doesn't satisfy that. – Christoph Oct 03 '21 at 17:07
  • My initial comment was incorrect (sorry everyone) there are exactly 8 as bubba stated.

    Christoph: Okay, I will grant that your (personal) definition does, indeed, uniquely identify the shaded area in the diagram. So feel free to use it. It does describe a subset of the actual spherical triangles. But it is not a good general definition, and I'd hate to see others use it and limit their view of the topic.

    A general definition of a spherical triangle would be: an area on the surface of a sphere bounded by three arcs connecting three points on the surface of the sphere.

     – Mark T Oct 04 '21 at 15:06
  • @MarkT You can't both agree with bubbas answer of 8 spherical triangles and at the same time claim those are only a subset of the actual spherical triangles. – Christoph Oct 05 '21 at 09:55