Three points on a sphere define eight spherical triangles
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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
spherical-geometry
edited Dec 17 '18 at 10:48
Arthur
118k7118201
asked Dec 17 '18 at 10:27
C.BaroniC.Baroni
51
$endgroup$
add a comment |
$begingroup$
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
spherical-geometry
edited Dec 17 '18 at 10:48
Arthur
118k7118201
asked Dec 17 '18 at 10:27
C.BaroniC.Baroni
51
$endgroup$
$begingroup$
Is there a word missing? Seems like it should say "each one of which is entirely situated in a hemisphere".
$endgroup$
– Don Hatch
Dec 18 '18 at 8:23
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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.
$endgroup$
– C.Baroni
Dec 18 '18 at 22:31
$begingroup$
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?
$endgroup$
– Don Hatch
Dec 19 '18 at 3:01
$begingroup$
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.
$endgroup$
– Don Hatch
Dec 19 '18 at 3:14
$begingroup$
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.
$endgroup$
– Don Hatch
Dec 19 '18 at 3:24
add a comment |
$begingroup$
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
spherical-geometry
edited Dec 17 '18 at 10:48
Arthur
118k7118201
asked Dec 17 '18 at 10:27
C.BaroniC.Baroni
51
$endgroup$
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
spherical-geometry
spherical-geometry
edited Dec 17 '18 at 10:48
Arthur
118k7118201
asked Dec 17 '18 at 10:27
C.BaroniC.Baroni
51
edited Dec 17 '18 at 10:48
Arthur
118k7118201
asked Dec 17 '18 at 10:27
C.BaroniC.Baroni
51
edited Dec 17 '18 at 10:48
Arthur
118k7118201
edited Dec 17 '18 at 10:48
Arthur
118k7118201
edited Dec 17 '18 at 10:48
Arthur
118k7118201
118k7118201
asked Dec 17 '18 at 10:27
C.BaroniC.Baroni
51
asked Dec 17 '18 at 10:27
C.BaroniC.Baroni
51
asked Dec 17 '18 at 10:27
C.BaroniC.Baroni
51
51
$begingroup$
Is there a word missing? Seems like it should say "each one of which is entirely situated in a hemisphere".
$endgroup$
– Don Hatch
Dec 18 '18 at 8:23
$begingroup$
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.
$endgroup$
– C.Baroni
Dec 18 '18 at 22:31
$begingroup$
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?
$endgroup$
– Don Hatch
Dec 19 '18 at 3:01
$begingroup$
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.
$endgroup$
– Don Hatch
Dec 19 '18 at 3:14
$begingroup$
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.
$endgroup$
– Don Hatch
Dec 19 '18 at 3:24
add a comment |
$begingroup$
Is there a word missing? Seems like it should say "each one of which is entirely situated in a hemisphere".
$endgroup$
– Don Hatch
Dec 18 '18 at 8:23
$begingroup$
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.
$endgroup$
– C.Baroni
Dec 18 '18 at 22:31
$begingroup$
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?
$endgroup$
– Don Hatch
Dec 19 '18 at 3:01
$begingroup$
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.
$endgroup$
– Don Hatch
Dec 19 '18 at 3:14
$begingroup$
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.
$endgroup$
– Don Hatch
Dec 19 '18 at 3:24
$begingroup$
Is there a word missing? Seems like it should say "each one of which is entirely situated in a hemisphere".
$endgroup$
– Don Hatch
Dec 18 '18 at 8:23
$begingroup$
Is there a word missing? Seems like it should say "each one of which is entirely situated in a hemisphere".
$endgroup$
– Don Hatch
Dec 18 '18 at 8:23
$begingroup$
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.
$endgroup$
– C.Baroni
Dec 18 '18 at 22:31
$begingroup$
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.
$endgroup$
– C.Baroni
Dec 18 '18 at 22:31
$begingroup$
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?
$endgroup$
– Don Hatch
Dec 19 '18 at 3:01
$begingroup$
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?
$endgroup$
– Don Hatch
Dec 19 '18 at 3:01
$begingroup$
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.
$endgroup$
– Don Hatch
Dec 19 '18 at 3:14
$begingroup$
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.
$endgroup$
– Don Hatch
Dec 19 '18 at 3:14
$begingroup$
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.
$endgroup$
– Don Hatch
Dec 19 '18 at 3:24
$begingroup$
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.
$endgroup$
– Don Hatch
Dec 19 '18 at 3:24
add a comment |
2 Answers
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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.
answered Dec 17 '18 at 11:12
bubbabubba
30.7k33188
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add a comment |
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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:
answered Dec 17 '18 at 11:23
ChristophChristoph
12.5k1642
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2 Answers
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$begingroup$
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.
answered Dec 17 '18 at 11:12
bubbabubba
30.7k33188
$endgroup$
add a comment |
$begingroup$
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.
answered Dec 17 '18 at 11:12
bubbabubba
30.7k33188
$endgroup$
add a comment |
$begingroup$
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.
answered Dec 17 '18 at 11:12
bubbabubba
30.7k33188
$endgroup$
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.
answered Dec 17 '18 at 11:12
bubbabubba
30.7k33188
answered Dec 17 '18 at 11:12
bubbabubba
30.7k33188
answered Dec 17 '18 at 11:12
bubbabubba
30.7k33188
answered Dec 17 '18 at 11:12
bubbabubba
30.7k33188
30.7k33188
add a comment |
add a comment |
$begingroup$
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:
answered Dec 17 '18 at 11:23
ChristophChristoph
12.5k1642
$endgroup$
add a comment |
$begingroup$
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:
answered Dec 17 '18 at 11:23
ChristophChristoph
12.5k1642
$endgroup$
add a comment |
$begingroup$
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:
answered Dec 17 '18 at 11:23
ChristophChristoph
12.5k1642
$endgroup$
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:
answered Dec 17 '18 at 11:23
ChristophChristoph
12.5k1642
answered Dec 17 '18 at 11:23
ChristophChristoph
12.5k1642
answered Dec 17 '18 at 11:23
ChristophChristoph
12.5k1642
answered Dec 17 '18 at 11:23
ChristophChristoph
12.5k1642
12.5k1642
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$begingroup$
Is there a word missing? Seems like it should say "each one of which is entirely situated in a hemisphere".
$endgroup$
– Don Hatch
Dec 18 '18 at 8:23
$begingroup$
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.
$endgroup$
– C.Baroni
Dec 18 '18 at 22:31
$begingroup$
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?
$endgroup$
– Don Hatch
Dec 19 '18 at 3:01
$begingroup$
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.
$endgroup$
– Don Hatch
Dec 19 '18 at 3:14
$begingroup$
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.
$endgroup$
– Don Hatch
Dec 19 '18 at 3:24