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Topic: Euclidian Constructions on a Sphere (Read 932 times) |
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Barukh
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Euclidian Constructions on a Sphere
« on: Dec 14th, 2005, 11:45pm » |
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Solve the following problems using only straightedge and compass.
1. Determine the radius of a given sphere.
2. Three points are given on a sphere. Construct the circle passing through these points.
3. Two points are given on a sphere. Construct a meridian (big circle) passing through these points.
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Icarus
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Re: Euclidian Constructions on a Sphere
« Reply #1 on: Dec 15th, 2005, 4:38pm » |
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This requires some interpretation. I take it you do not mean the construction to actually be "on" the sphere, but rather in three-dimensional space. Otherwise, (1) has no interpretation that is not meaningless, and (3) would be trivial, as our spherical "straight-edge" would always draw great-circles (this is the common name for them in English - meridians are great circles that pass through a set pair of poles).
Also, by (1), I assume you mean "construct a radius", since determining "the radius" would require some scale, and none is provided.
So I assume:
Straight-edge: given any two points I can construct the line passing through them.
Compass: given three non-colinear points, I can construct a circle centered on the first, passing through the second, and coplanar with the third.
For (2) The three points lie in a plane. Construct the circle in the plane passing through the three points. It is also the circle on the sphere that you want.
One way to approach (1) and (3) is to construct the center of the sphere. If you can do that, then you can connect it to an arbitrary point on the sphere for a radius, and draw the circle centered on it, passing through the two given points to get the great-circle desired.
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Barukh
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Re: Euclidian Constructions on a Sphere
« Reply #2 on: Dec 16th, 2005, 1:36am » |
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Icarus, your assumptions are correct. So, every straight-edge construction may be made only on the plane. However, using compass we can construct circles on the sphere as well.
Clarifying the statements of the problem, in 1) the radius should be constructed on the plane; in 2) and 3) circles should drawn on the sphere. Said that, the following is not sufficient:
Quote:| For (2) The three points lie in a plane. Construct the circle in the plane passing through the three points. It is also the circle on the sphere that you want. |
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Icarus
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Re: Euclidian Constructions on a Sphere
« Reply #3 on: Dec 16th, 2005, 7:58am » |
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 The radius should be drawn on what plane?
And the circles are in planes no matter if they are drawn on the sphere or not. Are you saying that all circles drawn in the construction must be on the sphere, even though the lines cannot be? The particular circle I described will be on the sphere, as well as on the plane I described it on. In fact, it represents the intersection of the plane and sphere.
Your clarification has only added to the confusion!
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Barukh
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Re: Euclidian Constructions on a Sphere
« Reply #4 on: Dec 16th, 2005, 11:47pm » |
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I couldn't imagine that my formulation will be so confusing... And actually my previous remark about the first question was really confusing!
Let me try to re-formulate in a more formal way:
1. Find two points on a sphere that are the endpoints of its diameter.
2. Construct an intersection of the given sphere with the plane defined by the 3 points lying on that sphere.
3. Construct an intersection of the given sphere with the plane passing through 2 given points on the sphere and its center.
As you see, all the constructions must be completed on the given sphere. However, to make some intermediate constructions possible (drawing a line with a straight-edge) an auxiliary plane may be used.
Hope this will make it more clear to everybody.
Icarus, I still don't see how your proposal solve 2).
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Deedlit
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Re: Euclidian Constructions on a Sphere
« Reply #5 on: Dec 17th, 2005, 4:24am » |
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It seems, then, there is a third construction we can do: constructing a plane. Which planes are we allowed to "draw"?
I thought at first that one could construct a plane through three given noncollinear points, or construct an arbitrary plane through fewer points. In that case, we can solve #1 as follows:
Construct an arbtrary plane that intersects the sphere. The intersection is a circle. Find the center and a diameter of that circle (classical). Construct two arbitrary (but nonparallel) planes through that diameter. Find the centers of the two new intersection circles. The plane through the three centers is a great circle of the sphere, so we can construct a radius or diameter.
But then, #2 is completely trivial, so perhaps this isn't what you mean. (#3 is simple enough after #1, since we can find the center of the sphere.)
If, on the other hand, we are given just one random auxillary plane (not necessarily going through anything), I don't see that there's much we can do with it.
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Barukh
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Re: Euclidian Constructions on a Sphere
« Reply #6 on: Dec 17th, 2005, 5:38am » |
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on Dec 17th, 2005, 4:24am, Deedlit wrote:| It seems, then, there is a third construction we can do: constructing a plane. Which planes are we allowed to "draw"? |
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The only "plane construction" allowed is placing one leg of the compass on the sphere and describing a circle passing through another point on the sphere.
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Deedlit
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Re: Euclidian Constructions on a Sphere
« Reply #7 on: Dec 17th, 2005, 6:08am » |
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I assume you mean the circle on the sphere that consists of points equidistant from the original point?
I can get #3 based on the first two: draw two diamaters to get the center, draw a line through one of the points and the center to get the polar opposite, then draw a circle through the two points and the polar opposite. 
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Icarus
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Re: Euclidian Constructions on a Sphere
« Reply #8 on: Dec 17th, 2005, 6:47am » |
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on Dec 16th, 2005, 11:47pm, Barukh wrote:| Icarus, I still don't see how your proposal solve 2). |
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How can you not see it? Those 3 points define a plane. In that plane, they are not colinear, therefore there is exactly one circle passing through them. Because the intersection of the sphere and plane is a circle which passes through the three points, it must be this same circle. So the circle in the plane that passes through the three points is also the circle in the sphere that passes through the three points.
So, to construct the circle in the sphere, construct instead the circle in the plane, following the well-known means. What do you not follow here?
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Barukh
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Re: Euclidian Constructions on a Sphere
« Reply #9 on: Dec 17th, 2005, 10:27am » |
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on Dec 17th, 2005, 6:47am, Icarus wrote:
OK, let’s see: given sphere S; 3 points on it A, B, C and an auxiliary plane P. Construct 3 points A’, B’, C’ on P such that A’B’ = AB, B’C’ = BC, A’C’ = AC – no problem. Construct (the unique) circle G’ on P passing through points A’, B’, C’ – so far, so good. But now, Icarus, please tell me: how are you going to construct the circle G on the sphere S congruent to G’?!
Am I missing something?
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Barukh
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Re: Euclidian Constructions on a Sphere
« Reply #10 on: Dec 17th, 2005, 10:30am » |
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Guys, are you kidding, or I have gone mad?!
on Dec 17th, 2005, 6:08am, Deedlit wrote:| I assume you mean the circle on the sphere that consists of points equidistant from the original point? |
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Quote:| I can get #3 based on the first two: draw two diamaters to get the center, draw a line through one of the points and the center to get the polar opposite, then draw a circle through the two points and the polar opposite. |
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How are you going to construct the line passing through the center of the sphere, if it's not accessible?
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Deedlit
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Re: Euclidian Constructions on a Sphere
« Reply #11 on: Dec 18th, 2005, 4:23am » |
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on Dec 17th, 2005, 10:30am, Barukh wrote:Guys, are you kidding, or I have gone mad?!
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I guess we're all starting to go a little mad here...
I'd say it's just that you had some strong intuitions on what would be the natural interpretation of certain statements, and ours were a little different. Happens sometimes. And at least in my case, I needed to read a little more carefully...
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I assume you mean the circle on the sphere that consists of points equidistant from the original point?
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Wow, I was sure I got at least that part right! In fact, I'm still kinda sure whatever you say, so I'm gonna assume it anyway.
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How are you going to construct the line passing through the center of the sphere, if it's not accessible? |
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I think I thought that one of your replies was agreeing with Icarus' 3-D statement. But the meaning of "auxillary plane" finally dawned on me.
You didn't say exactly how information is transferred back and forth between the sphere and the plane, but I'm assuming it's just copying lengths using a noncollapsing compass.
Assuming that I've got it right now, here's a somewhat grungy and unsatisfying solution:
#1 - Draw a random circle on the sphere, and draw a circle with the same radius on the plane. Pick three points at random on the sphere's circle to form a triangle, and copy the lengths of the three sides onto the plane. Construct the circumcircle.
Assuming the radius of the sphere's circle subtends an arc of theta, the two circles we've drawn on the plane have radii 2r sin (theta/2) and r sin theta. It's easy to recover r now, if you know how to perform basic arithmetic operations using straightedge and compass.
Bisect r sin theta to get (r sin theta)/2 = r sin (theta/2) cos (theta/2)
Bisect 2r sin (theta/2) to get r sin (theta/2)
(r sin (theta/2) cos (theta/2) ) (r sin (theta/2) cos (theta/2) ) / (r sin (theta/2)) = r sin (theta/2) cos^2 (theta/2)
r sin (theta/2) - r sin (theta/2) cos^2 (theta/2) = r sin^3 (theta/2)
sqrt ( (r sin (theta/2)) (r sin^3 (theta/2)) ) = r sin^2 (theta/2)
( r sin (theta/2) cos (theta/2) ) ( r sin (theta/2) cos (theta/2) ) /( r sin^2 (theta/2)) = r cos^2 (theta/2)
r sin^2 (theta/2) + r cos^2 (theta/2) = r
r + r = 2r
Now draw a circle of radius 2r on the sphere.
#2 - Same idea. Copy the triangle onto the plane, and the circle will have radius r sin theta. From #1 we can get the radius of the sphere r. So we need 2r sin (theta/2) - we just use algebra like above. Then draw two circles of radii 2r sin (theta/2) from two different points, and the intersection will be the center of the required circle.
#3 - Just draw a circle of radius 2r from one of the two points. This gives two points and a polar opposite, and we can just draw a circle through all three.
Edit: Bah, I originally thought that the radius of the spherical circle was r tan theta, rather than 2r sin (theta/2). And I used r in the calculation. Basically, it's good it's not there anymore. 
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Icarus
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Re: Euclidian Constructions on a Sphere
« Reply #12 on: Dec 18th, 2005, 12:54pm » |
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on Dec 17th, 2005, 10:27am, Barukh wrote:
OK, let’s see: given sphere S; 3 points on it A, B, C and an auxiliary plane P. Construct 3 points A’, B’, C’ on P such that A’B’ = AB, B’C’ = BC, A’C’ = AC – no problem. Construct (the unique) circle G’ on P passing through points A’, B’, C’ – so far, so good. But now, Icarus, please tell me: how are you going to construct the circle G on the sphere S congruent to G’?!
Am I missing something?
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Who said anything about some "auxillary plane P"? The plane I am talking about is the plane passing through A, B, and C! That is the plane I am doing my construction in, and the circle I construct is EXACTLY the circle you are asking for, not just congruent to it.
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kokopelli
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Re: Euclidian Constructions on a Sphere
« Reply #13 on: Dec 18th, 2005, 1:14pm » |
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...he must be talking about the turboprop plane rather than the turbine kind.
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Barukh
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Re: Euclidian Constructions on a Sphere
« Reply #14 on: Dec 19th, 2005, 9:08am » |
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Deedlit, I am glad you got the statement right!
You've also got the idea of the solution: use two different circles associated with one compass construction.
I apologize for all the confusion I caused by not specifying the statement precisely from the beginning.
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Icarus
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Re: Euclidian Constructions on a Sphere
« Reply #15 on: Dec 19th, 2005, 3:33pm » |
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So let me see if I understand the tools you are allowing us:
(1) A sphere.
(2) An arbitrary fixed "auxillary" plane (unrelated to the sphere).
(3) Given two points on the sphere, to draw a circle in the sphere with one of the points as its "spherical" center (one of the intersections of its 3D axis of symmetry with the sphere), and the passing through the other point.
(4) Given two points on the plane, to draw a circle centered at one and passing through the other.
(5) Given two points on the plane, to draw the line through them.
(6) Given two points A,B on the plane and a point C on the sphere, to create a circle on the sphere whose 3D distance from C is AB. (By 3D distance, I mean the straight line distance between two points, as opposed to distance following the sphere.)
(7) Given two points A,B on the sphere and a point C on the plane, to draw a circle in the plane centered at C with radius AB (the 3D distance from A to B).
So in fact what we have is a 2D spherical construction using compass only, except that we are also allowed to transfer distances to the auxillary plane where both compass and straight-edge can both be used. (But the transfer is straight 3D distance, not distance along the sphere.)
A very odd set of tools (and not at all like the assumptions I listed in my first post which you agreed at the time were correct).
It seems to me that the key to this is to solve (3), then. If you can do that, then you no longer have a "compass only" construction on the sphere, but a "compass and straight-edge" construction, and can do (2) in exactly the same way it is done in the plane :
Given P, Q, R, draw the circles C(P, Q) (centered at P, passing through Q) and C(Q,P), and then draw the great-circle passing through the two intersection points. Do the same for Q,R. The intersections of the two great-circles are the two centers of the circle passing through P, Q, and R (circles on the sphere always have two antipodal centers).
And (1) is trivial: to find the antipole of A, choose two arbitrary points B, C, and draw the great-circles through A and B, and through A and C. The other intersection of the great circles is the antipode of A.
So the hard part is (3): given two point P, Q, how do you find the centers of the great-circle passing through P, Q.
It has been stated elsewhere that any construction in the plane that can be done with compass and straight-edge can also be done with compass alone. Does the same technique transfer to the sphere? If so, then you should be able to do this without any need for the auxillary plane.
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Barukh
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Re: Euclidian Constructions on a Sphere
« Reply #16 on: Dec 20th, 2005, 4:43am » |
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on Dec 19th, 2005, 3:33pm, Icarus wrote:So let me see if I understand the tools you are allowing us:
(1) A sphere.
(2) An arbitrary fixed "auxillary" plane (unrelated to the sphere).
(3) Given two points on the sphere, to draw a circle in the sphere with one of the points as its "spherical" center (one of the intersections of its 3D axis of symmetry with the sphere), and the passing through the other point.
(4) Given two points on the plane, to draw a circle centered at one and passing through the other.
(5) Given two points on the plane, to draw the line through them.
(6) Given two points A,B on the plane and a point C on the sphere, to create a circle on the sphere whose 3D distance from C is AB. (By 3D distance, I mean the straight line distance between two points, as opposed to distance following the sphere.)
(7) Given two points A,B on the sphere and a point C on the plane, to draw a circle in the plane centered at C with radius AB (the 3D distance from A to B). |
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Exactly! I ought to put it this way in my original post!
Quote:| A very odd set of tools (and not at all like the assumptions I listed in my first post which you agreed at the time were correct). |
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Somehow, when I first saw this problem, it seemed to me the only natural way, so my first reply was like unconscious…
Quote:| It has been stated elsewhere that any construction in the plane that can be done with compass and straight-edge can also be done with compass alone. Does the same technique transfer to the sphere? If so, then you should be able to do this without any need for the auxillary plane. |
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Interesting question! I don’t know the answer yet…
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Deedlit
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Re: Euclidian Constructions on a Sphere
« Reply #17 on: Dec 20th, 2005, 5:58am » |
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on Dec 19th, 2005, 9:08am, Barukh wrote:Deedlit, I am glad you got the statement right!
You've also got the idea of the solution: use two different circles associated with one compass construction. |
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Hmmm, is there something wrong with the solution? If it's how to turn the algebra into constructions, I can explain that.
But anyway, I have a nicer and more motivated solution, and it may be what you had in mind. Basically, we construct a cross section of the sphere.
For the following, "planar radius of the circle" refers to the normal radius of a circle in the plane, and "spherical radius of the circle" refers to the the compass radius we need to draw the circle on the sphere.
For #2, first we construct the diameter, same as before. Construct a right triangle with the radius of the sphere is the hypotenuse, and the planar radius of our circle as one of the legs. Call it ABC with AB = sphere radius and BC = planar radius of circle. Now find D on AC with AD = AB; then BD will be the desired spherical radius of the circle.
For #1, we use the same picture, except we start with lengths BC and BD, so construct triangle BCD. Now copy angle CDB to point B, and extend the line until it intersects line CD. The intersection is point A, and AB is our desired radius. Double, and draw a circle of that radius on the sphere.
#3 is the same.
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Barukh
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Re: Euclidian Constructions on a Sphere
« Reply #18 on: Dec 20th, 2005, 11:22pm » |
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Deedlit, that's exactly what I had in mind! Nicely done! 
on Dec 20th, 2005, 5:58am, Deedlit wrote:| For #1, we use the same picture, except we start with lengths BC and BD, so construct triangle BCD. Now copy angle CDB to point B... |
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In other words, construct the line through point B perpendicular to DB (because BCD is a right triangle).
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Icarus
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Re: Euclidian Constructions on a Sphere
« Reply #19 on: Dec 21st, 2005, 6:19pm » |
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Once you've constructed the appropriate length to set your compass at to get a great-circle on the sphere, there are much easier constructions for #1, #2, #3.
#1: Draw two arbitrary great circles on the sphere. Their two intersections are the endpoints of a diameter.
More generally, given an arbitrary point A, you can construct it's antipode by drawing a great-circle about it, choosing two arbitrary (but not antipodal) points on this circle and drawing great-circles about each. These two great-circles will intersect in A and its antipode.
#3: Given two points A, B, draw great-circles around each. The great-circle around the intersection passes through A and B.
#2: Given A, B, C. Draw circles centered at each of A, B, and passing through the other. The intersections are P, Q. Draw circles centered at each of B, C, and passing through the other. The intersections are X, Y. Draw (by the method of #3), great-circles through P and Q, and through X and Y. The intersections of these two great circles are the two centers of the circle passing through A, B, and C. (In total, you draw 10 circles on the sphere in this construction.)
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