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Topic: Circles Through the Point on a Torus (Read 1666 times) |
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Barukh
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Circles Through the Point on a Torus
« on: Jul 22nd, 2005, 4:16am » |
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Given a torus and a point P on its surface.
How many circles can be drawn through P so that every circle lies entirely on the surface of the torus?
P.S. Inspired by the following thread.
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JocK
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Re: Circles Through the Point on a Torus
« Reply #1 on: Jul 22nd, 2005, 5:46am » |
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I see 2 trivial solutions and 2 non-trivial ones 
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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raprap
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Re: Circles Through the Point on a Torus
« Reply #2 on: Jul 22nd, 2005, 8:31pm » |
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Forgive me I've just got home after downing several beers at the local Biker Bar, but right now I see two through P.
One created by an intersection of the torus and one from a plane perpendicular to the axial radius of the torus, and one by the intersection of a plane parallel to this axial radius.
If you wanted an ellipse, there are an infinite number ranging from the perpendiculars to the parallel plane intersection.
Rap c∫;?/
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Barukh
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Re: Circles Through the Point on a Torus
« Reply #3 on: Jul 22nd, 2005, 10:46pm » |
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Rap,
I wanted circles, not ellipses.
Maybe, you need more beers to see more circles? 
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raprap
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Re: Circles Through the Point on a Torus
« Reply #4 on: Jul 23rd, 2005, 3:55am » |
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 Early the next morning, with less alcohol on the brain
I am imagining a perfect torus, as a perfectly circular O-ring.
I’m still only seeing two circles through P. Forgive my lack of implicit rigor I’m not a mathematician and my definitions may not be formally correct.
From my limited understanding is correct if P is a point in space it is defined as an intersection of three planes. If P happens to be a point on a torus then intersection of any of those planes with the torus will describe a two dimensional shape. Interestingly sometimes these planar intersections create two shapes as they may transverse the doughnut hole, but P is only on one of these shapes, so I will tend to ignore this ghost.
When I use the term axial, I’m referring to the line through center of the torus hole and perpendicular to the plane of the torus. Moreover there is a point on this line (I’ll call it the center--C) where a plane perpendicular to this line intersects the torus to create two concentric circles. I’ll refer to the circle halfway between theses two concentric circles as the major torus circle and its diameter as the major torus diameter.
Now going back to the line through C. This line is defined by the intersection of two planes. Either of these planes is perpendicular to the major torus circle, s.t. the intersection of either of these planes and this major torus circle is two points equidistant from C. The intersection of the torus and this plane mow traces two circles, each with a center at these equidistant points and a diameter the same as the difference in the radius of the two concentric axial circles. I’ll call this diameter the minor torus diameter.
First P Plane.
Now put a point P on the torus. A plane includes P and perpendicular to the line through the major torus center intersects the torus creating two concentric circles one of which includes P. If I construct a plane through C that is not perpendicular to the torus and includes P I see torus intersections that are ellipses not circles. So I only circle see one circle traced for the first P plane.
Second P plane
This is the also an intersection of the torus and the plane that includes the point C. However this plane can include the line perpendicular to the axial diameter circle. If it does include this perpendicular line and the point P the torus traces two circles, each of the minor torus radii, one of which includes P. Any plane that includes P and C but is not include the line perpendicular to the major torus circle intersects the torus to make two ellipses, one of which includes P, So again I only see one circle traced for the second P plane.
Third P plane
This may be more difficult to describe. This plane is through P and intersects the first P plane s.t, it creates a line that intersects the torus at two or four points (none of which are P but all lie in a line) or one point (that is P). The latter case is trivially not a circle, since the only intersection of the plane and the torus I see that that includes P is the point P if P is on the outside of the torus (outside axial circle) and a two ellipses joined at P if on the inside of the torus (inside the axial circle).
The former case; however’ is more interesting, because if the plane intersecting the torus describes two points equidistant from P this distance is the same as the minor torus diameter, the intersection of the plane with the torus traces a circle. If P is on the outside of the torus (same as above) only one circle is described, if P on the inside two circles are traced one of which includes P. Anything else traces an ellipse, a barbell shaped obloid, or the point P.
Now as to how many planes meet this criteria, I’m not sure but as a conjecture I’d say there are an infinite number, or two at right angles to each other that intersect in the plane of the torus axial plane, or one.
Right now my gut tells me one circle with two possibilities for the third P plane.
Note: This is an interesting mental exercise, but it also makes me wonder if my now sober (sort of) wetware geometer’s sketchpad comes anywhere close to the software version.
Rap
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JocK
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Re: Circles Through the Point on a Torus
« Reply #6 on: Jul 23rd, 2005, 5:16am » |
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The easiest way to imagine the circles under discussion is - I think - the following:
| hidden: | Consider a circle of a given radius c. We define the axis of the circle as the line through the centre and perpendicular to the plane of the circle. One can construct half-planes bounded by this circle axis. Let's refer to these as 'axial wings'. Obviously, each axial wing intersects the circle perpendicularly.
The question now is: can you position a second circle of the same radius c such that when you intersect both circles with an arbitrary axial wing of the first circle, the distance a between the two intersection points is constant?
Positioning a large number of such secondary circles each with 'wing distance' a from the original circle leads to a torus with inner radius c-a and outer radius c+a. |
Hmmmm... having some alcohol in the brain might help to make sense out of the above... 
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| « Last Edit: Jul 23rd, 2005, 5:22am by JocK » |
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Grimbal
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Re: Circles Through the Point on a Torus
« Reply #8 on: Jul 25th, 2005, 2:41pm » |
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I knew about these circles, but I was always wondering whether these were perfect circles and not just approximations, maybe ellipses. I never found a proof for or against the case. I never tried it on paper, only mentally, on the bus or so.
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Barukh
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Re: Circles Through the Point on a Torus
« Reply #9 on: Jul 27th, 2005, 12:33am » |
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Correct, Rap!  Isn't that beautiful?
Grimbal, you may want to consider the following proof (formulas 28 downward).
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