[math-fun] circles, ellipses, foci
Take a circle & look at it along its axis; it appears as a circle. Now tilt the axis of the circle at an angle alpha; it now appears as an ellipse. Consider the foci of the ellipse. Is there anything interesting and/or cool about the relationship of the foci and the angle alpha? (I don't know any interesting answer; I'm just curious.)
Does everyone know about the Dandelin spheres? A plane intersects a cone in a conic section. The spheres inscribed in the cone and touching the plane do so at the foci. For proof, note that the tangents to a sphere from a point are equal in length, and use the `pins and string' construction for the ellipse (or hyperbola, or parabola -- focus-directrix for this last.) R. On Mon, 11 Nov 2013, Henry Baker wrote:
Take a circle & look at it along its axis; it appears as a circle.
Now tilt the axis of the circle at an angle alpha; it now appears as an ellipse.
Consider the foci of the ellipse.
Is there anything interesting and/or cool about the relationship of the foci and the angle alpha?
(I don't know any interesting answer; I'm just curious.)
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Re: drinking the Dandelin-whines: Yes, indeed. But I haven't been able to "see" how a circle transforms into a Dandelin ellipse merely by rotating it somehow in 3-space. If one takes a slightly different Dandelin configuration, wherein equal spheres are embedded in a circular cylindrical tube, then the inner spheres touch an oblique plane/ellipse at the foci. But I'm still having trouble finding a cool relationship between the locations of the foci and the angle of the plane. Perhaps Dandelin spheres in 4-space ? At 09:29 AM 11/11/2013, rkg wrote:
Does everyone know about the Dandelin spheres? A plane intersects a cone in a conic section. The spheres inscribed in the cone and touching the plane do so at the foci. For proof, note that the tangents to a sphere from a point are equal in length, and use the `pins and string' construction for the ellipse (or hyperbola, or parabola -- focus-directrix for this last.) R.
On Mon, 11 Nov 2013, Henry Baker wrote:
Take a circle & look at it along its axis; it appears as a circle.
Now tilt the axis of the circle at an angle alpha; it now appears as an ellipse.
Consider the foci of the ellipse.
Is there anything interesting and/or cool about the relationship of the foci and the angle alpha?
(I don't know any interesting answer; I'm just curious.)
Yes. What a beautiful and ingenious way to prove the counterintuitive fact that the intersection of a plane and a cone is an ellipse (generically)! --Dan On 2013-11-11, at 9:29 AM, rkg wrote:
Does everyone know about the Dandelin spheres? A plane intersects a cone in a conic section. The spheres inscribed in the cone and touching the plane do so at the foci. For proof, note that the tangents to a sphere from a point are equal in length, and use the `pins and string' construction for the ellipse (or hyperbola, or parabola -- focus-directrix for this last.) R.
On Mon, 11 Nov 2013, Henry Baker wrote:
Take a circle & look at it along its axis; it appears as a circle.
Now tilt the axis of the circle at an angle alpha; it now appears as an ellipse.
Consider the foci of the ellipse.
Is there anything interesting and/or cool about the relationship of the foci and the angle alpha?
(I don't know any interesting answer; I'm just curious.)
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Intersect a unit-radius cylinder with a plane F inclined at angle A to its axis, and touching a unit sphere inscribed in the cylinder at the focus of your ellipse. Now section by the plane perpendicular to F through the axis. By elementary plane trigonometry, the distance from ellipse centre to focus equals tan A , and the major axis of the ellipse equals sec A . Their ratio (the eccentricity) equals sin A . WFL On 11/11/13, Dan Asimov <dasimov@earthlink.net> wrote:
Yes. What a beautiful and ingenious way to prove the counterintuitive fact that the intersection of a plane and a cone is an ellipse (generically)!
--Dan
On 2013-11-11, at 9:29 AM, rkg wrote:
Does everyone know about the Dandelin spheres? A plane intersects a cone in a conic section. The spheres inscribed in the cone and touching the plane do so at the foci. For proof, note that the tangents to a sphere from a point are equal in length, and use the `pins and string' construction for the ellipse (or hyperbola, or parabola -- focus-directrix for this last.) R.
On Mon, 11 Nov 2013, Henry Baker wrote:
Take a circle & look at it along its axis; it appears as a circle.
Now tilt the axis of the circle at an angle alpha; it now appears as an ellipse.
Consider the foci of the ellipse.
Is there anything interesting and/or cool about the relationship of the foci and the angle alpha?
(I don't know any interesting answer; I'm just curious.)
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participants (4)
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Dan Asimov -
Fred Lunnon -
Henry Baker -
rkg