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Say you have a unit-circle with its center at $(0,0)$, and you "cut out" the upper-right quadrant. You rotate this segment around the Y-axis and the orthographic projection is the upper-right segment of a vertical ellipse. Now you rotate it horizontally, which gives you a section of a parabola $(a<0)$; As these $3$ shapes are all -segments of- conic sections, is there a way to go back from this parabola (segment) to the initial circle's equation if the parabola's $a,b,c$ parameters are known?

It may even be related to how a unit circle forms a sinusoid with period $2\pi$ and amplitude $1$.

Widawensen
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MisterH
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  • I can't view your video clip. But orthographic projections of circles and ellipses are always circles or ellipses, so I don't know where the parabola came from. A perspective projection of a parabola can be a circle. Given a parabola, it is possible to find the view-point from which it appears to be circular. Is that what you meant? – bubba Oct 05 '13 at 04:57
  • Yes, the view-point in 3D space, and the radius of that circle after rotating back parallel to the XY-plane. – MisterH Oct 06 '13 at 14:35
  • This is the clip: vimeo * com / 49066857 – MisterH Oct 06 '13 at 22:08
  • The vimeo site is blocked here in China (which is where I live). But I think I know what you mean. Since we're discussing a view-point in 3D space, the projection must be a perspective one, right? Orthographic projections are what you get when the view-point is at infinity. Please confirm. – bubba Oct 07 '13 at 00:26
  • Yes, it is a perspective projection. – MisterH Oct 07 '13 at 09:53
  • Another way to see the clip is on this link from Geogebratube, but that requires you to install the app into your browser / computer. – MisterH Oct 07 '13 at 12:16
  • I looked at the Geogebratube clip. There, you're analysing the parabolas that pass through three given points. That topic has already been treated thoroughly in this discussion: http://math.stackexchange.com/questions/482063/parabolas-through-three-points/485594#485594 – bubba Oct 08 '13 at 04:17
  • So how about the first part of the question: how do you find the center and the radius of that circle? (I edited the second part out because it was confusing). – MisterH Oct 08 '13 at 21:20

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If you rotate a unit circle so that one point of the circle becomes the "eye point" for your perspective view, the remaining circle points will indeed form a parabola. If you know the parameters of your perspective view (left-right and up-down field of view, location of the center of projection, distance to view-plane, coordinate system on view plane) and you know the equation of the (perspectively-projected) parabola in film-plane coordinates, then you can certainly find the center and radius of the original circle, because you've just performed a sequence of linear and projective transformations. But to get a better answer than that, you'll need to be explicit about what are the inputs and outputs of the computation you wish to perform. If you describe and name everything, I'll do the algebra. (Bad news: your job is probably the harder one!)

John Hughes
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