9887

Twin Parabolas

Twin parabolas in three-dimensional space are two congruent parabolas lying in two perpendicular planes such that the vertex of the one is the focus of the other. This Demonstration shows that a cone with its apex at some point on one of the parabolas and with its axis tangent to the parabola at that point can be constructed (by choosing an adequate opening angle) so that it contains the other parabola. In fact, every point of one parabola can serve as the apex of some cone containing the other parabola, and there is no other possible cone apex with this property anywhere else.

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The parabola is a special conic section. Define a host cone for a parabola as one that contains the parabola as a conic section. Given a parabola, one may ask for all of its host cones. There is a whole family of host cones, and the locus of the apexes of such cones is another parabola, the "host parabola". Interestingly, the host parabola of the host parabola is the original parabola, which leads to the name twin parabolas: two congruent parabolas lying in two perpendicular planes such that the vertex of the one is the focus of the other.
A suggested standardized analytic Cartesian representation of twin parabolas is the following:
1. (in the - plane): , and
2. (in the - plane): ,
where is the focal length, which can be chosen as 1 in some unit system.
One can easily find a host cone constructed on the second parabola at a point . The cone's axis is the tangent line at the point and the opening angle is , where and are the vertex and the focus of the second parabola, respectively. This Demonstration shows this property of twin parabolas.
This property of the twin parabolas (that one is the host of the other) is a limiting case of an ellipse-hyperbola pair.

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