Abstract
In this note, the ideas employed in [1] to treat the problem of an ellipsoid intersected by a plane are applied to the analogous problem of a hyperboloid being intersected by a plane. The curves of intersection resulting in this case are not only ellipses but rather all types of conics: ellipses, hyperbolas and parabolas. In text books of mathematics usually only cases are treated, where the planes of intersection are parallel to the coordinate planes. Here the general case is illustrated with intersecting planes which are not necessarily parallel to the coordinate planes.
Highlights
IntroductionIn case D r, D r 0 and D s, D s 0 the line of intersection reduces to
Let a hyperboloid be given with the three positive semi axes a, b, c x12 a2
Remark 1: In the special case that the plane of intersection of the hyperboloid is parallel to the x-y-plane, i.e
Summary
In case D r, D r 0 and D s, D s 0 the line of intersection reduces to. For D r, D r 0 and D s, D s 0 the line of intersection is of the form. If D q, D r 0 holds, (13) represents a parabola in the variables t and u. The Grassmann expansion theorem for the double cross product x y z x, z y x, y z
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