Abstract
It is well known that the line of intersection of an ellipsoid and a plane is an ellipse. In this note simple formulas for the semi-axes and the center of the ellipse are given, involving only the semi-axes of the ellipsoid, the componentes of the unit normal vector of the plane and the distance of the plane from the center of coordinates. This topic is relatively common to study, but, as indicated in [1], a closed form solution to the general problem is actually very difficult to derive. This is attemped here. As applications problems are treated, which were posed in the internet [1,2], pertaining to satellite orbits in space and to planning radio-therapy treatment of eyes.
Highlights
Let an ellipsoid be given with the three positive semiaxes a, b, c x12 a2 x22 b2 x32 c2 (1) (2)Inserting the components of x into the equation of the ellipsoid (1) leads to the line of intersection as a quadratic form in the variables t and u
It is well known that the line of intersection of an ellipsoid and a plane is an ellipse
As applications problems are treated, which were posed in the internet [1,2], pertaining to satellite orbits in space and to planning radio-therapy treatment of eyes
Summary
As q is an interior point of the ellipsoid the righthand side of Equation (2) is positive. If the vectors D1r and D1s are linearly independent, this is equivalent with the linear independence of the vectors r and s , the matrix in (3) is positive definite and the line of intersection is an ellipse. Let r and s be unit vectors orthogonal to the unit normal vector n of the plane r, r r12 r22 r32 1. Condition (7) ensures that the 2 2 matrix in (3) has diagonal form. The line of intersection reduces to an ellipse in translational form t t0 2. Where is the distance of plane (2) from the origin. The Grassmann expansion theorem for the double cross product x y z x, z y x, y z
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