intersection of quadratic surface and plane
The intersection of a sphere with a plane (http://planetmath.org/IntersectionOfSphereAndPlane) is a circle, similarly the intersection of any surface of revolution formed by the revolution of an ellipse or a hyperbola about its axis with a plane perpendicular to the axis of revolution is a circle of latitude.
is cut with the plane (i.e. the -plane), we substitute to the equation of the ellipsoid, and thus the intersection curve satisfies the equation
which an ellipse. Actually, all plane intersections of the ellipsoid are ellipses, which may be in special cases circles.
As another exaple of quadratic surface we take the hyperbolic paraboloid
Cutting it e.g. with the plane , which is parallel to the -plane, the substitution yields the equation
If we cut the surface (2) with the plane , the result is the hyperbola having the projection
in the -plane. But cutting with gives , i.e. the pair of
lines which is a degenerate conic.
Let us then consider the general equation
of quadratic surface and an arbitrary plane
where at least one of the coefficients , , is distinct from zero. Their intersection equation is obtained, supposing that e.g. , by substituting the solved form
of (4) to the equation (3). We then apparently have the equation of the form
which a quadratic curve (http://planetmath.org/QuadraticCurves) or some of the degenerated cases of them.
|Title||intersection of quadratic surface and plane|
|Date of creation||2013-03-22 18:31:38|
|Last modified on||2013-03-22 18:31:38|
|Last modified by||pahio (2872)|