Modeling and study of properties of surfaces equidistant to a sphere and a plane

Abstract

In the present paper geometric locus of points (GLP) equidistant to a sphere and a plane is considered; the properties of the acquired surfaces are studied. Four possible cases of mutual location of a sphere and a plane are considered: the plane passing through the center of the sphere, the plane intersecting the sphere, the plane tangent to the sphere and the plane passing outside the sphere. GLP equidistant to a sphere and a plane constitutes two co-axial co-focused paraboloids of revolution. General properties of the acquired paraboloids were studied: the location of foci, vertices, axis and directing planes, distance between the sphere center and the vertices, the distance between the vertices. GLP for each case of mutual location of a plane and a sphere constitutes: in case one passes through the center of other, two co-axial co-focused oppositely directed paraboloids of revolution symmetrical with respect to the given plane; in case they intersect each other, two co-axial co-focused oppositely directed non-symmetrical paraboloids; in case they are tangent to each other, a paraboloid and a straight line passing through the tangency point; in case they have no common points, a pair of co-axial co-focused mutually directed paraboloids of revolution.

Geometric modeling of advanced cellular structures with skeletal graphs

“…The furthest equidistant surfaces Φ + and Φ − from the middle plane Φ will be placed at h/2 from it in the directions of ⃗ n Φ and −⃗ n Φ , respectively. Constructing the equidistant surfaces Φ + and Φ − is not always straightforward [78]. If Φ is defined in parametric form ⃗ Φ(p, q), then the vector ⃗ n Φ can be found as…”