Loci of Points Equally Spaced from Two Given Geometrical Figures. Part 2: Loci of Points Equally Spaced from a Point and a Conical Surface

Abstract

In this paper are studied surfaces which are loci of points (LOP) equally spaced from a point and a conical surface under a variety of the point and conical surface’ mutual arrangement. Mathematical models of such surfaces are studied, and mathematical analysis of their properties is performed, as well as 3D models of considered surfaces are constructed. Possible cases of mutual arrangement for the point and the conical surface: • the point is at the conical surface’s vertex; • the point is on the conical surface; • the point is inside the conical surface: –– on the axis, –– not on the axis; • the point is outside the conical surface. The point is on the vertex of the conical surface Γ — the obtained conical surface Ω has the same vertex, whose generatrixes are perpendicular to the generatrixes of the surface Γ. The point is on the conical surface Γ — LOP equally spaced from the surface Γ and the point O separates into a straight-line l and a surface Φ of 4th order. The line l is located in the axial plane passing through the point O and is perpendicular to the generatrix of the conical surface Γ. Obtained surface Φ has a symmetry plane passing through the axis of the conical surface Γ and the point O. Many sections of the obtained surface Φ are Pascal snails. The point is inside the conical surface on the axis. Obtained surface α is a rotation surface, and the axis z is its axis of rotation. All the sections of the surface by planes perpendicular to the axis z are circles. Point is outside the conical surface. A very interesting surface Ω has been obtained, with the following properties: the surface Ω has a support plane, which is tangent to the surface Ω on a hyperbole; the surface Ω has 2 symmetry planes; there are a circle, parabola and Pascal’s snail among the surface Ω sections. In this paper have been considered analogues between surfaces of LOP equally spaced from the cylindrical surface and the point, and from the conical surface and the point.