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.

Loci of points (LOP) equally spaced from two given geometrical figures are considered. Has been proposed a method, giving the possibility to systematize the loci, and the key to their study. The following options have been considered. A locus equidistant from N point and l straight line. N belongs to l. We have a plane that is perpendicular to l and passing through N. N does not belong to l – parabolic cylinder. A locus equidistant from F point and a plane. In the general case, we have a paraboloid of revolution. The F point belongs to the given plane. We get a straight line perpendicular to the plane and passing through the F point. A locus equidistant from a point and a sphere. The point coincides with the sphere center. We get the sphere with a radius of 0.5 R. The point lies on the sphere. We get the straight line passing through the sphere center and the point. The point does not coincide with the sphere center, but is inside the sphere. We get the ellipsoid. The point is outside the sphere. We have parted hyperboloid of rotation. A locus equidistant from a point and a cylindrical surface. The point lies on the cylindrical surface’s axis. We get the surface of revolution which generatix is a parabola. The point lies on the generatrix of the cylindrical surface of rotation. We get a straight line, perpendicular to that generatrix and passing through the cylinder axis. The point does not lie on the axis, but is located inside the cylindrical surface. We get the surface with a horizontal sketch line – the ellipse, and a front sketch lines – two different parabolas. The point is outside the cylindrical surface. A locus consists of two surfaces. The one with the positive Gaussian curvature, and the other – with the negative one.

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.