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.

The article deals with the geometric locations of points equidistant from two spheres. In all variants of the mutual position of the spheres, the geometric places of the points are two surfaces. When the centers of the spheres coincide with the locus of points equidistant from the spheres, there will be spheres equal to the half-sum and half-difference of the diameters of the original spheres. In three variants of the relative position of the initial spheres, one of the two surfaces of the geometric places of the points is a two-sheet hyperboloid of revolution. It is obtained when: 1) the spheres intersect, 2) the spheres touch, 3) the outer surfaces of the spheres are removed from each other. In the case of equal spheres, a two-sheeted hyperboloid of revolution degenerates into a two-sheeted plane, more precisely, it is a second-order degenerate surface with a second infinitely distant branch. The spheres intersect - the second locus of the points will be the ellipsoid of revolution. Spheres touch - the second locus of points - an ellipsoid of revolution, degenerated into a straight line, more precisely into a zero-quadric of the second order - a cylindrical surface with zero radius. The outer surfaces of the spheres are distant from each other - the second locus of points will be a two-sheet hyperboloid of revolution. The small sphere is located inside the large one - two coaxial confocal ellipsoids of revolution. In all variants of the mutual position of spheres of the same diameters, the common geometrical place of equidistant points is a plane (degenerate surface of the second order) passing through the middle of the segment perpendicular to it, connecting the centers of the original spheres. The second locus of points equidistant from two spheres of the same diameter can be either an ellipsoid of revolution (if the original spheres intersect), or a straight (cylindrical surface with zero radius) connecting the centers of the original spheres when the original spheres touch each other, or a two-sheet hyperboloid of revolution (if continue to increase the distance between the centers of the original spheres).

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 have been investigated the loci equidistant from sphere and plane, and properties of obtained surfaces have been studied. Four options for possible mutual arrangement of plane and sphere have been considered: the plane passes through the center of the sphere; the plane intersects the sphere; the plane is tangent to the sphere; the plane passes outside the sphere. In all options of the mutual arrangement of the sphere and the plane, the loci are two surfaces - two coaxial confocal paraboloids of revolution. The general properties of the obtained paraboloids of revolution have been studied: foci and vertices positions, axes of rotation, the distance from the sphere center to the vertices of the paraboloids, the distance between the vertices of the paraboloids, and the position of the directorial planes have been defined. Have been derived equations for the surfaces of the loci equidistant from the sphere and the plane: various paraboloids of revolution. The loci in each of the four options for the possible mutual arrangement of the plane and the sphere are as follows. 1. The original plane passes through the sphere center – two coaxial confocal multidirectional paraboloids of revolution symmetric relative to the original plane. 2. The initial plane intersects the sphere – two coaxial confocal multidirectional but not symmetrical paraboloids of revolution, since the circle of intersection of the plane and the sphere does not coincide with the diameter of the sphere great circle. 3. The plane is tangent to the sphere – a paraboloid of revolution and a straight line (more precisely, a second order zero-quadric – a cylindrical surface with zero radius) passing through the tangency point of the plane and the sphere and the sphere center. 4. The plane passes outside the sphere – the equidistant loci will be two coaxial confocal unidirectional paraboloids of revolution.