Loci of Points Equidistant From Two Geometric Objects. Part 6: Loci of Points Equidistant From a Sphere and a Cylindrical Surface of Equal Diameters

This article examines the loci of points (LoPs) equidistant from a sphere and a cylindrical surface of equal diameter. The properties of the resulting LoPs surfaces are studied. When constructing a LoPs equidistant from a cylindrical surface Γ and a sphere Δ, four sheets of surfaces are always obtained. The first sheet is when both surfaces are increasing, the second when both are decreasing. Two more sheets are formed when one of the given surfaces is increasing and the other is decreasing, and vice versa. Four possible positions of the sphere and cylindrical surface are considered: 1. 6.5.1.1. The center of the sphere Δ is on the axis of the cylindrical surface Γ (a = 0). The LoPs are the two-sheeted plane Σ 6.5.1.1 and the perpendicular paraboloid of revolution (symmetric) Ψ6.5.1.1 . One of the surfaces is imaginary. 2. 6.5.1.2. The sphere Δ and the cylindrical surface Γ intersect (0 < a < R): the LoPs is one imaginary and two real surfaces: • a two-sheeted parabolic cylindrical surface λ6.5.1.2 ; • an asymmetric perpendicular paraboloid Ψ6.5.1.1 . Option 6.5.1.1. is a special case of option 6.5.1.2. 3. 6.5.1.3. A sphere and a cylindrical surface of equal diameter touch (have external tangency) (a = R). In this case, all four surfaces are real, but one is represented by a line—zero quadric of the second order. The LoPs are: • a two-sheeted parabolic cylindrical surface λ. The surface λ is a LoPs equidistant from the sphere Δ and the cylindrical surface Γ in all four cases of their relative positions. 5. 6.5.1.4. The sphere Δ and the cylindrical surface Γ are at a distance from each other (a > R), the LoPs are three real surfaces with four sheets: • a two-sheeted parabolic surface λ; • a quartic surface Ψ, its frontal and horizontal outlines are a parabola and a branch of a hyperbola, respectively; • a quartic surface Σ, a parabola, and a branch of a hyperbola are the frontal and horizontal outlines of Σ, respectively.