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 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.