Although points which are equidistant from L and S must satisfy the implicit equation, some of the following cases will show that there may be solutions of the implicit equation which are not equidistant points.
In this picture (and the following pictures), S is the
transparent blue sphere, L is the red line, and the surface of
equidistant points is green.
This surface is the rotationally symmetric case, where C is on L. When the radius is c = 1, the equation is
It is equal to the surface of revolution generated by rotating the parabola x = (z2 - 1)/2 around the z-axis.
In this picture, the line L does not pass through the
center of the sphere, but meets it at two points, corresponding to
c > 1 in the equation
The horizontal cross-sections (by planes perpendicular to L) are ellipses, all with the same eccentricity (except at the singular points). Each cross-section by a plane containing L is a union of two parabolas.
This shows the line tangent to the sphere, corresponding to
c = 1. The implicit equation simplifies to the quartic:
The horizontal cross-sections are parabolas (except at the plane through the origin). Each cross-section by a plane containing L is a union of two parabolas tangent to L at the origin (except the plane containing C, which intersects the surface along the double line and one parabola).
There are some solutions of the implicit equation which are not solutions of the equation b = |a - c|. Points on the negative x-axis (where the surface crosses itself) are equidistant from L and S, as are points on the line segment connecting the origin (0,0,0) to the center C = (1,0,0). However, points of the form (x,0,0), with x > 1, satisfy z4 - 4xz2 - 4y2 = 0, but are not equidistant from L and S.
The shape can be parametrized by quadratic polynomials: x=u2-v2, y=2uv, z=2u, so it is a Steiner surface (of Type 4) and also falls into a classification of quadratically parametrized affine surfaces.
The remaining case is that the sphere and line are disjoint,
corresponding to 0 < c < 1. The surface has two components.
The left half of the surface is the locus of equidistant points, which all lie outside the sphere, so the equation is b = a - c. Each of its horizontal cross-sections is one branch of a hyperbola.
The constant c appearing in the implicit equation
There are two degenerate cases, the first being that the sphere
could have radius c=0 and center C=(1,0,0). The surface
becomes a parabolic cylinder (dark green), defined by the quadric
The second case is where c=0 and C=(0,0,0). The locus of points equidistant from L and S is the xy-plane, z = 0 (light green).
These loci can also be considered as "line-sphere bisector" surfaces, in the sense of the following article:
M. Peternell, Geometric Properties of Bisector Surfaces, Graphical Models and Image Processing 62 (2000), 202-236.