8.1.3.6 Orthotomics

Orthotomic curves and surfaces are used to display the angle between the position vector of a curve (or surface) and the normal of the curve or surface respectively. Orthotomic curves and surfaces are useful for indicating the presence of inflection points [146]. A -orthotomic curve of a planar curve with respect to a point , not on or any of its tangents, is defined as

where is the unit normal vector of and is a scaling factor chosen for appropriate visualization. The tangent vector of an orthotomic curve is zero (and the orthotomic curve usually has a cusp-like singularity) at any parameter value of at which the curve has an inflection point. An illustrative example is shown in Fig. 8.6.

Figure 8.6: Orthotomics of two curves (adapted from [3]). Note that the orthotomic on the right shows the inflection point of the curve. The thin curves are the orthotomics of the thick curves

A -orthotomic surface of a surface with respect to a point , not on or any of its tangent planes, is defined as

where is the unit normal vector of the surface and is a scaling factor. An orthotomic surface has a singularity, i.e. a degenerate tangent plane, at all values of at which the Gaussian curvature of the surface vanishes or changes sign.