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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 $ \sigma$ -orthotomic curve $ {\bf y}_{\sigma}(t)$ of a planar curve $ {\bf r}(t)$ with respect to a point $ {\bf p}$ , not on $ {\bf r}(t)$ or any of its tangents, is defined as

$\displaystyle {\bf y}_{\sigma}(t) = {\bf p} + \sigma[({\bf r}(t) - {\bf p}) \cdot{\bf n}(t)]{\bf n}(t)\;,$     (8.17)

where $ {\bf n}(t)$ is the unit normal vector of $ {\bf r}(t)$ and $ \sigma$ 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 $ t$ at which the curve $ {\bf r}(t)$ 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
\begin{figure}\vspace*{5mm} \centerline{ \psfig{file=fig/orthotomic.ps,height=2in} }\end{figure}

A $ \sigma$ -orthotomic surface $ {\bf y}_{\sigma}(u,v)$ of a surface $ {\bf r}(u,v)$ with respect to a point $ {\bf p}$ , not on $ {\bf r}(u,v)$ or any of its tangent planes, is defined as

$\displaystyle {\bf y}_{\sigma}(u,v) = {\bf p} + \sigma[( {\bf r}(u,v) - {\bf p } ) \cdot{\bf N}(u,v)]{\bf N}(u,v)\;,$     (8.18)

where $ {\bf N}(u,v)$ is the unit normal vector of the surface $ {\bf r}(u,v)$ and $ \sigma$ is a scaling factor. An orthotomic surface has a singularity, i.e. a degenerate tangent plane, at all values of $ u,v$ at which the Gaussian curvature of the surface $ {\bf r}(u,v)$ vanishes or changes sign.

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Next: 8.1.3.7 Curvature lines Up: 8.1.3 Second-order interrogation methods Previous: 8.1.3.5 Focal curves and   Contents   Index
December 2009