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EricBrechner (talk | contribs) m →Parallel curve to a curve with a corner: Removed the {{in use}} template since the edit is complete. |
EricBrechner (talk | contribs) →Normal fans: Forgot to normalize the normal fan |
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[[File:Normal fan for defining parallel curves.png|thumb|Normal fan for defining parallel curves around a sharp corner]]
However, we can define a normal fan<ref name="Brechner1990"/> <math>\vec n_f(\alpha)</math> that provides an [[Interpolation|interpolant]] between <math>\vec n(t_c^-)</math> and <math>\vec n(t_c^+)</math>, and use <math>\vec n_f(\alpha)</math> in place of <math>\vec n(t_c)</math> at the sharp corner:
:<math>\vec n_f(\alpha) = \frac{(1 - \alpha)\vec n(t_c^-) + \alpha\vec n(t_c^+)}{\lVert (1 - \alpha)\vec n(t_c^-) + \alpha\vec n(t_c^+) \rVert},\quad</math>where <math>0 < \alpha < 1</math>.
The resulting definition of the parallel curve <math>\vec x_d(t)</math> provides the desired behavior:
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