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SPHERICAL INDICATRIX

The spherical indicatrix of *curvature* of a 3D curve is the trajectory (included in the sphere with center *O* and radius 1) of the point *P* such that where is the tangent (unit) vector of the curve under consideration.

If the indicatrix is parametrized by the curvilinear abscissa *s*, then the speed of the point *P* is equal to the curvature:
(see the notations); therefore, the curvilinear abscissa of the indicatrix of curvature is the angle of curvature j.

The indicatrix of curvature is a circle iff the curve is a helix.

The spherical indicatrix of *torsion* of a 3D curve is the trajectory (included in a sphere with center *O* and radius 1) of the point *Q* such that where is the binormal (unit) vector of the curve under consideration.

If the indicatrix is parametrized by the curvilinear abscissa *s*, then the speed of the point *Q* is equal to the torsion: ; therefore, the curvilinear abscissa of the indicatrix of torsion is the angle of curvature y.

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© Robert FERRÉOL 2018