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DUPIN INDICATRIX
Charles Dupin (1784-1873): French economist, mathematician and politician. |
When a plane parallel to the tangent plane at a point M of a surface tends to this plane, its section with the surface tends to become the homothetic image of a curve which, after proper normalization, is the Dupin indicatrix of the point of the surface.
When the point is not a planar point, the Dupin indicatrix is the union of two conics, with equation in the tangent plane with the frame of the principal directions, where
R_{1}
and R_{2} are the principal radii of curvature of the surface at M. These principal directions are given by the two curvature lines passing by the point.
When R_{1} and R_{2} are finite and have the same sign, the indicatrix is an ellipse, and the point is said to be elliptic (and called umbilic when R_{1} = R_{2}). | |
When R_{1} et R_{2} are finite and have opposite signs, the indicatrix is the union of two conjugate hyperbolas, and the point is said to be hyperbolic; the two asymptotes of these hyperbolas are the asymptotic tangents at M, tangent to the two asymptotic lines passing by M. | |
When R_{1} or R_{2} is infinite, the indicatrix is the union of two parallel lines, and the point is said to be parabolic or torsal. |
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© Robert FERRÉOL 2017