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Curvature of general surfaces was first studied by Euler. In 1760 [4] he proved a formula for the curvature of a plane section of a surface and in 1771 [5] he considered surfaces represented in a parametric form. Monge laid down the foundations of their theory in his classical memoir L'application de l'analyse à la géometrie which
The Monge gauge has two obvious limitations: If the average surface is not plane, then the Monge gauge only makes sense on length scales smaller than the curvature of the average surface. And the Monge gauge fails completely if the surface is so strongly bent that there are overhangs (points x,y corresponding to more than one z).
The normal curvature, k n, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, k g, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τ r, measures the rate of change of the surface ...
Gaspard Monge, Comte de Péluse (French pronunciation: [ɡaspaʁ mɔ̃ʒ kɔ̃t də pelyz]; 9 May 1746 [2] – 28 July 1818) [3] was a French mathematician, commonly presented as the inventor of descriptive geometry, [4] [5] (the mathematical basis of) technical drawing, and the father of differential geometry. [6]
The product k 1 k 2 of the two principal curvatures is the Gaussian curvature, K, and the average (k 1 + k 2)/2 is the mean curvature, H. If at least one of the principal curvatures is zero at every point, then the Gaussian curvature will be 0 and the surface is a developable surface. For a minimal surface, the mean curvature is zero at every ...
The Monge cone at a given point (x 0, ..., x n) is the zero locus of the equation in the tangent space at the point. The Monge equation is unrelated to the (second-order) Monge–Ampère equation . References
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R 3. It permits the calculation of curvature and metric properties of a surface such as length and area in a manner consistent with the ambient space.
A point p in a Riemannian submanifold is umbilical if, at p, the (vector-valued) Second fundamental form is some normal vector tensor the induced metric (First fundamental form). Equivalently, for all vectors U , V at p , II( U , V ) = g p ( U , V ) ν {\displaystyle \nu } , where ν {\displaystyle \nu } is the mean curvature vector at p .
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