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Differential Geometry of Curves and Surfaces

Differential Geometry of Curves and Surfaces,M. Do Carmo

Differential Geometry of Curves and Surfaces   (Citations: 467)
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Published in 1976.
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    • ...Frenet-Serret Equations: Given a curvature (s) > 0 and a torsion (s), according to the fundamental theorem of the local theory of curves [9], there exists a unique (up to rigid motion) spatial curve, parameterized by the arc-length s, dened by its Frenet-Serret equations, as follows:...
    • ...ds2 (s) 0. According to the fundamental theorem of local theory of curves, for every dierential function with (s) > 0 and (s), there exists a regular parameterized curve, where (s) is the curvature, (s) is the torsion, and s is the arc-length parameterization [9]...
    • ...Note that this could be done since all the properties of the Frenet-Serret Equations hold for every parameterization, not necessarily the arc-length parameterization [9] ...

    Gur Hararyet al. 3D Euler spirals for 3D curve completion

    • ...Given these derivatives, the curvature and the torsion are computed according to their denition [5] (where (x;y;z)):...
    • ...du is calculated according to the Frenet-Serret equations [5]...

    David Ben-Haimet al. Piecewise 3D Euler spirals

    • ...[2, 13] Using the Gauss-Bonnet theorem for surfaces with boundaries, see, e.g., [41], the bending energy contribution that involves the integral over the Gaussian curvature can be reformulated as an integral over the domain boundaries...

    Erwin Gutledereret al. Polymorphism of vesicles with multi-domain patterns

    • ...It is well known that, beside the trivial case of planes, helicoid is the unique (noncylindrical) ruled minimal surface (see [2], [1])...

    Nguyen Minh Hoang. Ruled minimal surfaces in $\Bbb R^3$ with density $e^z$

    • ...(For all the relevant concepts on the differential geometry of surfaces, we refer the reader to [7].)...

    A. Kumaret al. Active Contours for visual tracking: a geometric gradient based approa...

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