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  2. Curvature - Wikipedia

    en.wikipedia.org/wiki/Curvature

    is equal to one. This parametrization gives the same value for the curvature, as it amounts to division by r 3 in both the numerator and the denominator in the preceding formula. The same circle can also be defined by the implicit equation F(x, y) = 0 with F(x, y) = x 2 + y 2 – r 2. Then, the formula for the curvature in this case gives

  3. Radius of curvature - Wikipedia

    en.wikipedia.org/wiki/Radius_of_curvature

    In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof. [1] [2] [3]

  4. Torsion of a curve - Wikipedia

    en.wikipedia.org/wiki/Torsion_of_a_curve

    Assume that this is a regular parametrization and that the curvature of the curve does not vanish. Analytically, r(t) is a three times differentiable function of t with values in R 3 and the vectors ′ (), ″ are linearly independent. Then the torsion can be computed from the following formula:

  5. Intrinsic equation - Wikipedia

    en.wikipedia.org/wiki/Intrinsic_equation

    The intrinsic quantities used most often are arc length, tangential angle, curvature or radius of curvature, and, for 3-dimensional curves, torsion. Specifically: The natural equation is the curve given by its curvature and torsion.

  6. Curl (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Curl_(mathematics)

    The curl of a vector field F, denoted by curl F, or , or rot F, is an operator that maps C k functions in R 3 to C k−1 functions in R 3, and in particular, it maps continuously differentiable functions R 3 → R 3 to continuous functions R 3 → R 3. It can be defined in several ways, to be mentioned below:

  7. Differential geometry of surfaces - Wikipedia

    en.wikipedia.org/wiki/Differential_geometry_of...

    The development of calculus in the seventeenth century provided a more systematic way of computing them. [3] 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.

  8. Differentiable curve - Wikipedia

    en.wikipedia.org/wiki/Differentiable_curve

    Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus.

  9. Menger curvature - Wikipedia

    en.wikipedia.org/wiki/Menger_curvature

    Let x, y and z be three points in R n; for simplicity, assume for the moment that all three points are distinct and do not lie on a single straight line.Let Π ⊆ R n be the Euclidean plane spanned by x, y and z and let C ⊆ Π be the unique Euclidean circle in Π that passes through x, y and z (the circumcircle of x, y and z).

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