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  2. Euler spiral - Wikipedia

    en.wikipedia.org/wiki/Euler_spiral

    The graph on the right illustrates an Euler spiral used as an easement (transition) curve between two given curves, in this case a straight line (the negative x axis) and a circle. The spiral starts at the origin in the positive x direction and gradually turns anticlockwise to osculate the circle.

  3. First fundamental form - Wikipedia

    en.wikipedia.org/wiki/First_fundamental_form

    Theorema egregium of Gauss states that the Gaussian curvature of a surface can be expressed solely in terms of the first fundamental form and its derivatives, so that K is in fact an intrinsic invariant of the surface. An explicit expression for the Gaussian curvature in terms of the first fundamental form is provided by the Brioschi formula.

  4. Curvature of a measure - Wikipedia

    en.wikipedia.org/wiki/Curvature_of_a_measure

    A Dirac measure δ a supported at any point a has zero curvature. If μ is any measure whose support is contained within a Euclidean line L, then μ has zero curvature. For example, one-dimensional Lebesgue measure on any line (or line segment) has zero curvature. The Lebesgue measure defined on all of R 2 has infinite curvature.

  5. Principal curvature - Wikipedia

    en.wikipedia.org/wiki/Principal_curvature

    When a line of curvature has a local extremum of the same principal curvature then the curve has a ridge point. These ridge points form curves on the surface called ridges. The ridge curves pass through the umbilics. For the star pattern either 3 or 1 ridge line pass through the umbilic, for the monstar and lemon only one ridge passes through. [3]

  6. Motion graphs and derivatives - Wikipedia

    en.wikipedia.org/wiki/Motion_graphs_and_derivatives

    Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. (Velocity is on the y-axis and time on the x-axis. Multiplying the velocity by the time, the time cancels out, and only displacement remains.)

  7. Curvature - Wikipedia

    en.wikipedia.org/wiki/Curvature

    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 ...

  8. Dupin's theorem - Wikipedia

    en.wikipedia.org/wiki/Dupin's_theorem

    A curvature line is a curve on a surface, which has at any point the direction of a principal curvature (maximal or minimal curvature). The set of curvature lines of a right circular cylinder consists of the set of circles (maximal curvature) and the lines (minimal curvature). A plane has no curvature lines, because any normal curvature is zero.

  9. Frenet–Serret formulas - Wikipedia

    en.wikipedia.org/wiki/Frenet–Serret_formulas

    A space curve; the vectors T, N, B; and the osculating plane spanned by T and N. In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space, or the geometric properties of the curve itself irrespective of any motion.

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