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Curvature Curvature Presentation Transcript

  • A circle of radius r has a curvature of size 1/r.Therefore, small circles have large curvatureand large circles have small curvature. Thecurvature of a line is 0. In general, an objectwith zero curvature is "flat."
  • CurvatureThe act of curvingThe state of being curved. The ratio of the change in the angle of a tangent that moves along curve from point to pointThe limit of the ratio of the change in the angle of a tangent as arc length approaches zeroThe reciprocal of the radius of a circle.
  • Let C:𝑟 = 𝑟(𝑠) be a space curve and P be a point on it,then curvature at 𝑃 is defined as rate of rotation oftangent (change in the direction of tangent) at 𝑃. Itsmagnitude is denoted by 𝜅 (kappa) and defined by 𝛿𝜃 𝑑𝜃 𝜅 = 𝑙𝑖𝑚 𝛿𝑠 = 𝑑𝑠 𝛿𝑠 →0Where 𝛿𝜃 is the angle between tangents at points 𝑃 and𝑄 on the curve along arc length 𝛿𝑠. tangent 𝛿𝜃 tangent C:𝑟 = 𝑟(𝑠)
  • More precisely, curvature is•Scalar measure of bending nature of the curve•Degree of curving in a line•Change in the direction of tangent line•Arc rate of rotation of tangent line from point to point•Change in principal normal along tangent direction
  • Curvature measures the rate at which a space curve 𝒓(t) changes direction.The direction of curve is given by the unit tangent vector 𝒓(𝒕) 𝒕(𝒕) = 𝒓(𝒕)which has length 1 and is tangent to 𝒓(t).The picture below shows the unit tangent vector 𝒕 to the curve 𝒓(t) =(2cos(t), sin(t), 0)at several points.Obviously, if 𝒓(t) is a straight line, the curvature is 0. Otherwise the curvature is non-zero.To be precise, curvature is defined to be themagnitude of the rate of change of the unittangent vector with respect to arc length: 𝒅𝒕 𝒅𝒕 𝒌= 𝒅𝒓 𝒅𝒕
  • Note 1. Straight line has zero curvature 2. A circle has constant curvature 3. A circular helix has constant curvature 4. The curvature of small circle is large and vice versa 1 5. The radius of curvature is denoted by 𝜌, i.e 𝜅 = 𝜌