Curvature

Curvature

• 1.
  A circle ofradius r has a curvature of size 1/r. Therefore, small circles have large curvature and large circles have small curvature. The curvature of a line is 0. In general, an object with zero curvature is "flat."
• 2.
  Curvature The act ofcurving The 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.
• 3.
  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 of tangent (change in the direction of tangent) at 𝑃. Its magnitude is denoted by 𝜅 (kappa) and defined by 𝛿𝜃 𝑑𝜃 𝜅 = 𝑙𝑖𝑚 𝛿𝑠 = 𝑑𝑠 𝛿𝑠 →0 Where 𝛿𝜃 is the angle between tangents at points 𝑃 and 𝑄 on the curve along arc length 𝛿𝑠. tangent 𝛿𝜃 tangent C:𝑟 = 𝑟(𝑠)
• 4.
  More precisely, curvatureis •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
• 5.
  Curvature measures therate 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 the magnitude of the rate of change of the unit tangent vector with respect to arc length: 𝒅𝒕 𝒅𝒕 𝒌= 𝒅𝒓 𝒅𝒕
• 6.
  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