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# Curvature

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### Transcript of "Curvature"

1. 1. 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."
2. 2. 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.
3. 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 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)
4. 4. 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
5. 5. 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
6. 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
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