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# Curvature

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### 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.ο± The ratio of the change in the angle of a tangent that moves along curve from point to pointο±The limit of the ratio of the change in the angle of a tangent as arc length approaches zeroο±The reciprocal of the radius of a circle.
3. 3. Let C:π = π(π ) be a space curve and P be a point on it,then curvature at π is defined as rate of rotation oftangent (change in the direction of tangent) at π. Itsmagnitude is denoted by π (kappa) and defined by πΏπ ππ π = πππ πΏπ  = ππ  πΏπ  β0Where πΏπ is the angle between tangents at points π andπ on the curve along arc length πΏπ . tangent πΏπ tangent C:π = π(π )
4. 4. More precisely, curvature isβ’Scalar measure of bending nature of the curveβ’Degree of curving in a lineβ’Change in the direction of tangent lineβ’Arc rate of rotation of tangent line from point to pointβ’Change in principal normal along tangent direction
5. 5. Curvature measures the rate at which a space curve π(t) changes direction.The direction of curve is given by the unit tangent vector π(π) π(π) = π(π)which has length 1 and is tangent to π(t).The picture below shows the unit tangent vector π to the curve π(t) =(2cos(t), sin(t), 0)at several points.Obviously, if π(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: ππ ππ π= ππ ππ
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 π, i.e π = π
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