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- 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. CurvatureThe act of curvingThe state of being curved.๏ฑ The ratio of the change in the angle of a tangent that moves along curve from point to point๏ฑThe limit of the ratio of the change in the angle of a tangent as arc length approaches zero๏ฑThe reciprocal of the radius of a circle.
- 3. Let C:๐ = ๐(๐ ) be a space curve and P be a point on it,then curvature at ๐ is defined as rate of rotation oftangent (change in the direction of tangent) at ๐. Itsmagnitude is denoted by ๐ (kappa) and defined by ๐ฟ๐ ๐๐ ๐ = ๐๐๐ ๐ฟ๐ = ๐๐ ๐ฟ๐ โ0Where ๐ฟ๐ is the angle between tangents at points ๐ and๐ on the curve along arc length ๐ฟ๐ . tangent ๐ฟ๐ tangent C:๐ = ๐(๐ )
- 4. More precisely, curvature isโขScalar measure of bending nature of the curveโขDegree of curving in a lineโขChange in the direction of tangent lineโขArc rate of rotation of tangent line from point to pointโขChange in principal normal along tangent direction
- 5. Curvature measures the rate at which a space curve ๐(t) changes direction.The direction of curve is given by the unit tangent vector ๐(๐) ๐(๐) = ๐(๐)which has length 1 and is tangent to ๐(t).The picture below shows the unit tangent vector ๐ to the curve ๐(t) =(2cos(t), sin(t), 0)at several points.Obviously, if ๐(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: ๐ ๐ ๐ ๐ ๐= ๐ ๐ ๐ ๐
- 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 ๐, i.e ๐ = ๐

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