A circle of radius r has a curvature of size 1/r.Therefore, small circles have large curvatureand large circles have small...
CurvatureThe act of curvingThe state of being curved.๏ฑ The ratio of the change in the angle of a  tangent that moves along...
Let C:๐‘Ÿ = ๐‘Ÿ(๐‘ ) be a space curve and P be a point on it,then curvature at ๐‘ƒ is defined as rate of rotation oftangent (chang...
More precisely, curvature isโ€ขScalar measure of bending nature of the curveโ€ขDegree of curving in a lineโ€ขChange in the direc...
Curvature measures the rate at which a space curve ๐’“(t) changes direction.The direction of curve is given by the unit tang...
Note   1.   Straight line has zero curvature   2.   A circle has constant curvature   3.   A circular helix has constant c...
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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.๏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. 3. Let C:๐‘Ÿ = ๐‘Ÿ(๐‘ ) be a space curve and P be a point on it,then curvature at ๐‘ƒ is defined as rate of rotation oftangent (change in the direction of tangent) at ๐‘ƒ. Itsmagnitude is denoted by ๐œ… (kappa) and defined by ๐›ฟ๐œƒ ๐‘‘๐œƒ ๐œ… = ๐‘™๐‘–๐‘š ๐›ฟ๐‘  = ๐‘‘๐‘  ๐›ฟ๐‘  โ†’0Where ๐›ฟ๐œƒ is the angle between tangents at points ๐‘ƒ and๐‘„ on the curve along arc length ๐›ฟ๐‘ . tangent ๐›ฟ๐œƒ tangent C:๐‘Ÿ = ๐‘Ÿ(๐‘ )
  4. 4. More precisely, curvature isโ€ขScalar measure of bending nature of the curveโ€ขDegree of curving in a lineโ€ขChange in the direction of tangent lineโ€ขArc rate of rotation of tangent line from point to pointโ€ขChange in principal normal along tangent direction
  5. 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’Œ= ๐’…๐’“ ๐’…๐’•
  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 ๐œŒ, i.e ๐œ… = ๐œŒ

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