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A circle of radius r has a curvature of size 1/r.
Therefore, small circles have large curvature
and large circles have sma...
Curvature
The act of curving
The state of being curved.

๏ฑ The ratio of the change in the angle of a
  tangent that moves ...
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 of
tangent (cha...
More precisely, curvature is
โ€ขScalar measure of bending nature of the curve
โ€ขDegree of curving in a line
โ€ขChange in the di...
Curvature measures the rate at which a space curve ๐’“(t) changes direction.
The direction of curve is given by the unit tan...
Note
   1.   Straight line has zero curvature
   2.   A circle has constant curvature
   3.   A circular helix has constan...
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Curvature

  1. 1. A circle of radius 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. 2. Curvature The act of curving The 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 of tangent (change in the direction of tangent) at ๐‘ƒ. Its magnitude 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†’0 Where ๐›ฟ๐œƒ is the angle between tangents at points ๐‘ƒ and ๐‘„ on the curve along arc length ๐›ฟ๐‘ . tangent ๐›ฟ๐œƒ tangent C:๐‘Ÿ = ๐‘Ÿ(๐‘ )
  4. 4. More precisely, curvature is โ€ขScalar measure of bending nature of the curve โ€ขDegree of curving in a line โ€ขChange in the direction of tangent line โ€ขArc rate of rotation of tangent line from point to point โ€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 the magnitude of the rate of change of the unit tangent 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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    Nov. 11, 2018
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    Nov. 14, 2014

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