circle of curvature
Let a given plane curve have a definite curvature (http://planetmath.org/CurvaturePlaneCurve) in a point of . The circle of curvature of in the point is the circle which has the radius and which has with the curve the common tangent in and which in a neighbourhood of is on the same side as the curve.
When the curve is given in the parametric form
the coordinates of the center of curvature belonging to the point of the curve are
Example. Since the curvature of the parabola in the origin is , the corresponding radius of curvature is and the center of curvature .
Furthermore, it is possible to define the circle of curvature without first knowing about curvature of the curve. (In fact, using this definition, one could reverse the procedure and define curvature as the radius of the circle of curvature.) We may define the circle of curvature a point of as the unique circle passing through which makes a second-order contact with at .
|Title||circle of curvature|
|Date of creation||2013-03-22 16:59:46|
|Last modified on||2013-03-22 16:59:46|
|Last modified by||rspuzio (6075)|
|Defines||radius of curvature|
|Defines||center of curvature|
|Defines||centre of curvature|