Curvature - Department of Mathematics and Statistics曲率的数学与统计学系.pdf

CurvatureCurvature Outline of Hass, Weir, Thomas – Section 10.4 UnitUnit TangentTangent VectorVector If r t is a smooth curve, then the tangent vector to the curve  (as a function of t) is v t  dr/dt and the unit tangent vector  is TT tt  11 vv tt .    |v t | DefinitionDefinition The curvature of a curve, r t , as a function of t is defined  to be 1 dT  t  .   dtdt ||vv tt || IfIf ss denotesdenotes arcarc lengthlength, thenthen 1 dt dT dT  t  |v t | t        dsds dtdt dsds ||vv tt || and thus the curvative is the magnitude of the rate of change ofof thethe unitunit tangenttangent vectorvector withwith respectrespect toto arcarc lengthlength.. ExamplesExamples 1) Show that a circle of radius a has curvature 1/a. 2) Find the curvature function for the parabola y x 2 . 3) Let a and b be positive constants. Show that the helix, x  a cos t  y  a sin t  z  bt