Yan - Bin Jia Oct 3 , 2017
نویسنده
چکیده
We want to find a measure of how ‘curved’ a curve is. Since this ‘curvature’ should depend only on the ‘shape’ of the curve, it should not be changed when the curve is reparametrized. Further, the measure of curvature should agree with our intuition in simple special cases. Straight lines themselves have zero curvature. Large circles should have smaller curvature than small circles which bend more sharply. The (signed) curvature of a curve parametrized by its arc length is the rate of change of direction of the tangent vector. The absolute value of the curvature is a measure of how sharply the curve bends. Curves which bend slowly, which are almost straight lines, will have small absolute curvature. Curves which swing to the left have positive curvature and curves which swing to the right have negative curvature. The curvature of the direction of a road will affect the maximum speed at which vehicles can travel without skidding, and the curvature in the trajectory of an airplane will affect whether the pilot will suffer “blackout” as a result of the g-forces involved. In this lecture we will primarily look at the curvature of plane curves. The results will be extended to space curves in the next lecture.
منابع مشابه
Yan - Bin Jia Oct 17 , 2017
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Geodesics are the curves in a surface that make turns just to stay on the surface and never move sideways. A bug living in the surface and following such a curve would perceive it to be straight. We will begin with a definition of geodesics, then present various method for finding geodesics on surfaces, and later reveal their relationships to shortest paths. The term geodesic comes from the sci...
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تاریخ انتشار 2017