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13 Nov 2019
Another important consideration is that we want to ensure that small changes in the parameter we choose results in small changes in the location on the curve. We can think of the parameter t as a time parameter, and the given parameterization of a curve not only tells us what the curve is, but how quickly it is traced out. If the curve is being traced out very quickly (ex: Consider the curve r(t) - e,y(t) -t for large values of t) small changes in time may produce large distances travelled, and we want to ensure that we are comparing the tangent vector at a given point to a nearby tangent vector. One way to ensure that small changes in the input parameter do not yield large distances travelled along the curve is to parametrize the curve by arclength. With this in mind, we can now define the curvature by: Let r(s) describe a smooth curve parameterized by arclength, and T(s) denote the unit tangent vector. Then, the curvature, denoted by k, is the magnitude of the rate of change of the unit tangent vector with respect to arclength; i.e. dT ds R(s) IV Consider the helix given byãì cos( curvature in two ways sin(t2), 2t2 ã. We will compute the
Another important consideration is that we want to ensure that small changes in the parameter we choose results in small changes in the location on the curve. We can think of the parameter t as a time parameter, and the given parameterization of a curve not only tells us what the curve is, but how quickly it is traced out. If the curve is being traced out very quickly (ex: Consider the curve r(t) - e,y(t) -t for large values of t) small changes in time may produce large distances travelled, and we want to ensure that we are comparing the tangent vector at a given point to a nearby tangent vector. One way to ensure that small changes in the input parameter do not yield large distances travelled along the curve is to parametrize the curve by arclength. With this in mind, we can now define the curvature by: Let r(s) describe a smooth curve parameterized by arclength, and T(s) denote the unit tangent vector. Then, the curvature, denoted by k, is the magnitude of the rate of change of the unit tangent vector with respect to arclength; i.e. dT ds R(s) IV Consider the helix given byãì cos( curvature in two ways sin(t2), 2t2 ã. We will compute the
Bunny GreenfelderLv2
24 Jan 2019