2.4. REGULAR CURVES179Remark 2.4.5.Suppose−→p(t)is a reparametrization of−→q(s), withrecalibration functiont(s). Then the velocity vectors at correspondingparameter values are linearly dependent: ift0=t(s0)then−→v−→p(t0) =t′(s0)−→v−→q(s0).Intuitively, reparametrizing a curve amounts to speeding up or slowingdown the process of tracing it out. Since the speed with which we traceout a curve is certainlynotan intrinsic property of the curve itself, we cantry to eliminate the effects of such speeding up and slowing down byconcentrating on theunit tangent vectordetermined by aparametrization−→p(t),−→T−→p(t) =−→v−→p(t)vextendsinglevextendsingle−→v−→p(t)vextendsinglevextendsingle.The unit tangent vector can be used as the direction vector for the tangentline of−→p(t) att=t0. Remark2.4.5suggests that the unit tangent isunchanged if we compute it using a reparametrization of−→p(t). This isalmost
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