Engineering Calculus Notes 191

# Engineering Calculus Notes 191 - 179 2.4 REGULAR CURVES...

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2.4. REGULAR CURVES 179 Remark 2.4.5. Suppose −→ p ( t ) is a reparametrization of −→ q ( s ) , with recalibration function t ( s ) . Then the velocity vectors at corresponding parameter values are linearly dependent: if t 0 = t ( s 0 ) then −→ v −→ p ( t 0 ) = t ( s 0 ) −→ v −→ q ( s 0 ) . Intuitively, reparametrizing a curve amounts to speeding up or slowing down the process of tracing it out. Since the speed with which we trace out a curve is certainly not an intrinsic property of the curve itself, we can try to eliminate the effects of such speeding up and slowing down by concentrating on the unit tangent vector determined by a parametrization −→ p ( t ), −→ T −→ p ( t ) = −→ v −→ p ( t ) vextendsingle vextendsingle −→ v −→ p ( t ) vextendsingle vextendsingle . The unit tangent vector can be used as the direction vector for the tangent line of −→ p ( t ) at t = t 0 . Remark 2.4.5 suggests that the unit tangent is unchanged if we compute it using a reparametrization of −→ p ( t ). This is almost
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