Arc Length Parametrization

Differential Geometry · Axiom Academy

LESSON Arc Length Parametrization Understanding curves through their natural geometric parameter Given a smooth curve r ( t ) parametrized by t ∈ [ a , b ], the arc length from t = a to any point t is computed by integrating the speed: Here, r' ( u ) is the velocity vector and | r' ( u )| is the speed (magnitude of velocity). The arc length s ( t ) measures the actual distance traveled along the curve. 2. Arc Length as a Natural Parameter We can use arc length itself as the parameter for our curve. When we do this, we're parametrizing the curve by the distance traveled along it, starting from some reference point. If the original curve is r ( t ) with arc length function s ( t ), we can (in principle) invert this relationship to get t = t ( s ), and then define: The most important property of arc length parametrization is that the curve has unit speed —that is, the magnitude of the velocity vector is always 1: Proof: By the chain rule, if α ( s ) = r ( t ( s )), then: Since s = ∫ a t | r' ( u )| d u , we have d s /d t = | r' ( t )|. Therefore d t /d s = 1/| r' ( t )| and: 4. How to Reparametrize by Arc Length To reparametrize a curve r ( t ) by arc length, follow these steps: Example: Consider the helix r ( t ) = (cos t , sin t , t ). 5. Why Arc Length Parametrization is Useful Arc length parametrization is fundamental in differential geometry because it separates the geometry of the curve from the speed at which we traverse it.

This is the written version of the interactive lesson above. See the full Differential Geometry course.