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# FunctionInterpolation over an open interval

I'm trying to obtain an arc-length parametrization of a spline curve as per https://mathematica.stackexchange.com/a/8456/6986. To do so, I need to calculate partial derivatives of such curve and then perform a FunctionInterpolation. The spline curve is continuous over its whole domain, but not differentiable at its knots (left-side derivatives are different from the right-side ones). Therefore, I'd need to create several FunctionInterpolations, each defined on an open interval between two consecutive knots.

Is that possible? And if so, how? If not, how can I overcome this limitation?

Here is the actual data I'm working on:

curve=BSplineCurve[{{198.2063059205538`,402.121623269958`},{191.2621031932776`,243.2810494404048`},{350.2070352491573`,246.3653323645185`},{307.7702781407043`,406.7480444184375`},{624.1169877141547`,382.0737810255268`},{367.1817351705904`,154.607913753287`},{408.0753377407309`,28.15231386462074`},{192.0336877673071`,89.837972346897`}},SplineDegree->3