2.6: Reconstruction from Curvature

Given a function κg(t) and a positive length function s(t), can we always find a curve X(t) such that the arclength of X(t) is s(t) and the signed curvature is κg(t)? It turns out that we can always find a curve with such properties which can be made unique by a choice of boundary conditions on X(t) and T(t). From κg(t) and s(t) we can obtain an expression θ'(t) = κg(t)s'(t) for the angle between T(t) and the x-axis, and then by integrating we get θ(t). We then have

x'(t) = cos(θ(t))s'(t)

y'(t) = sin(θ(t))s'(t)

Integrating these expressions with respect to t we obtain the coordinate functions x(t) and y(t) which determine X(t).


This demo allows you to reconstruct a curve, given its curvature function κg(t) and its length function s(t). By default, the demo uses the initial condition X(0) = 0. The graphs of κg(t) and s(t) are shown in two separate windows. The reconstructed curve, X(t), is colored according to the value of signed geodesic curvature. To verify that the curve has the correct speed function, use the t0 tapedeck in the control panel to make the point X(t0) trace out the curve. Compare the motion of this point with the graph of s(t).



1. Investigate what happens if we start with a constant function κg(t) = 1/R and a constant speed s'(t) = 1. What if k(t) = t for all t?

2. What happens if we interpolate between the curvature function of a known curve and a constant curvature function. For example, what if we apply the reconstruction process to a curvature function κ(t) = 1 - v + v*κg(t) for a given function κg(t)?
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