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Reconstruction from Curvature

Given a function kg(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 kg(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 kg(t) and s(t) we can obtain an expression q'(t) =kg(t)s '(t) for the angle between T(t) and the x-axis, and then by integrating we get q(t) . We then have

    x'(t)=cos(q(t)) s'(t)
and
    y'(t)=sin(q(t)) s'(t)
Integrating these expressions with respect to t we obtain the coordinate functions x(t) and y(t) which determine X(t) .

Reconstruction from Curvature Demo

One a curvature function has been typed in, the curve will be drawn beginning at some arbitrary initial point. Thie initial point [0,0] has been chosen for simplicity.

The curve in this demo takes a little bit longer to draw than in some demos because it is being reconstructed from speed and curvature functions one small increment at a time.

While the set of equations one gets from for the coordinate curves in terms of the curvature and arclength are integrable and can therefore be solved explicitly, this is not what the demo does. The demo takes an arbitrary initial point and from there it constructs an approximation of the curve with the given curvature.

As its title suggests, this demo can reconstruct a curve in the plane simply from its curvature function and its speed function. The resulting curve will be unique up to an isometry in the plane, i.e. a rotation and/or a translation..

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

    What happens if we interpolate between the curvature function of a known curve and a constant curvature function, i.e. if we apply the reconstruction process to a curvature function k(t)=1-v+vk g(t) for a given function kg(t) ? .

Exploration

Plane Curves Exploration

There are two control panels, one for choosing a curve and domain and one for controlling the display of tangents, normal lines, osculating circles, parallel curves and the evolute to that curve. This demo is a useful tool for studying the relationships between various objects associated with families of curves.

This demo is very extensive and includes almost all of the important features of the above demos for studying the properties of curves.


Next: Inversion with Respect to a Circle