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Isaac Newton Institute for Mathematical Sciences

Euler-Lagrange Equations for Missing Curve Problem

Author: Jun Zhao (University of Kent)

Abstract

Motivation For two given paths (smooth curves), how to find a best path (missing smooth curve) which connects the two given path under certain criterion?

Continuous Euler-Lagrange Equation The solution of the missing curve is not unique, even a straight line can be a solution. Since the human vision system is Mother Nature's biological device for carrying out intelligent visual perception, the best solution must be able to fool human eyes. Since the solution needs to be equivariant under certain group actions, it describes a variational problem: $\mathcal L=\int{L}$, where $\mathcal L$ is a functional and $L$ is Lagrangian. The very famous Lagrangian used to solve the missing curve problem is $L=\kappa^2{\rm d}s$, where $\kappa$ is the curvature and $s$ is the arc length. The corresponding Euler-Lagrange equation for this variational problem is $\kappa_{ss}+\frac{1}{2}\kappa^3=0$, and the solution to this Euler-Lagrange equation is called the Euler's elastica. Since the solution is equivariant under the actions of rotation and translation, it is much easier to transform the original curves to the ''normalized position'' by using the moving frame method and find the solution, then transform them back to the original position. For the planar curves, the moving frames can be obtained by solving the normalization equations. But for the space curves, it's really hard to solve the moving frames directly. But we can manage to obtain the Euler-Lagrange equations without solving the moving frames explicitly by using the syzygies (differential relationship) between the invariants. Once we get the Euler-Lagrange equations, we can use different methods to solve the non-linear equation system, e.g. Newton methods. But the calculation is extremely complex in the space curve case and even in the planar curve case, the denominators which contain square roots in the curvature term will make the computation difficult, so we need another method to solve the problem easily without significant loss of accuracy.

Discrete Euler-Lagrange Equation In the industrial field of computer graphic design, using polynomials to approximate actual curves is very practical and B-spline basis function curves has proved to be one of the most efficient methods due to their outstanding properties. We will use B-spline curves to simplify our variational problems especially because of its affine invariance property, which implies, the curve after transformation can be constructed by transform the control points. Hence we can convert the differential invariants problem to the finite difference invariants problem which is much easier. Instead of finding the solution curve, we only need to find the control points of the solution B-spline curve, and use them to construct the solution curve. This leads us to the problem of finding and solving discrete Euler-Lagrange equations. In the planar curve case, the Lagrangian can be obtained by solving the moving frames and using the replacement theorem. In the space curve case, the computation will be very complex if we use the so called symbolic replacement theorem to find the discrete invariants. Construct alternate difference invariants, we will be able to get the discrete syzygies, hence find the discrete Euler-Lagrange equations.