Some plane curvature approximations

Mjolsness, R. C.; Swartz, Blair

doi:10.1090/S0025-5718-1987-0890263-2

Some plane curvature approximations
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by R. C. Mjolsness and Blair Swartz PDF

Math. Comp. 49 (1987), 215-230 Request permission

Abstract:

Second-order accurate approximations to the curvature function along a sufficiently smooth plane curve are presented, the curve being given in finite form (and thus, approximately) by $N + 2$ points taken along its full length. The curvature estimates are continuous and invariant under translation and rotation, and they are based on local information—so are easy to implement computationally. In particular, second-order accurate estimates of surface tension forces halfway between immediate neighbors in the curve’s mesh can thereby be made for hydrodynamic simulations. The construction makes use of any of the common techniques one might contemplate for using the information present in three adjacent points (of the $N + 2$ points) in order to estimate the curve’s curvature near those three points. It may do this because each of these techniques yields a number which is, to within second order in the distances between the three points, the value of the true curvature function at the same place, namely, at the arithmetic mean of the location of the three points as measured along the curve. The asymptotic form, displaying all terms through the second order, of error estimates for these techniques is provided, along with comparison of gross properties and numerical examples. Finally, continuous, locally second-order accurate, global approximation to the curvature function is obtained by interpolation of successive local estimates between the locations of successive means. A related result is given for the simpler but analogous situation concerning the nth-order difference quotient of a function of one variable. The broken line interpolant of successive nth difference quotients, between the successive mean values of their stencil points, provides a continuous, locally second-order accurate, global approximation to the nth derivative. It also coincides, between two successive stencil means, with the nth derivative of the polynomial interpolant of the $n + 2$ data points associated with the two successive stencils.

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