@MISC{Horn83thecurve, author = {B. K. P. Horn}, title = {The Curve Of Least Energy}, year = {1983} }

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Abstract

Here we search fi)r the curve which has the smallest integral of the square of curvature, while passing through two given points with given orientation. This is the true shape of a spline used in lofting. In computer-aided design, curves have been sought which maximize "smoothness". The curve discussed here is the one arising in this way from a commonly used measure of smoothness. The human visual system may use such a curve when it constructs a subjective contour.

...of the simple semicircle approximation. We have also given a method for finding approximations, consisting of circular arcs, to the curve of least energy. Note that the curve found here is extensible =-=[18]-=- in the sense that if the least energy curve with orientation a at A and orientation /S at B passes through the point C with orientation y, then the segments from A to C and from C to B are themselves...

...onstructed may have near-minimal energy. (This does not mean that it necessarily lies close to the curve of minimum energy, as we shall see.) Brady et al. used cubic polynomial approximations instead =-=[3]-=-. PREVIEW We first consider a special case. Here the curve must pass through the points (—1, 0) and (+1, 0) in the xy plane with vertical orientation at both points. We first determine the optimum cur...