Saturday, May 20, 2017

Is the Minimum for the Four Point Fermat Problem Where We Thought?

  I've been trying to convince myself that minimum in the four point Fermat problem of the last blog is not slightly displaced from the point c. The plot below shows changes in the sum L=Σℓi of the lengths of the links from the known points to the unknown point x for changes along two lines, u and v through the point (0.700,0.700) in the plane of x.

Below 0.700 both lines decrease and increase above this value. The slopes are fairly linear on each side. But it's difficult to be certain that point c is the actual minimum just going by the data because of the discontinuity in the slope. Notice that the angle from horizontal is not the same for both lines. It may be possible for the slope on the right to be decrease also but at a lower rate. But under the circumstances it does look like c is the actual point of intersection for the two line segments.

Supplemental (May 21): Obviously we can't use an extension of Newton's method to solve this type of minimum problem since the gradients are not zero at the minimum. Fermat's theory of maxima and minima is not a general theory. When doing searches for curve fits one often encounters local minima that appear to be line segments. This might happen if the minimum is paraboloidal in shape and the contour lines are elliptical.

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