I had the same idea as MechEng83 last night, then started with his solution this evening (also in a spreadsheet, which I cleaned up just a bit and posted). My goal was to minimize the ratio of the standard deviation of the angle with its mean. (Actually, I used the sine of the angle, much easier to calculate with a cross product.) The point kept marching up and to the right. I eventually hit paydirt.

The center point where all of the angular displacements are the same is

(1.16, -0.3)

At this point, the angular displacements are all exactly 1 degree (to within the precision of the numbers as given). Using the mathematical convention (x axis is 0 degrees and the y axis is 90 degrees), the points span the range from 139 to 199 degrees, inclusive.

While the radial function increases monotonically as the points are given, it appears to be somewhat concave down. Of the stock excel fits, polynomial (quadratic by default) is better than the others, but is not an exact match.

An explanation of the spreadsheet is in order. The top two rows were used to create the “origin” for the polar coordinate system. Each cell to the right of C1 and C2 was added with another power of 1/10 to create C1 and C2. This let me manually migrate towards a minimum value for the scaled deviation. Row3 just has the results I was trying to optimize on. Row 4 is headers for the table. Below this, columns A and B are Ether’s data. Column C is the distance from the origin (not used). Columns D and E are the x and y values relative to the origin specified by (C1,C2). Column F is the distance of the point from the point (C1,C2). Column G is the change in the sine of the angle from (C1,C2) to this point and the next. Column H is the angle in degrees from (C1,C2) to Ether’s point, x towards y, in the range from 0 to 360.

OBTW, I’ve been working MQ9 for the cube extension off and on. I think I have all of the terms integrated on scratch paper, but I’m still writing them up and checking. I definitely have the mean distance from the origin to an arbitrary point complete, but the random line has eight times as many terms to start, and several of those are quite a bit nastier than the distance from the origin.

MQ10-polar-origin.xlsx (28.3 KB)

MQ10-polar-origin.xlsx (28.3 KB)