Math Quiz 10

[Ether]

Originally Posted by GeeTwo

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.

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.

Impressive work Gus.

Of the stock excel fits, polynomial (quadratic by default) is better than the others, but is not an exact match.

Try rotating the polar axis and fitting the power function r = θ^b

[Ether]

Using the analysis results from Gus' last post, and the hint in post 13, the final solution step is quite straightforward.

Open a LibreOffice spreadsheet and populate columns A and B with the XY data. Label the data in column A x_data. Label the data in column B y_data.

In column C put degrees from 139 to 199 stepping by 1.

In column D put radians equivalent of column C: colD=colC/180*pi()

Initialize cell O2 to zero

Label cell P2 "_b" and initialize it to one.

Label cell L2 "_a" and put the formula = O2/180*pi()

In column E put the 2-parameter polar equation model:
r = (colD - _a)^_b

In column G put formula colE*COS(colD)+1.16
Label the data cells in this column x_model

In column H put the formula colE*SIN(colD)-0.3
Label the data cells in this column y_model

In cell M2 put the formula SUMXMY2(x_data,x_model) + SUMXMY2(x_data,x_model). This is the objective function for the Solver.

Tools | Solver
Target Cell M2
Minimum
By changing cells O2 : P2
Solve
click continue a couple of times to refine the value

[GeeTwo]

When I saw θ^b, my mind read "theta flat" .

In any case, I found the answer a different way:
Assuming that r=θ^b, we know that dr/dθ = bθ^b-1, so that r/(dr/dθ) = θ/b. I therefore calculated the slope of r using the numbers given. The intercept of the line with r=0 is therefore (a good approximation to) θ₀, and the slope is (approximately) 1/b. My spreadsheet gave a θ₀ of 2.007223652 radians, or 115.0054438 degrees, and a b=0.499973651. These did not work out exactly, but a bit of tweaking found that 115 degrees and 0.5 worked "exactly".

Using the mathematical polar coordinate system centered on 1.16, -0.3 and rotated 115 degrees counterclockwise, the equation is simply:

r = √θ

Going back to calculate the original series:

D = [139..199]
θ₁ = D * π / 180
θ = (D-115) * π / 180
r = √θ
x = cosθ₁ + 1.16
y = sinθ₁ - 0.3

An updated spreadsheet with a "double check" tab is attached. The maximum error in the reconstructed x and y is less than 10^-14.