**HP Prime and TI-84 Plus: Fitting Points to a Circle**

**Introduction**

The program
CIRCFIT attempts to fit a collection of points to the circle:

x^2 + y^2 = r^2

The center is
assumed to be the origin (0,0).

The estimated
radius is measured in three ways:

Maximum: The radius is determined by the point that is
furthest away from the origin.

Arithmetic
Mean: The radius is determined by the
arithmetic mean (Σx/n) of all the radii.

Geometric
Mean: The radius is determined by the
geometric mean ((Πx)^(1/n)) of all the radii.

For each estimated
radius, the root mean square (RMS, see formula below) is calculated. The object is to get the RMS as low as
possible. I tested several fits and its
seems that the radius determined by the arithmetic mean gives the lowest RMS.

RMS = √( Σ((y_i
– mean)^2) / n)

**HP Prime Program: CIRCFIT**

EXPORT CIRCFIT(lx,ly)

BEGIN

// "CIRCLE FIT TEST"

// "EWS 2016-12-09"

LOCAL n,lr,r,s;

LOCAL a,b,g,h;

n:=SIZE(lx);

lr:=√(lx^2+ly^2);

PRINT();

r:=(MAX(lr));

s:=√(ΣLIST((lr-r)^2)/n);

PRINT("Maximum radius:
"+√r);

PRINT("RMS: "+s);

PRINT("---------");

a:=mean(lr);

b:=√(ΣLIST((lr-a)^2)/n);

PRINT("Arithmetic mean:
"+√a);

PRINT("RMS: "+b);

PRINT("---------");

g:=n NTHROOT ΠLIST(lr);

h:=√(ΣLIST((lr-g)^2)/n);

PRINT("Geometric mean:
"+√g);

PRINT("RMS: "+h);

END;

**TI-84 Plus Program CIRCFIT**

"CIRCLE FIT TEST"

"EWS 2016-12-09"

Input "LIST X:",L₁

Input "LIST Y:",L₂

dim(L₁)→N

√(L₁²+L₂²)→L₃

max(L₃)→R

√(sum((L₃-R)²)/N)→S

Disp "MAX RADIUS:",√(R)

Disp "RMS:",S

Pause

mean(L₃)→A

√(sum((L₃-A)²)/N)→B

Disp "ARITH. AVG.:",√(A)

Disp "RMS:",B

Pause

N

^{x}√(prod(L₃))→G
√(sum((L₃-G)²)/N)→H

Disp "GEOM. AVG.:",√(G)

Disp "RMS:",H

Examples

Each of the results are rounded to 5
digits.

Example 1:

X |
Y |

0.99 |
0.09 |

0.56 |
-0.82 |

-0.36 |
0.72 |

-0.96 |
-0.08 |

Maximum Radius: 0.99704, RMS: 0.09579

Arithmetic Mean: 0.96894, RMS: 0.07826

Geometric Mean: 0.96714, RMS:
0.07834

Example 2:

X |
Y |

1.02 |
0.01 |

0.96 |
-0.05 |

0.86 |
0.16 |

0.77 |
0.24 |

0.64 |
-0.33 |

0.03 |
1.00 |

Maximum Radius: 1.00997, RMS:
0.16356

Arithmetic Mean: 0.94724, RMS:
0.10798

Geometric Mean: 0.94362, RMS: 0.10819

This blog is property of Edward
Shore, 2016