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
Nx√(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