Friday, December 9, 2016

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

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:",L
Input "LIST Y:",L
dim(L)→N
√(L²+L²)→L
max(L)→R
√(sum((L-R)²)/N)→S
Disp "MAX RADIUS:",√(R)
Disp "RMS:",S
Pause
mean(L)→A
√(sum((L-A)²)/N)→B
Disp "ARITH. AVG.:",√(A)
Disp "RMS:",B
Pause
Nx√(prod(L))→G
√(sum((L-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




2 comments:

  1. I guess about this the best blog I have read all this hour.
    merchant cash advance calculator

    ReplyDelete
  2. Very helpful suggestions that help in the optimizing website.
    I really like you post.Thanks for sharing.
    Please click this post,if you wanna join casino online. Thank you
    gclub
    gclub casino online
    จีคลับ

    ReplyDelete

HHC 2017 In Review

HHC 2017 In Review Hello Nashville, TN! HHC 2017 took place on September 16 and 17, 2017 in Brentwood, TN.  If you have not go...