Linear Algebra and the C Language/a04y
Install and compile this file in your working directory.
/* ------------------------------------ */
/* Save as: c00c.c */
/* ------------------------------------ */
#include "v_a.h"
/* ------------------------------------ */
/* ------------------------------------ */
#define RA R5
#define CA C5
#define Cb C1
/* ------------------------------------ */
/* ------------------------------------ */
int main(void)
{
double txy[6] ={
1, -2,
2, -3,
3, 6 };
double ab[RA*(CA+Cb)]={
/* x**2 y**2 x y e = 0 */
+1.00, +0.00, +0.00, +0.00, +0.00, +1.00,
+0.00, +1.00, +0.00, +0.00, +0.00, +1.00,
+1.00, +4.00, +1.00, -2.00, +1.00, +0.00,
+4.00, +9.00, +2.00, -3.00, +1.00, +0.00,
+9.00, +36.00, +3.00, +6.00, +1.00, +0.00,
};
double **xy = ca_A_mR(txy, i_mR(R3,C2));
double **Ab = ca_A_mR(ab, i_Abr_Ac_bc_mR(RA,CA,Cb));
double **A = c_Ab_A_mR(Ab, i_mR(RA,CA));
double **b = c_Ab_b_mR(Ab, i_mR(RA,Cb));
double **A_T = transpose_mR(A, i_mR(CA,RA));
double **A_TA = mul_mR(A_T,A, i_mR(CA,CA));
double **invA_TA = inv_mR(A_TA, i_mR(CA,CA));
double **invA_TAA_T = mul_mR(invA_TA,A_T, i_mR(CA,RA));
double **x = mul_mR(invA_TAA_T,b, i_mR(CA,Cb));
/* x = inv(A_TA)A_T b */
clrscrn();
printf("\n");
printf(" Find the coefficients a, b, c, d, of a circle \n\n");
printf(" ax**2 + ay**2 + bx + cy + d = 0 \n\n");
printf(" that passes through these three xy. \n\n");
printf(" x y");
p_mR(xy, S5,P0,C6);
printf("\n");
printf(" Using the given xy, we obtain this matrix.\n");
printf(" (a = 1. This is my choice)\n\n");
printf(" x**2 y**2 x y ");
p_mR(Ab, S7,P2,C6);
stop();
clrscrn();
printf(" A_T:");
p_mR(A_T, S10,P2,C7);
printf(" A_TA:");
p_mR(A_TA, S10,P2,C7);
stop();
clrscrn();
printf(" inv(A_TA):");
p_mR(invA_TA, S10,P4,C7);
printf(" inv(A_TA)A_T:");
p_mR(invA_TAA_T, S10,P4,C7);
printf("\n x = inv(A_TA)A_T b:");
p_mR(x, S10,P4,C7);
stop();
clrscrn();
printf("\n x = inv(A_TA)A_T b:");
p_mR(x, S10,P4,C7);
printf(" The coefficients a, b, c, d, e, of the curve are: \n\n"
" %+.2fx**2 %+.2fy**2 %+.2fx %+.2fy %+.2f = 0\n\n"
,x[R1][C1],x[R2][C1],x[R3][C1],x[R4][C1],x[R5][C1]);
stop();
f_mR(xy);
f_mR(A);
f_mR(b);
f_mR(Ab);
f_mR(A_T);
f_mR(A_TA);
f_mR(invA_TA);
f_mR(invA_TAA_T);
f_mR(x);
return 0;
}
/* ------------------------------------ */
/* ------------------------------------ */
Presentation :
Find the coefficients a, b, c, d of the circle,
ax**2 + ay**2 + bx + cy + d = 0
which crosses these three points
(x[1],y[1]) (x[2],y[2]) (x[3],y[3])
Using the three points we obtain the matrix.
(a)x**2 (a)y**2 (b)x (c)y (d) = 0
x[1]**2 y[1]**2 x[1] y[1] 1 0
x[2]**2 y[2]**2 x[2] y[2] 1 0
x[3]**2 y[3]**2 x[3] y[3] 1 0
This system has three lines and four unknowns (a, b, c, d).
It is a homogeneous system, so it has an infinite number of solutions.
To find a solution, I chose to set a = 1.
We therefore have five rows and five unknowns.
(a)x**2 (a)y**2 x y
1 0 0 0 0 1
0 1 0 0 0 1
x[1]**2 y[1]**2 x[1] y[1] 1 0
x[2]**2 y[2]**2 x[2] y[2] 1 0
x[3]**2 y[3]**2 x[3] y[3] 1 0
All that remains is to solve the system.
Screen output example:
Find the coefficients a, b, c, d, of a circle
ax**2 + ay**2 + bx + cy + d = 0
that passes through these three XY.
x y
+1 -2
+2 -3
+3 +6
Using the given XY, we obtain this matrix.
(a = 1. This is my choice)
x**2 y**2 x y
+1.00 +0.00 +0.00 +0.00 +0.00 +1.00
+0.00 +1.00 +0.00 +0.00 +0.00 +1.00
+1.00 +4.00 +1.00 -2.00 +1.00 +0.00
+4.00 +9.00 +2.00 -3.00 +1.00 +0.00
+9.00 +36.00 +3.00 +6.00 +1.00 +0.00
Press return to continue.
A_T :
+1.00 +0.00 +1.00 +4.00 +9.00
+0.00 +1.00 +4.00 +9.00 +36.00
+0.00 +0.00 +1.00 +2.00 +3.00
+0.00 +0.00 -2.00 -3.00 +6.00
+0.00 +0.00 +1.00 +1.00 +1.00
A_TA :
+99.00 +364.00 +36.00 +40.00 +14.00
+364.00 +1394.00 +130.00 +181.00 +49.00
+36.00 +130.00 +14.00 +10.00 +6.00
+40.00 +181.00 +10.00 +49.00 +1.00
+14.00 +49.00 +6.00 +1.00 +3.00
Press return to continue.
inv(A_TA) :
+1.0000 -0.0000 -3.2000 -0.2000 +1.8000
-0.0000 +1.0000 -7.2000 -2.2000 -1.2000
-3.2000 -7.2000 +63.5400 +16.2400 +0.0400
-0.2000 -2.2000 +16.2400 +4.9400 +2.7400
+1.8000 -1.2000 +0.0400 +2.7400 +10.5400
inv(A_TA)*A_T :
+1.0000 +0.0000 -0.0000 -0.0000 -0.0000
+0.0000 +1.0000 -0.0000 -0.0000 -0.0000
-3.2000 -7.2000 -0.9000 +0.8000 +0.1000
-0.2000 -2.2000 +0.1000 -0.2000 +0.1000
+1.8000 -1.2000 +2.1000 -1.2000 +0.1000
x = inv(A_TA)*A_T*b :
+1.0000
+1.0000
-10.4000
-2.4000
+0.6000
Press return to continue.
x = inv(A_TA)*A_T*b :
+1.0000
+1.0000
-10.4000
-2.4000
+0.6000
The coefficients a, b, c, d, e, of the curve are :
+1.00x**2 +1.00y**2 -10.40x -2.40y +0.60 = 0
Press return to continue.
Copy and paste in Octave:
function xy = f (x,y)
xy = +1.00*x^2 +1.00*y^2 -10.40*x -2.40*y +0.60;
endfunction
f (+1,-2)
f (+2,-3)
f (+3,+6)