Mathc matrices/c23q
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c00s.c |
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/* ------------------------------------ */
/* Save as : c00s.c */
/* ------------------------------------ */
#include "v_a.h"
/* ------------------------------------ */
/* ------------------------------------ */
#define RA R5
#define CA C5
#define Cb C1
/* ------------------------------------ */
/* ------------------------------------ */
void fun(void)
{
double xy[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(xy,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 = i_mR(CA,RA);
double **A_TA = i_mR(CA,CA); // A_T*A
double **invA_TA = i_mR(CA,CA); // inv(A_T*A)
double **invA_TAA_T = i_mR(CA,RA); // inv(A_T*A)*A_T
double **x = i_mR(CA,Cb); // x = inv(A_T*A)*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(transpose_mR(A,A_T),S10,P2,C7);
printf(" A_TA :");
p_mR(mul_mR(A_T,A,A_TA),S10,P2,C7);
stop();
clrscrn();
printf(" inv(A_TA) :");
p_mR(inv_mR(A_TA,invA_TA),S10,P4,C7);
printf(" inv(A_TA)*A_T :");
p_mR(mul_mR(invA_TA,A_T,invA_TAA_T),S10,P4,C7);
printf("\n x = inv(A_TA)*A_T*b :");
p_mR(mul_mR(invA_TAA_T,b,x),S10,P4,C7);
stop();
clrscrn();
printf("\n x = inv(A_TA)*A_T*b :");
p_mR(x,S10,P2,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); // A_T*A
f_mR(invA_TA); // inv(A_T*A)
f_mR(invA_TAA_T); // inv(A_T*A)*A_T
f_mR(x);
}
/* ------------------------------------ */
int main(void)
{
fun();
return 0;
}
/* ------------------------------------ */
/* ------------------------------------ */
Calculons les coefficients a, b, c, d d'un cercle,
ax**2 + ay**2 + bx + cy + d = 0
Qui passe par ces trois points.
(x[1],y[1]) (x[2],y[2]) (x[3],y[3])
En utilisant ces trois points nous avons cette matrice.
(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
Ce système a trois lignes et quatre inconnues. Il est homogène, donc il a une infinité de solution.
Pour trouver une solution j'ai choisi que a = 1.
Nous obtenons cette matrice.
(a)x**2 (a)y**2
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
Il suffit de resoudre le système. Exemple de sortie écran :
-----------------------------------
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 :
+1.00
+1.00
-10.40
-2.40
+0.60
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.