SMath routines

d453ee08-0f5c-484c-9755-73d72091d335 276 4 5 false false 0 decimal

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Simple solutions to two problems

1.The construction of a sphere with respect to four given points 2.The construction of a circle with respect to three given points To calculate the coordinates of the center and radius using the solver AlgLib In the first problem, a system of four sphere equations for four given points is solved. In the second - a system of three equations of a circle and the equation of a plane for three given points.

SMath routines

2 f U V MeshPlot n.p 5 : n.u U length 1 - : n.v V length 1 - : r 1 : c 13 range : 15 mat M n.p n.u * n.v * n.u n.v 1 - (* + 3 matrix : F n.u 1 + n.v 1 + matrix : 12 mat u 1 n.u 1 + (range : v 1 n.v 1 + (range : F u v el U u el V v el f eval : 13 mat u 1 n.u range v 1 n.v range M r c el F u v el c el : M r 1 + c el F u v 1 + el c el : M r 2 + c el F u 1 + v 1 + el c el : M r 3 + c el F u 1 + v el c el : M r 4 + c el F u v el c el : r r n.p + : 61 line for k 0 n.v 2 - (range n u 1 - (n.p n.v * n.v + 1 - (* n.p v 1 - (* + k n.p * - : M r c el M n c el : r r 1 + : 13 mat for 21 line for M eval 51 line :

2 f u.1 u.2 v.1 v.2 n.u n.v MeshPlot u# v# f u.1 u.2 n.u LinSpace v.1 v.2 n.v LinSpace MeshPlot 11 line :

a b n LinSpace d b a - n 1 - / : v n 1 matrix : k 1 n range : v k el k 1 - (d * : 14 mat a v + eval 21 line :

1 f\032E\ a\032E\ N\032E\ εx\032E\ 1 J\032E\ NR x\0324\ a\032E\ : k\0324\ 1 N\032E\ range t\0324\ x\0324\ x\0324\ J\032E\ (1 -^x\0324\ f\032E\ * - eval : t\0324\ x\0324\ - min abs εx\032E\ < break x\0324\ t\0324\ : if 21 line for k\0324\ N\032E\ < t\0324\ NR: MaxItersRached. error if 31 line :

1.The construction of a sphere with respect to four given points

Координаты заданных точек

1

A 13 - 4 - 13 mat :

B 215413 mat :

C 411513 mat :

D 8 - 2 - 013 mat :

Solver:

O\0026\r X x xo Clear k 1 x f rows range X k el x k el : xo k el 1 : 12 mat for J X f X Jacob : x J J : 12 mat x# f xo 251011 -^x# J NR 41 line :

P x D2 P 1 el x 1 el - (2^P 2 el x 2 el - (2^+ P 3 el x 3 el - (2^+ x 4 el 2^- :

x1, x2, x3- coordinates of the center, x4-radius x f A x D2 B x D2 C x D2 D x D2 41 mat :

O O\0026\r 13 range el : 0.2447 3.9281 4.8186 31 mat

r O\0026\r 4 el : 11.24

plot

Поворот на угол α вокруг оси OX α Ω1 1000 α cos α sin - 0 α sin α cos 33 mat :

Поворот на угол φ вокруг оси OY φ Ω2 φ cos 0 φ sin 010 φ sin - 0 φ cos 33 mat :

γ 0.8232 - 0.4194 - 0.3827 0.5677 0.6187 - 0.543 0.009 0.6643 0.7474 33 mat :

R circle k 125 range x k el R π 12 / k * cos * eval : y k el R π 12 / k * sin * eval : z k el 0 : 31 line for x y z augment 21 line :

u v F u cos v cos * u cos v sin * u sin 31 mat :

ψ1 k 13 range a k el B k el A k el - : b k el C k el B k el - : c k el A k el C k el - : 31 line for axis a 2 el a 3 el 1 b 2 el b 3 el 1 c 2 el c 3 el 133 mat det a 3 el a 1 el 1 b 3 el b 1 el 1 c 3 el c 1 el 133 mat det a 1 el a 2 el 1 b 1 el b 2 el 1 c 1 el c 2 el 133 mat det stack : eaxis axis axis i el (2^i 13 sum sqrt / : (direction cosine matrices ≡ d eaxis 2 el (2^eaxis 3 el (2^+ sqrt eval : (We design a unit vector on the plane yOz ≡ ψ1 axis 1 el sign d acos * - Ω2 axis 2 el sign eaxis 3 el d 1015 -^+ / acos * Ω1 * eval : 51 line :

u v Fsp O u v F r * + :

ps u v F 0.04 * r * :

u v F1 A transpose ps + :

u v F2 B transpose ps + :

u v F3 C transpose ps + :

u v F4 D transpose ps + :

n 6 :

PA u v F1 02 π * 02 π * n n MeshPlot γ * :

PC u v F3 02 π * 02 π * n n MeshPlot γ * :

PB u v F2 02 π * 02 π * n n MeshPlot γ * :

PD u v F4 02 π * 02 π * n n MeshPlot γ * :

P clr Point P γ * (1 el P γ * (2 el . 24 clr augment :

SPH u v Fsp 02 π * 02 π * 1616 MeshPlot γ * :

3D PLOT

2D PLOT

1 t x3 2 t x3 1 t x1 2 t x1 1 t x2 2 t x2 1 t x3 2 t x3 4 2 mat 1 t x1 2 t x1 1 t x4 2 t x4 2 2 mat cil1 B1γ transpose 1 el B1γ transpose 2 el 1 t x4 2 t x4 2 2 mat stack 1 t x1 2 t x1 . 15 brown 1 5 mat 1 t x2 2 t x2 . 15 brown 1 5 mat 1 t x3 2 t x3 . 15 brown 1 5 mat C1γ transpose 1 el C1γ transpose 2 el C2γ transpose 1 el C2γ transpose 2 el C3γ transpose 1 el C3γ transpose 2 el C1γ transpose 1 el C1γ transpose 2 el 4 2 mat 1 t x2 2 t x2 1 t x5 2 t x5 2 2 mat cil2 B2γ transpose 1 el B2γ transpose 2 el 1 t x5 2 t x5 2 2 mat stack t C1 t C2 . 15 green 1 5 mat 1 t x4 2 t x4 . 20 1 5 mat 1 t x5 2 t x5 . 20 1 5 mat 1 t x3 2 t x3 1 t x6 2 t x6 2 2 mat cil3 B3γ transpose 1 el B3γ transpose 2 el 1 t x6 2 t x6 1 t x3 2 t x3 3 2 mat stack 1 t x6 2 t x6 . 20 1 5 mat Sy γ * 1 col Sy γ * 2 col augment XYZ t C1 t C2 t C1 t C2 0.1 - 2 2 mat TextXYZ t 4 < path3 t C1 t C2 0.1 - . 13 green 1 5 mat : t 0 ≡ path3 t C1 t C2 0.1 - . 13 green 1 5 mat : path3 path3 t C1 t C2 0.1 - . 13 green 1 5 mat stack : if if 21 1 sys PA PB PC PD SPH 5 1 sys A blue Point B red Point C green Point D violet Point SPH 1 SPH rows 1 2 submatrix 5 1 sys

2.The procedure for constructing a circle by three points

x1, x2, x3- coordinates of the center, x4-radius x f A x D2 B x D2 C x D2 x 1 el x 2 el x 3 el 1 A 1 el A 2 el A 3 el 1 B 1 el B 2 el B 3 el 1 C 1 el C 2 el C 3 el 144 mat det 41 mat :

O O\0026\r 13 range el : 2.2577 3.9606 4.4939 31 mat

r O\0026\r 4 el : 11.0534

Csp r circle ψ1 * :

Χ3 Csp 1 col O 1 el + Csp 2 col O 2 el + Csp 3 col O 3 el + augment γ * :

3D PLOT

2D PLOT

Circle and normal(rotate/zoom by mouse)

PA PB PC Χ3 5 1 sys A blue Point B red Point C green Point Χ3 1 Χ3 rows 1 2 submatrix 5 1 sys PA PB PC SPH Χ3 6 1 sys A blue Point B red Point C green Point SPH 1 SPH rows 1 2 submatrix Χ3 1 Χ3 rows 1 2 submatrix 6 1 sys