<p>Euler sustem Create mesh</p>

b4e06859-87a4-4148-8286-3e5d0db178f0 128 4 5 decimal

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The procedure for constructing a circle by three points in 3D

The calculation algorithm 1. The three points are vertices of the triangle. Write three linear equations: two equations of planes passing through the midpoints of the two sides of the triangle perpendicular to these sides: f1 = 0, f3 = 0 and the equation of the plane which contains the triangle: f2 = 0. 2. Solve the system of equations and find the center of a circle with coordinates Cen xCen, ... zCen. 3. Find the circle radius R of the equation of a sphere with radius R and center at Cen 4.Circle is constructed by matrix transformation, and not Dragilev Method

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 :

Normal L zum ax u v F 0.5 v sin * 0.5 v cos * u 31 mat : u v F1 4 u - 4 / 1.5 * v cos * 4 u - 4 / 1.5 * v sin * 1.5 u * L + 31 mat : P u v F 0 L 02 π * 112 CreateMesh : C u v F1 0402 π * 1612 CreateMesh : C P stack zum * 51 line :

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 F1 1 u cos * v cos * 1 u cos * v sin * 1 u sin * 31 mat :

zum S u v F1 02 π * 02 π * 88 CreateMesh zum * eval :

Координаты заданных точек

p1 81 - 4 - 13 mat transpose :

p2 2 - 3 - 513 mat transpose :

p3 3 - 4 - 11 - 13 mat transpose :

Cir3p3d k 13 range a k el p2 k el p1 k el - : b k el p3 k el p2 k el - : c k el p1 k el p3 k el - : 31 line for f 1 el a 1 el p1 1 el p2 1 el + 2 / x 1 el - (* a 2 el p1 2 el p2 2 el + 2 / x 2 el - (* + a 3 el p1 3 el p2 3 el + 2 / x 3 el - (* + : f 2 el x 1 el p1 1 el - x 2 el p1 2 el - x 3 el p1 3 el - p2 1 el p1 1 el - p2 2 el p1 2 el - p2 3 el p1 3 el - p3 1 el p1 1 el - p3 2 el p1 2 el - p3 3 el p1 3 el - 33 mat det : (Уравнение плоскости треугольника ≡ f 3 el c 1 el p1 1 el p3 1 el + 2 / x 1 el - (* c 2 el p1 2 el p3 2 el + 2 / x 2 el - (* + c 3 el p1 3 el p3 3 el + 2 / x 3 el - (* + : cntr f 1 el 0 ≡ f 2 el 0 ≡ f 3 el 0 ≡ 13 mat transpose x 1 el x 2 el x 3 el 13 mat transpose roots : (центр окружности ≡ rad p1 1 el cntr 1 el - (2^p1 2 el cntr 2 el - (2^+ p1 3 el cntr 3 el - (2^+ sqrt eval : (радиус окруж ≡ We find the transformation matrix ψ1 of the horizontal plane in a plane passing through three given points: ψ1 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 ≡ α2 eaxis 3 el d 1015 -^+ / acos eval : (Find the angles α2, β2 ≡ β2 d acos eval : ψ1 axis 1 el 0 ≥ (axis 2 el 0 ≥ (& β2 - Ω2 α2 Ω1 * eval axis 1 el 0 ≤ (axis 2 el 0 ≥ (& β2 Ω2 α2 Ω1 * eval axis 1 el 0 ≤ (axis 2 el 0 ≤ (& β2 Ω2 α2 - Ω1 * eval β2 - Ω2 α2 - Ω1 * eval if if if : ψ1 71 line : Circle and normal after transformation plotCircle center radius Normal rad circle ψ1 * cntr rad 20 0.2 ax ψ1 * 24 mat 111 line :

M Cir3p3d : C3 M 1 col 2 el : cntr M 2 col 2 el : Normal M 4 col 2 el : poγ cntr transpose γ * eval : 51 line

Circle and normal(rotate/zoom by mouse)

0.3 S γ * 1 col p1 transpose γ * 1 col 1 el + 0.3 S γ * 2 col p1 transpose γ * 2 col 1 el + 0.2 S γ * 3 col p1 transpose γ * 3 col 1 el + augment C3 γ * 1 col poγ 1 el + C3 γ * 2 col poγ 2 el + C3 γ * 3 col poγ 3 el + augment Normal γ * 1 col poγ 1 el + Normal γ * 2 col poγ 2 el + Normal γ * 3 col poγ 3 el + augment 0.3 S γ * 1 col p2 transpose γ * 1 col 1 el + 0.3 S γ * 2 col p2 transpose γ * 2 col 1 el + 0.2 S γ * 3 col p2 transpose γ * 3 col 1 el + augment 0.3 S γ * 1 col p3 transpose γ * 1 col 1 el + 0.3 S γ * 2 col p3 transpose γ * 2 col 1 el + 0.2 S γ * 3 col p3 transpose γ * 3 col 1 el + augment 9 1 sys

Cir3p3d plotCircle center radius 7.881 2.0104 2.0801 7.0659 1.9089 4.1119 5.7693 1.6772 5.8635 4.0795 1.3313 7.2155 2.1117 0.8946 8.0758 0 0.397 8.3857 2.1117 - 0.1277 - 8.1242 4.0795 - 0.6437 - 7.309 5.7693 - 1.1158 - 5.9957 7.0659 - 1.5119 - 4.2738 7.881 - 1.8049 - 2.2607 8.159 - 1.975 - 0.0935 7.881 - 2.0104 - 2.0801 - 7.0659 - 1.9089 - 4.1119 - 5.7693 - 1.6772 - 5.8635 - 4.0795 - 1.3313 - 7.2155 - 2.1117 - 0.8946 - 8.0758 - 1.4987 1015 -^* - 0.397 - 8.3857 - 2.1117 0.1277 8.1242 - 4.0795 0.6437 7.309 - 5.7693 1.1158 5.9957 - 7.0659 1.5119 4.2738 - 7.881 1.8049 2.2607 - 8.159 1.975 0.0935 - 7.881 2.0104 2.0801 253 mat 0.1286 - 2.9334 - 3.1836 - 31 mat 8.3951 23 mat