5076e8f4-d8cf-46b6-95bf-e179d076e0b7 27 4 5 decimal

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The construction of a sphere with respect to four given points

http://mathworld.wolfram.com/Circumsphere.html

1. We compose the equation of the circle for each of the four points.

2.We subtract the fourth equation for each of the first three equation to get three linear equations with three unknown coordinates of the center

3.Solving the system of three linear equations.

p1 12 - 1 - 13 mat :

p2 215413 mat :

p3 411513 mat :

p4 8 - 2 - 213 mat :

Solving the system of three linear equations D p4 transpose (2^p1 transpose (2^- p4 transpose (2^p2 transpose (2^- p4 transpose (2^p3 transpose (2^- 31 mat : (The matrix of coefficients of the system ≡ a 2 p4 p1 - p4 p2 - p4 p3 - stack * : (Vector of right parts ≡ C a 1 -^D * : (Coordinates of the sphere center ≡ L C p1 transpose - : r L transpose L * (1 el sqrt : (The radius of the sphere ≡ γ 0.8232 - 0.4194 - 0.3827 0.5677 0.6187 - 0.543 0.009 0.6643 0.7474 33 mat : u v F1 u cos v cos * u cos v sin * u sin 31 mat : zum S u v F1 02 π * 02 π * 88 CreateMesh zum * γ * eval : r Sp u v F1 02 π * 02 π * 2020 CreateMesh r * γ * eval : 101 line

r 10.7724

C transpose 1.4402 - 5.1496 6.6795 13 mat

Finding a Sphere From Four Points

0.2 S 1 col p1 γ * 1 col 1 el + 0.2 S 2 col p1 γ * 2 col 1 el + 0.2 S 3 col p1 γ * 3 col 1 el + augment 0.2 S 1 col p2 γ * 1 col 1 el + 0.2 S 2 col p2 γ * 2 col 1 el + 0.2 S 3 col p2 γ * 3 col 1 el + augment 0.2 S 1 col p3 γ * 1 col 1 el + 0.2 S 2 col p3 γ * 2 col 1 el + 0.2 S 3 col p3 γ * 3 col 1 el + augment r Sp 1 col C transpose γ * 1 col 1 el + r Sp 2 col C transpose γ * 2 col 1 el + r Sp 3 col C transpose γ * 3 col 1 el + augment 0.2 S 1 col p4 γ * 1 col 1 el + 0.2 S 2 col p4 γ * 2 col 1 el + 0.2 S 3 col p4 γ * 3 col 1 el + augment p1 C transpose p2 C transpose p3 C transpose p4 stack γ * 7 1 sys