{"id":15276,"date":"2022-10-11T01:12:12","date_gmt":"2022-10-10T19:42:12","guid":{"rendered":"https:\/\/jassweb.com\/solved\/solved-sphere-center-point-and-radius-from-3-points-on-the-surface\/"},"modified":"2022-10-11T01:12:12","modified_gmt":"2022-10-10T19:42:12","slug":"solved-sphere-center-point-and-radius-from-3-points-on-the-surface","status":"publish","type":"post","link":"https:\/\/jassweb.com\/solved\/solved-sphere-center-point-and-radius-from-3-points-on-the-surface\/","title":{"rendered":"[Solved] Sphere center point and radius from 3 points on the surface"},"content":{"rendered":"

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Express that the center of the sphere is equidistant to the three given points and coplanar with them (assuming that the three given points are on a great circle).<\/p>\n

(X - Xa)\u00b2 + (Y - Ya)\u00b2 + (Z - Za)\u00b2 = R\u00b2    \n(X - Xb)\u00b2 + (Y - Yb)\u00b2 + (Z - Zb)\u00b2 = R\u00b2    \n(X - Xc)\u00b2 + (Y - Yc)\u00b2 + (Z - Zc)\u00b2 = R\u00b2    \n\n|X  Y  Z  1|\n|Xa Ya Za 1|\n|Xb Yb Zb 1| = 0\n|Xc Yc Zc 1|\n<\/code><\/pre>\n
Subtracting the first equation from the second and the third, you get rid of the quadratic terms.<\/p>\n
(2X - Xb - Xa)(Xb - Xa) + (2Y - Yb - Ya)(Yb - Ya) + (2Z - Zb - Za)(Zb - Za) = 0\n(2X - Xc - Xa)(Xc - Xa) + (2Y - Yc - Ya)(Yc - Ya) + (2Z - Zc - Za)(Zc - Za) = 0\n<\/code><\/pre>\n
Now you have an easy linear system of 3 equations in 3 unknowns.<\/p>\n
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For conciseness you can translate the three points so that Xa=Ya=Za=0<\/code>, and the equations simplify as<\/p>\n
|X  Y  Z |\n|Xb Yb Zb| = 0\n|Xc Yc Zc|\n\n(2X - Xb) Xb + (2Y - Yb) Yb + (2Z - Zb) Zb = 0\n(2X - Xc) Xc + (2Y - Yc) Yc + (2Z - Zc) Zc = 0\n<\/code><\/pre>\n
or<\/p>\n
(Yb Zc - Yc Zb) X + (Zb Xc - Zc Xb) Y + (Xb Yc - Xc Yb) Z = 0\n       2 Xb     X +        2 Yb     Y +        2 Zb     Z = Xb\u00b2 + Yb\u00b2 + Zb\u00b2\n       2 Xc     X +        2 Yc     Y +        2 Zc     Z = Xc\u00b2 + Yc\u00b2 + Zc\u00b2\n<\/code><\/pre>\n
Then, R\u00b2 = X\u00b2 + Y\u00b2 + Z\u00b2<\/code>, and don’t forget to translate back.<\/p>\n<\/p><\/div>\n
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solved Sphere center point and radius from 3 points on the surface <\/p>\n","protected":false},"excerpt":{"rendered":"
[ad_1] Express that the center of the sphere is equidistant to the three given points and coplanar with them (assuming that the three given points are on a great circle). (X – Xa)\u00b2 + (Y – Ya)\u00b2 + (Z – Za)\u00b2 = R\u00b2 (X – Xb)\u00b2 + (Y – Yb)\u00b2 + (Z – Zb)\u00b2 = … Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[320],"tags":[1999,739,357],"class_list":["post-15276","post","type-post","status-publish","format-standard","hentry","category-solved","tag-3d","tag-geometry","tag-math"],"yoast_head":"\n[Solved] Sphere center point and radius from 3 points on the surface - JassWeb<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/jassweb.com\/solved\/solved-sphere-center-point-and-radius-from-3-points-on-the-surface\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"[Solved] Sphere center point and radius from 3 points on the surface - JassWeb\" \/>\n<meta property=\"og:description\" content=\"[ad_1] Express that the center of the sphere is equidistant to the three given points and coplanar with them (assuming that the three given points are on a great circle). (X - Xa)\u00b2 + (Y - Ya)\u00b2 + (Z - Za)\u00b2 = R\u00b2 (X - Xb)\u00b2 + (Y - Yb)\u00b2 + (Z - Zb)\u00b2 = ... 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(X - Xa)\u00b2 + (Y - Ya)\u00b2 + (Z - Za)\u00b2 = R\u00b2 (X - Xb)\u00b2 + (Y - Yb)\u00b2 + (Z - Zb)\u00b2 = ... 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