re sphere from points in article hcrlgw crl hitachi co jp steve hcrlgw steven collins writes in article qkgbuinns n shelley u washington edu bolson carson u washington edu edward bolson writes boy this will be embarassing if it is trivial or an faq given points non coplanar how does one find the sphere that is center and radius exactly fitting those points i know how to do it for a circle from points but do not immediately see a straightforward way to do it in d i have checked some geometry books graphics gems and farin but am still at a loss please have mercy on me and provide the solution wouldn t this require a hyper sphere in space points over specifies a sphere as far as i can see unless that is you can prove that a point exists in space that is equi distant from the points and this may not necessarily happen correct me if i m wrong which i quite possibly am steve sorry call the four points a b c and d any three of them must be non collinear otherwise all three could not lie on the surface of a sphere and all four must not be coplaner otherwise either they cannot all lie on a sphere or they define an infinity of them a b and c define a circle the perpendicular bisectors of ab bc and ca meet in a point p say which is the centre of this circle this circle must lie on the surface of the desired sphere consider the normal to the plane abc passing through p all points on this normal are equidistant from a b and c and its circle in fact it is a diameter of the desired sphere take the plane containing this normal and d if d lies on the normal any plane containing the normal will do this plane is at right angles to the abc one let e be the point there are normally two of them on the circumference of the abc circle which lies in this plane we need a point q on the normal such that eq dq but the intersection of the perpendicular bisector of ed and the normal is such a point and it exists since d is not in the plane abc and so ed is not at right angles to the normal algorithm is the sphere well defined check that a and b are not coincident failure find the line ab and check that c does not lie on it failure find the plane abc and check that d does not lie in it failure yes find its centre find the perpendicular bisectors of ab and ac find their point of intersection p find the normal to the plane abc passing through p line n find the plane containing n and d find the point e on the abc circle in this plane if d lies on n take e as a find the perpendicular bisector of ed line l find the point of intersection of n and l q q is the centre of the desired sphere pictures in the plane abc a p b c at right angles to abc in the plane containing n and d e d line n p q numerically if ed ep then q will be very close to p relative to the radius of the abc circle and subject to error it s best to choose d so that the least of ad bd and cd is larger than for any other choice bob douglas computing services university of oxford internet bob oxford ac uk address banbury road oxford ox nn uk telephone
