Metric <-> Imperial <-> Metric
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This Java Script calculator finds the center and radius of a sphere given four points in Cartesian coordinates. To operate the calculator, enter the x-y-z coordinates for four points. Press the Compute button to obtain the solution. On invalid entries, a popup window will display an error message
| Notes From analytic geometry, we know that there is a unique sphere that passes through four non-coplanar points if, and only if, they are not on the same plane. If they are on the same plane, either there are no spheres through the 4 points, or an infinite number of them if the 4 points are on a circle. Given 4 points,
how does
one find the center and radius of a sphere exactly fitting those points?
They can be found by solving the following determinant equation:
Evaluating the cofactors for the first row of the determinant can give us a solution. The determinant equation can be written as an equation of these cofactors: This can be converted to the canonical form of the equation of a sphere:(x2 + y2 + z2)·M11 - x·M12 + y·M13 - z·M14 + M15 = 0 Completing the squares in x and y and z gives:x2 + y2 + z2 - (M12/M11)·x + (M13/M11)·y - (M14/M11)·z + M15/M11 = 0 x0 = 0.5·M12/M11 y0 = -0.5·M13/M11 z0 = 0.5·M14/M11 r02 = x02 + y02 + z02 - M15/M11 Note
that there is no solution when M11 is equal
to zero. In this case, the points are not on a sphere; they may
all be on a plane or three points may be on a straight line. |