Finding a Sphere from Geometric Conditions

From Geometric Information to an Equation

A sphere can be described in several ways: its center and radius may be given directly, two endpoints may form a diameter, or one or more points may be required to lie on its surface. Each description imposes equations on the coordinates of the center and the radius. The main task is to translate the information carefully and then check that the equation satisfies it.

For a center \(C=(h,k,\ell)\) and radius \(r>0\), a point \(P=(x,y,z)\) lies on the sphere exactly when its distance from \(C\) is \(r\). By the three-dimensional distance formula, this condition is

This equation is useful even when the center or radius is not stated directly: substitute the coordinates of any specified point on the sphere, and the result is an equation the unknown quantities must satisfy.

When the Center Is Known

If the center is known and a point on the sphere is supplied, the radius is the distance from the center to that point. The radius is nonnegative, while it is often most convenient to calculate its square first. The resulting equation can then be written directly in center-radius form.

When the Endpoints of a Diameter Are Known

If \(A\) and \(B\) are the endpoints of a diameter, the center is the midpoint of the segment \(AB\). The radius is half the distance from \(A\) to \(B\). In particular, if \(A=(x_1,y_1,z_1)\) and \(B=(x_2,y_2,z_2)\), then

The midpoint and distance formulas provide the center and radius; these are then placed in the sphere equation.

When Points Must Determine the Sphere

Suppose the center and radius are both unknown, but points on the sphere are given. One way to find the sphere is to use the general expanded equation

Here \(A\), \(B\), \(C\), and \(D\) are unknown constants. Substituting each point on the sphere gives a linear equation in those four constants. Once they are found, completing squares gives the center and radius. This use of completing squares is a way to read the geometric information from the equation, not a separate condition on the points.

Why Subtracting Point Equations Helps

Each point substitution contains the same quadratic part of the unknown point coordinates, but those coordinates are fixed numbers. Subtracting one point equation from another cancels \(D\) and leaves an equation linear in \(A\), \(B\), and \(C\). This reduces the algebra and has a geometric meaning: points on the same sphere are equally distant from its center.

To see the reason, let \(P=(x_1,y_1,z_1)\) and \(Q=(x_2,y_2,z_2)\) both lie on a sphere centered at \((h,k,\ell)\). Their squared distances to the center are equal. Subtracting those squared-distance expressions cancels \(h^2+k^2+\ell^2\), giving

This is linear in the unknown center coordinates \(h,k,\ell\). Thus equal-distance conditions become linear equations when one squared-distance equation is subtracted from another. The same cancellation appears when subtracting substitutions into the expanded sphere equation.

Conditions Must Be Compatible

Not every collection of geometric conditions describes a sphere, and conditions do not always specify exactly one sphere. For example, two endpoints alone do not determine a unique sphere unless they are stated to be the endpoints of a diameter: many spheres can pass through two given points. Likewise, when several points are specified, the equations obtained by substitution must be consistent. If they are inconsistent, no sphere passes through all the points.

Four points that do not lie in a single plane determine at most one sphere. One way to see the uniqueness is to subtract the first point equation from the other three. The resulting three equations are linear in \(A,B,C\). For points not in one plane, these equations determine at most one triple \((A,B,C)\); the first point equation then determines \(D\). If the points do lie in one plane, the same equations need not determine a unique sphere. Therefore, count and placement of points both matter.

These checks are useful before a long calculation: identify exactly what is given, write the equations implied by those conditions, and check that the solution satisfies every condition rather than just one of them.

A Common Mistake: Using the Diameter as the Radius

If two points are diameter endpoints, their full separation is the diameter, not the radius. For the endpoints \(A=(-2,1,4)\) and \(B=(4,3,0)\), the squared distance is \(56\), but this is the squared diameter length. The correct squared radius is \(56/4=14\). Using \(56\) as \(r^2\) would produce a sphere twice as large in radius as the correct one.

The check exposes the error: the center is \((1,2,2)\), and each endpoint has squared distance \(14\) from it. An equation with \(r^2=56\) would not contain either endpoint. When a diameter is given, calculate its length and divide by two before using it as the radius, or divide the squared diameter length by four to obtain \(r^2\).