Equation of a Sphere Through Four Points (Vector Form)

In 3D geometry, the equation of a sphere is an important concept that can be determined if we know certain conditions, such as the center and radius of the sphere. However, when we are given four non-coplanar points, we can also find the equation of the sphere that passes through these points using vector algebra.

General Equation of a Sphere

The general equation of a sphere with center at point ( C(x_0, y_0, z_0) ) and radius ( r ) is given by:

[ (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2 ]

In vector form, if ( \vec{r} ) is the position vector of a general point ( P(x, y, z) ) on the sphere, and ( \vec{c} ) is the position vector of the center ( C ), the equation becomes:

[ ||\vec{r} - \vec{c}||^2 = r^2 ]

Where ( ||\vec{v}|| ) denotes the magnitude of vector ( \vec{v} ).

Determining the Equation Through Four Points

To find the equation of a sphere that passes through four non-coplanar points ( A, B, C, ) and ( D ), we use their position vectors ( \vec{a}, \vec{b}, \vec{c}, ) and ( \vec{d} ), respectively.

Steps to Find the Equation

1. Determine the vectors connecting the points: Find the vectors ( \vec{ab}, \vec{ac}, \vec{ad} ).
2. Find normal vectors: Calculate two normal vectors to the plane determined by these vectors, for example, ( \vec{n}_1 = \vec{ab} \times \vec{ac} ) and ( \vec{n}_2 = \vec{ab} \times \vec{ad} ).
3. Solve for the center: The center of the sphere ( \vec{c} ) lies on the line that is perpendicular to the plane formed by ( A, B, ) and ( C ) and passes through the midpoint of any side, say ( \vec{m}{ab} = \frac{\vec{a} + \vec{b}}{2} ). Similarly, it lies on the line that is perpendicular to the plane formed by ( A, B, ) and ( D ) and passes through ( \vec{m}{ab} ). The intersection of these two lines gives the center ( \vec{c} ).
4. Calculate the radius: Use the distance formula to find the radius ( r ) by calculating the distance between the center ( \vec{c} ) and any of the four points, say ( ||\vec{c} - \vec{a}|| ).

Equation Derivation

Given four points with position vectors ( \vec{a}, \vec{b}, \vec{c}, ) and ( \vec{d} ), the sphere equation can be derived by setting up the system of equations:

[ \begin{align*} ||\vec{r} - \vec{c}||^2 &= ||\vec{a} - \vec{c}||^2 \ ||\vec{r} - \vec{c}||^2 &= ||\vec{b} - \vec{c}||^2 \ ||\vec{r} - \vec{c}||^2 &= ||\vec{c} - \vec{c}||^2 \ ||\vec{r} - \vec{c}||^2 &= ||\vec{d} - \vec{c}||^2 \end{align*} ]

By solving this system, we can find the center ( \vec{c} ) and radius ( r ).

Example

Let's consider four points with the following position vectors:

• ( \vec{a} = \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} )
• ( \vec{b} = \begin{pmatrix} 2 \ 3 \ 4 \end{pmatrix} )
• ( \vec{c} = \begin{pmatrix} 3 \ 2 \ 5 \end{pmatrix} )
• ( \vec{d} = \begin{pmatrix} 4 \ 3 \ 2 \end{pmatrix} )

1. Calculate the vectors connecting the points:

   • ( \vec{ab} = \vec{b} - \vec{a} )
   • ( \vec{ac} = \vec{c} - \vec{a} )
   • ( \vec{ad} = \vec{d} - \vec{a} )

2. Find normal vectors:

   • ( \vec{n}_1 = \vec{ab} \times \vec{ac} )
   • ( \vec{n}_2 = \vec{ab} \times \vec{ad} )

3. Solve for the center ( \vec{c} ) using the intersection of perpendicular lines from the midpoints.

4. Calculate the radius ( r ) using the distance formula.

Important Points and Differences

Aspect	Description
Number of Points	A minimum of four non-coplanar points are required to uniquely determine a sphere.
Position Vectors	Position vectors of the points are used to derive the sphere's equation.
Normal Vectors	Normal vectors to the planes are crucial to find the center of the sphere.
System of Equations	A system of equations is set up based on the distances from the center to the points.
Uniqueness	The sphere passing through four non-coplanar points is unique.

In conclusion, the equation of a sphere through four non-coplanar points can be derived using vector algebra by finding the center and radius of the sphere. This method involves calculating vectors between points, finding normal vectors, solving for the center, and determining the radius. The process is systematic and can be applied to any set of four non-coplanar points to find the unique sphere that passes through them.