Unit 4: The Sphere in Space

A sphere is the locus of a point in three-dimensional space that moves such that its Euclidean distance from a fixed point (the center) remains constant (the radius).

1. Standard Center-Radius Form

Let $C(a, b, c)$ be the center and $R > 0$ be the radius. If $P(x, y, z)$ is any point on the sphere, then by the distance formula:

Expanding this expression:

2. General Second-Degree Equation of a Sphere

The general equation of second degree in $x, y, z$:

represents a sphere if and only if:

1. The coefficients of $x^2, y^2, z^2$ are equal: $A = B = C \neq 0$.
2. The product terms vanish: $F = G = H = 0$ (no $yz, zx, xy$ terms).

Dividing through by $A$, the standard general equation of a sphere is:

Completing the square in each variable:

From this canonical form, we deduce:

• Center: $(-u, -v, -w)$
• Radius: $R = \sqrt{u^2 + v^2 + w^2 - d}$

Classification based on the radicand:

• If $u^2 + v^2 + w^2 - d > 0$: Real sphere with non-zero radius.
• If $u^2 + v^2 + w^2 - d = 0$: Point sphere (degenerate sphere of radius 0).
• If $u^2 + v^2 + w^2 - d < 0$: Virtual / Imaginary sphere (no real points satisfy the equation).

3. Sphere with Given Diameter Endpoints

If $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ are the diametrically opposite extremities of a sphere, then for any point $P(x, y, z)$ on the surface, the vectors $\vec{AP}$ and $\vec{BP}$ are orthogonal (Thales' theorem in 3D):

In Cartesian components:

4. Sphere Passing Through Four Given Points

Four non-coplanar points $P_i(x_i, y_i, z_i)$ for $i = 1, 2, 3, 4$ uniquely specify a sphere. The equation can be represented compactly as a $5 \times 5$ determinant:

1. Plane Section of a Sphere

Every planar section of a sphere is a circle.

Let a sphere have center $C$ and radius $R$. Let a plane $\Pi$ be at perpendicular distance $p$ from $C$.

• If $p < R$: The intersection is a real circle with radius $r = \sqrt{R^2 - p^2}$.
• If $p = R$: The plane is tangent to the sphere, intersecting at a single point (radius 0).
• If $p > R$: The intersection is virtual (no real intersection).

Determining the Center and Radius of the Circular Section:

1. Center of Circle ($K$): The foot of the perpendicular dropped from sphere center $C(-u, -v, -w)$ onto the intersecting plane $\Pi: ax + by + cz + d = 0$.

2. Perpendicular distance $p$:

3. Radius of Circle: $r = \sqrt{R^2 - p^2}$.

4. Great Circle: When $p = 0$ (the intersecting plane passes through the center of the sphere), $r = R$. This is a great circle; all other sections with $0 < p < R$ are small circles.

2. Tangent Plane to a Sphere

Let $P(x_1, y_1, z_1)$ be a point lying on the sphere $S: x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$.

The normal to the tangent plane at $P$ is directed along the radius vector $\vec{CP} = (x_1 + u)\hat{i} + (y_1 + v)\hat{j} + (z_1 + w)\hat{k}$.

Using the point-normal form of a plane $\vec{CP} \cdot (\vec{r} - \vec{r}_1) = 0$ and the fact that $P$ satisfies the sphere equation, the tangent plane at $(x_1, y_1, z_1)$ is:

Rule of Thumb (Quadratic Substitution): Replace $x^2 \to x x_1$, $y^2 \to y y_1$, $z^2 \to z z_1$, $2x \to x + x_1$, $2y \to y + y_1$, $2z \to z + z_1$.

Condition of Tangency of a Plane

A plane $l x + m y + n z = p$ is tangent to the sphere $(x-a)^2 + (y-b)^2 + (z-c)^2 = R^2$ if and only if the perpendicular distance from the center $(a, b, c)$ to the plane equals the radius $R$:

1. Orthogonality Condition for Two Spheres

Two spheres are defined to be orthogonal if their tangent planes at any point of intersection are perpendicular to each other. Equivalently, the radii drawn to a point of intersection are mutually perpendicular.

Let $S_1$ and $S_2$ be two spheres with centers $C_1(-u_1, -v_1, -w_1)$ and $C_2(-u_2, -v_2, -w_2)$, and radii $R_1, R_2$:

At any point of intersection $P$, the triangle $C_1 P C_2$ is a right-angled triangle with hypotenuse $C_1 C_2$. By the Pythagorean Theorem:

Computing the square of the distance between centers:

Substitute $R_1^2 = u_1^2 + v_1^2 + w_1^2 - d_1$ and $R_2^2 = u_2^2 + v_2^2 + w_2^2 - d_2$:

Canceling common squared terms:

This is the fundamental condition for orthogonality of two spheres.

2. Radical Plane

The power of a point $P(x, y, z)$ with respect to a sphere $S = x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$ is the value $S(x, y, z)$. Geometrically, if tangents are drawn from $P$ to the sphere, the square of the length of each tangent equals the power of $P$.

The radical plane of two spheres $S_1 = 0$ and $S_2 = 0$ is the locus of points whose powers with respect to both spheres are equal:

Subtracting the two equations eliminates the quadratic terms $x^2 + y^2 + z^2$:

Fundamental Properties:

1. Perpendicularity to Line of Centers: The normal vector to the radical plane is $(u_1 - u_2, v_1 - v_2, w_1 - w_2)$. The vector joining centers $C_1$ and $C_2$ is $(u_1 - u_2, v_1 - v_2, w_1 - w_2)$. Hence, the radical plane is always perpendicular to the line joining the centers of the two spheres.

2. Intersection Circle: If two spheres intersect, their radical plane is the plane containing their common circular curve of intersection.

3. Radical Line and Radical Center

• Radical Line: For three spheres $S_1, S_2, S_3$, the three radical planes taken in pairs ($S_1 - S_2 = 0$, $S_2 - S_3 = 0$, $S_3 - S_1 = 0$) intersect in a single common line called the radical line.
• Radical Center: For four spheres whose centers are non-coplanar, the radical planes taken in pairs intersect at a single unique point called the radical center. Tangents drawn from the radical center to all four spheres are equal in length.