Spheres in Analytic Geometry: Tangent Planes, Tangent Cones and Point, Line, Plane Relationships

A sphere with center C = (a, b, c) and radius r has the equation (x − a)² + (y − b)² + (z − c)² = r²; a point, a line and a plane can each lie inside, on or outside it, decided by comparing a distance with r, and the tangent plane at a point P₀ on the sphere is (x₀ − a)(x − a) + (y₀ − b)(y − b) + (z₀ − c)(z − c) = r². In brief: for the sphere x² + y² + z² = 1 and the external point (0, 0, 3), the tangent length is √(9 − 1) = 2√2, the points of tangency form the circle z = 1/3, x² + y² = 8/9, and the tangent cone is 8x² + 8y² = (z − 3)².

This guide covers the standard and general equations, the position of a point, line and plane relative to a sphere, tangent planes, and tangent cones from an external point, with worked examples and a table.

What is the equation of a sphere?

The sphere consists of the points at the distance r from the center, so by the 3D distance formula its equation is (x − a)² + (y − b)² + (z − c)² = r². Expanding gives the general form x² + y² + z² + Dx + Ey + Gz + H = 0 with the center (−D/2, −E/2, −G/2) and r² = (D² + E² + G²)/4 − H, which must be positive.

Example 1: from the general form

x² + y² + z² − 4x + 2y − 4 = 0: the center is (2, −1, 0) and r² = (16 + 4 + 0)/4 + 4 = 9, so r = 3.

Example 2: through four points

Through (0, 0, 0), (2, 0, 0), (0, 2, 0), (0, 0, 2): from the origin H = 0, then 4 + 2D = 0 gives D = −2, and similarly E = G = −2. The sphere x² + y² + z² − 2x − 2y − 2z = 0 has the center (1, 1, 1) and r = √3. Four points not in a plane determine exactly one sphere.

Example 3: a sphere with a given diameter

The diameter from A(1, 2, 3) to B(5, 2, 7): the center is the midpoint (3, 2, 5) and the radius is half of |AB| = √32, that is r = 2√2, r² = 8. The sphere: (x − 3)² + (y − 2)² + (z − 5)² = 8.

How does a point relate to a sphere?

Compare the distance d = |PC| with r: d < r inside, d = r on the sphere, d > r outside. Equivalently, the sign of the power of the point S(P) = (x − a)² + (y − b)² + (z − c)² − r²: negative inside, zero on, positive outside. For an external point the power equals the square of the tangent length.

How does a line meet a sphere?

Compute the distance h from the center to the line: h < r gives a secant with two points, h = r a tangent with one point, h > r no common points. A secant chord has the length 2√(r² − h²).

Example 4: a secant

The sphere x² + y² + z² = 9 and the line (1, 1, 0) + t(1, 2, 2). Substituting: (1 + t)² + (1 + 2t)² + 4t² = 9, that is 9t² + 6t − 7 = 0 with the discriminant 36 + 252 = 288 > 0. Two points at t = (−1 ± 2√2)/3. The distance from the center to the line is |(1, 1, 0) × (1, 2, 2)|/3 = |(2, −2, 1)|/3 = 1, and the chord length is 2√(9 − 1) = 4√2 ≈ 5.66. Check by the parameters: |t₁ − t₂| · |d| = (4√2/3) · 3 = 4√2 ✓.

Example 5: a tangent line

The line (3, 0, 0) + t(0, 1, 0) and the sphere x² + y² + z² = 9: 9 + t² = 9 gives the single point t = 0, so the line touches the sphere at (3, 0, 0).

How does a plane meet a sphere?

Let h be the distance from the center to the plane. If h < r, the intersection is a circle of radius √(r² − h²) centered at the foot of the perpendicular from C. If h = r, the plane is tangent at that foot. If h > r, they are disjoint.

Example 6: a circle of intersection

The sphere x² + y² + z² = 9 and the plane z = 2: h = 2, so the circle has the radius √5 and the center (0, 0, 2).

Example 7: a tilted plane

The plane x + 2y + 2z = 6 and the same sphere: h = 6/3 = 2, the circle’s radius is √5, and the center is the foot (2/3)(1, 2, 2) = (2/3, 4/3, 4/3). Check: 2/3 + 8/3 + 8/3 = 6 ✓.

How do you find a tangent plane?

The tangent plane at a point P₀ of the sphere is perpendicular to the radius CP₀, so its normal is P₀ − C and its equation is (x₀ − a)(x − a) + (y₀ − b)(y − b) + (z₀ − c)(z − c) = r². Tangent planes parallel to a given plane with the normal n are at the points C ± r n/|n|.

Example 8: the tangent plane at a point

The sphere (x − 1)² + (y + 2)² + (z − 3)² = 14 and the point (2, 0, 6) on it: (1)² + (2)² + (3)² = 14 ✓. The tangent plane is 1(x − 1) + 2(y + 2) + 3(z − 3) = 14, that is x + 2y + 3z = 20. Check the point: 2 + 0 + 18 = 20 ✓.

Example 9: tangent planes parallel to a plane

The sphere x² + y² + z² = 9 and the direction n = (1, 2, 2), |n| = 3. The points of tangency are ±(1, 2, 2) and the planes are x + 2y + 2z = ±9.

What is the tangent cone from an external point?

From a point P outside the sphere with the center C, the tangent lines to the sphere form a cone with the apex P. The points of tangency lie on a circle in the polar plane (P − C) · (X − C) = r². The cone’s equation is S(X) · S(P) = [(P − C) · (X − C) − r²]², where S is the power function above. The tangent length from P is √S(P), and the half-angle α of the cone satisfies sin α = r/|PC|.

Example 10: a tangent cone

The sphere x² + y² + z² = 1 and P(0, 0, 3). S(P) = 9 − 1 = 8, the tangent length is √8 = 2√2, and the polar plane is 3z = 1. The cone: 8(x² + y² + z² − 1) = (3z − 1)², that is 8x² + 8y² + 8z² − 8 = 9z² − 6z + 1, so 8x² + 8y² = z² − 6z + 9 = (z − 3)². With sin α = 1/3, tan α = 1/√8, consistent with this equation. The points of tangency: z = 1/3 and x² + y² = 8/9.

Table of relationships

Where does the sphere equation come from?

The sphere was studied by Euclid and by Archimedes, who found its volume and area. The equation in coordinates appeared with the extension of analytic geometry to three dimensions in the eighteenth century.

Which mistakes are common?

Confusing r and r²

The right-hand side is r². For the general form, check that r² > 0.

Wrong tangent plane

The point must lie on the sphere. For an external point there is a tangent cone, not a single tangent plane.

Forgetting the second tangent plane

For a given normal direction, there are two tangent planes, at C + r n/|n| and C − r n/|n|.

Mixing the chord and the radius of the intersection circle

The circle in which a plane meets the sphere has the radius √(r² − h²), not r.