Plane of contact of sphere

Question

A plane of contact of sphere is locus of points of contact of all the tangent planes through a given point. I cannot visualize this. Please provide a 3D image illustrating the plane of contact of sphere

Answer 1

Let $O$ be a center of the sphere and $A$ be the given point on the sphere.

Let $K$ be a point in the space such that $AK\perp OA$.

Thus, $KA$ is a tangent line to the sphere and the set of all these points with $\{A\}$ gives a tangent plane to the sphere.

Answer 2

HINT.

If $P$ is a given external point, and the tangent plane through $P$ touches the given sphere at $T$, then $T$ belongs to the sphere of diameter $PO$, where $O$ is the center of the given sphere.

The locus you want is then the intersection between those two spheres.

Answer 3

Let your sphere be described by

$ p^T p = R^2 $

where $p = [x, y, z]^T$ and $R$ is the radius of the sphere. And let there be point $P_0$ outside the sphere, and let $q$ be a point of tangency, then this means that

$ (q - P_0)^T q = 0 $

Expanding the above expression yields

$ q^T q - P_0^T q = 0 $

But $q^T q = R^2 $, therefore,

$ P_0^T q = R^2 $

and this an equation of a plane whose normal vector is along the vector $P_0$

and passes through the point $Q_0 = R^2 \dfrac{P_0}{\| P_0 \|^2 } $

This is the plane of contact, and is shown in the following Geogebra worksheet in blue.