When a sphere and a plane intersects, the intersection can be described as either a point or a circle. Here, we will learn about the case when it’s a single a point. Go here to learn about intersection as a circle.
When the intersection between a tangent plane and a sphere surface is a point, we call this point the tangent point. To find this point, you do the following:
You can see that the vector extending from the center of the sphere to the tangent point have to be perpendicular to the tangent plane, because the angle between a tangent and the line from the tangent point to the center of the sphere is always °. That means this vector is parallel to the normal vector to the tangent plane, and you can create a line through the tangent point and the center of the sphere with that normal vector as a directional vector. To find the tangent point, you have to find the intersection between this line and the tangent plane.
Example 1
You have a spherical surface that has its center in and a tangent plane given by the equation
Find the tangent point.
You can see that is a normal vector to the plane. That means the parametric equation for the line through the center with the normal vector as its directional vector is
You insert for in the parametric equation and get
This point will be the intersection between the line through the center and the tangent plane, which also makes it the tangent point.
Formula
The tangent point where a plane with a normal vector intersects a sphere with radius and its center at is given by
Example 2
You will often be asked to show that a plane is tangent to a spherical surface and to find the tangent point. A plane is tangent to a spherical surface if the distance from the center of the sphere to the plane is equal to the radius of the sphere.
Say you have the sphere
and the plane
From the equation of the sphere, you can see that the radius is and the center is at . To show that the plane is tangent to the surface of the sphere, you use the formula for the distance between a point and a plane and check whether the answer is equal to the radius of the sphere, which we know is :
To find the tangent point where the surface of the sphere and the plane touches, you can do like in Example 1. You can also use the formula above for the tangent point of a spherical surface and a plane.
Note! Take a note of that in this case, the normal vector can send you in the wrong direction. Then you will end up at the opposite side of the sphere. For that reason, check whether your answer lies in the tangent plane. If it doesn’t, simply switch with .
In this case, the radius of the sphere and the distance between the center of the sphere and the plane are the same, giving you that
Put this into the equation of the tangent plane to check whether you have gone in the right direction. That gives you
Put this into the equation of the plane, and you get