Intersection of Plane with Sphere is Circle

Theorem

Let $S$ be a sphere of radius $R$ whose center is located for convenience at the origin.

Let $P$ be a plane which intersects $S$ but is not a tangent plane to $S$.

It is to be shown that $S \cap P$ is a circle.

Let $S$ and $P$ be embedded in a (real) cartesian space of $3$ dimensions.

Let this space be rotated until $P$ is parallel to the plane $z = 0$.

Thus from the Equation of Plane we have that $P$ can be described as:

$z = c$

where $\size c < R$ (or $P$ would not intersect $S$).

Let $A = \tuple {x, y, z}$ be an arbitrary point on $S \cap P$.

We have that:

$\ds x^2 + y^2 + z^2$

$=$

$\ds R^2$

$\ds \leadsto \ \ $

$\ds x^2 + y^2 + c^2$

$=$

$\ds R^2$

$\ds \leadsto \ \ $

$\ds x^2 + y^2$

$=$

$\ds R^2 - c^2$

$\ds $

$>$

$\ds 0$

as $-R < c < R$

The result follows from Equation of Circle.

$\blacksquare$

1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $2$. The spherical triangle.