Degree of a point

relative to a circle

with centre at a point

The number

One has if lies within the circle; if lies on the circle; if lies outside the circle. The degree of relative to a circle can be represented as the product of the vectors and , where and are the points of intersection of the circle and an arbitrary straight line passing through . In particular, the degree of a point relative to a circle is equal to the square of the length of the tangent drawn from to the circle.

The set of all circles in the plane relative to which a given point has an identical degree forms a net of circles. The set of points of identical degree relative to two non-concentric circles forms a radical axis.

The degree of a point relative to a sphere is defined in the same way. The set of all spheres relative to which a given point has identical degree is called a web of spheres. The set of all spheres relative to which the points of a straight line (the radical axis) have identical degree (different for different points) forms a net of spheres. The set of all spheres relative to which the points of a plane (the radical plane) have identical degree (different for different points) forms a bundle of spheres.

Comments

Customarily this notion is called the power of the point relative to the circle .

References

[a1]	J.L. Coolidge, "A treatise on the circle and the sphere" , Clarendon Press (1916)