Degree of a point

$M_0=(x_0,y_0)$ relative to a circle

$$(x-a)^2+(y-b)^2=R^2$$

with centre at a point $(a,b)$

The number

$$p=(x_0-a)^2+(y_0-b)^2-R^2.$$

One has $p<0$ if $M_0$ lies within the circle; $p=0$ if $M_0$ lies on the circle; $p>0$ if $M_0$ lies outside the circle. The degree of $M_0$ relative to a circle can be represented as the product of the vectors $\overrightarrow{M_0M_1}$ and $\overrightarrow{M_0M_2}$, where $M_1$ and $M_2$ are the points of intersection of the circle and an arbitrary straight line passing through $M_0$. In particular, the degree of a point $M_0$ relative to a circle is equal to the square of the length of the tangent drawn from $M_0$ 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 $M_0$ relative to the circle $(x-a)^2+(y-b)^2=R^2$.

References

• [a1] J.L. Coolidge, "A treatise on the circle and the sphere" , Clarendon Press (1916) Zbl 46.0921.02