# 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.