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Degree of a point

From Encyclopedia of Mathematics
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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)
How to Cite This Entry:
Degree of a point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degree_of_a_point&oldid=31693
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article