Radical axis
In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose power with respect to the circles are equal.
In detail: For two circles c1, c2 with centers M1, M2 and radii r1, r2 the powers of a point P with respect to the circles are Point P belongs to the radical axis, if If the circles have two points in common, the radical axis is the common secant line of the circles.
If the radii are equal, the radical axis is the line segment bisector of M1, M2.
In any case the radical axis is a line perpendicular to
[3] J. Steiner called the radical axis line of equal powers (German: Linie der gleichen Potenzen) which led to power line (Potenzgerade).
is a normal vector to the radical axis !)
The method described in the previous section for the construction of a pencil of circles, which intersect two given circles orthogonally, can be extended to the construction of two orthogonally intersecting systems of circles:[5][6] Let
Choosing the (red) radical axis as y-axis and line
Properties: a) Any two green circles intersect on the x-axis at the points
That means, the x-axis is the radical line of the green circles.
But, if one considers the real plane as part of the complex plane, then any two purple circles intersect on the y-axis (their common radical axis) at the points
the green circles are touching each other at the origin with the x-axis as common tangent and the purple circles have the y-axis as common tangent.
the two pencils of circles c) From the equations in b), one gets a coordinate free representation: Straightedge and compass construction: A system of orthogonal circles is determined uniquely by its poles
the equation describes the radical axis of
In detail: Introducing coordinates such that then the y-axis is their radical axis (see above).
gives the normed circle equation: Completing the square and the substitution
have the two points in common and the system of coaxal circles is elliptic.
Alternative equations: 1) In the defining equation of a coaxal system of circles there can be used multiples of the power functions, too.
The radical axis can be seen as a circle with an infinitely large radius.
3) In order to express the equal status of the two circles, the following form is often used: But in this case the representation of a circle by the parameters
Hence orthogonal systems of circles play an essential role with investigations on these mappings.
[9][10] b) In electromagnetism coaxal circles appear as field lines.
[11] Additional construction method: All points which have the same power to a given circle
have two points in common, which lie on the radical axis
In that case, the radical axis is simply the
A maximal collection of circles, all having centers on a given line and all pairs having the same radical axis, is known as a pencil of coaxal circles.
If the circles are represented in trilinear coordinates in the usual way, then their radical center is conveniently given as a certain determinant.
Specifically, let X = x : y : z denote a variable point in the plane of a triangle ABC with sidelengths a = |BC|, b = |CA|, c = |AB|, and represent the circles as follows: Then the radical center is the point The radical plane of two nonconcentric spheres in three dimensions is defined similarly: it is the locus of points from which tangents to the two spheres have the same length.
[12] The fact that this locus is a plane follows by rotation in the third dimension from the fact that the radical axis is a straight line.
The same definition can be applied to hyperspheres in Euclidean space of any dimension, giving the radical hyperplane of two nonconcentric hyperspheres.
The green circle intersects the three circles orthogonally.
