Radical axis

In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose powers with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of the two circles. In detail:

For two circles $c 1, c 2$ with centers $M 1, M 2$ and radii $r 1, r 2$ respectively, the powers of a point P with respect to the circles are

\Pi_1(P)=|PM_1|^2 - r_1^2,\qquad \Pi_2(P)= |PM_2|^2 - r_2^2.

Point P belongs to the radical axis, if

\Pi_1(P)=\Pi_2(P).

If the circles have two points in common, the radical axis is the common secant line of the circles.
If point P is outside the circles, P has equal tangential distance to the both circles.
If the radii are equal, the radical axis is the line segment bisector of $M 1, M 2$ .
In any case the radical axis is a line perpendicular to \overline{M_1M_2}.

On notations

The term radical axis was used by the French mathematician M. Chasles as axe radical.^[1]
J.V. Poncelet used the term chorde ideale.^[2]
J. Plücker introduced the term Chordale.^[3]
J. Steiner called the radical axis line of equal powers (German: Linie der gleichen Potenzen) which led to the term power line (Potenzgerade).^[4]

Properties

Geometric shape and its position

Let \vec x,\vec m_1,\vec m_2 be the position vectors of the points P,M_1,M_2. Then the defining equation of the radical line can be written as:

(\vec x-\vec m_1)^2-r_1^2=(\vec x-\vec m_2)^2-r_2^2 \quad \leftrightarrow
\quad 2\vec x\cdot(\vec m_2-\vec m_1)+\vec m_1^2-\vec m_2^2+r_2^2-r_1^2=0

From the right equation one gets

The pointset of the radical axis is indeed a line and is perpendicular to the line through the circle centers.

(\vec m_2-\vec m_1 is a normal vector to the radical axis!)

Dividing the equation by 2|\vec m_2-\vec m_1|, one gets the Hessian normal form. Inserting the position vectors of the centers yields the distances of the centers to the radical axis:

d_1 = \frac{d^2+{r_1}^2-{r_2}^2}{2d}\ ,\qquad d_2 = \frac{d^2+{r_2}^2-{r_1}^2}{2d},

with d = |M_1 M_2|=|\vec m_2-\vec m_1|.

(d_i may be negative if L is not between M_1,M_2.)

If the circles are intersecting at two points, the radical line runs through the common points. If they only touch each other, the radical line is the common tangent line.

Special positions

The radical axis of two intersecting circles is their common secant line.
The radical axis of two touching circles is their common tangent.
The radical axis of two non intersecting circles is the common secant of two convenient equipotent circles (see below Orthogonal circles).

Orthogonal circles

For a point P in the exterior of a circle c_i and the two tangent points S_i,T_i the equation |PS_i|^2=|PT_i|^2=\Pi_i(P) holds and S_i,T_i lie on the circle c_o with center P and radius \sqrt{\Pi_i(P)}. The circle c_o intersects the circles c_i orthogonally. Hence:

If P is a point of the radical axis, then the four points S_1,T_1, S_2,T_2 lie on the circle c_o, which intersects the given circles c_1,c_2 orthogonally.

The radical axis consists of all centers of circles, which intersect the given circles orthogonally.

System of orthogonal circles

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 c_1,c_2 be two apart lying circles (as in the previous section), M_1,M_2,r_1,r_2 their centers and radii respectively, and let g_{12} be their radical axis. Now, all the circles which have with c_1 the line g_{12} as a radical axis will be determined with their centers on the line \overline{M_1M_2}. If \gamma_2 is such a circle, whose center has distance \delta to the center M_1 and radius \rho_2. From the result in the previous section one gets the equation

d_1=\frac{\delta^2+r_1^2-\rho_2^2}{2\delta} ,\quadwhere d_1>r_1 are fixed.

With \delta_2=\delta-d_1 the equation can be rewritten as:

\delta_2^2=d_1^2-r_1^2+\rho_2^2.

If the radius \rho_2 is given, from this equation one finds the distance \delta_2 to the (fixed) radical axis of the new center. On the diagram, the color of the new circles is purple. Any green circle has its center on the radical axis and intersects the circles c_1,c_2 orthogonally and hence all the new circles (the purple ones) too. Choosing the radical axis (the red line) as y-axis and line \overline{M_1M_2} as x-axis, the two pencils of circles have the equations:

purple: \ \ \ (x-\delta_2)^2+y^2=\delta_2^2+r_1^2-d_1^2

green: \ x^2+(y-y_g)^2=y_g^2+d_1^2-r_1^2\ .

(the point\; (0,y_g) is the center of a green circle.)

Properties:
a) Any two green circles intersect on the x-axis at the points P_{1/2}=\big(\pm\sqrt{d_1^2-r_1^2},0\big), the poles of the orthogonal system of circles. That means, the x-axis is the radical axis of the green circles.
b) The purple circles have no points in common. 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 Q_{1/2}=\big(0,\pm i \sqrt{d_1^2-r_1^2}\big).

Special cases:
a) In case of d_1=r_1 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. Such a system of circles is called coaxal parabolic circles (see below).
b) Shrinking c_1 to its center M_1, i.e. r_1=0, the equations turn take a simpler form and one gets M_1=P_1.

Conclusion:
a) For any real w the pencil of circles

\;c(\xi):\; (x-\xi)^2+y^2-\xi^2-w=0\ :

has the property: The y-axis is the radical axis of c(\xi_1),c(\xi_2).

In case of w>0 the circles c(\xi_1),c(\xi_2) intersect at points P_{1/2}=(0,\pm\sqrt w).

In case of w<0 they have no points in common.

In case of w=0 they touch at (0,0) and the y-axis is their common tangent.

b) For any real w the two pencils of circles

c_1(\xi):\; (x-\xi)^2+y^2-\xi^2-w=0\ ,

c_2(\eta):\; x^2+(y-\eta)^2-\eta^2 + w=0 \

form a system of orthogonal circles. That means: any two circles c_1(\xi),c_2(\eta) intersect orthogonally.

c) From the equations in b), one gets a coordinate free representation:

For the given points P_1,P_2, their midpoint O and their line segment bisector g_{12} the two equations

|XM|^2=|OM|^2-|OP_1|^2\ ,

|XN|^2=|ON|^2+|OP_1|^2=|NP_1|^2with M on \overline{P_1P_2}, but not between P_1,P_2, and N on g_{12}

describe the orthogonal system of circles uniquely determined by P_1,P_2 which are the poles of the system.

For P_1=P_2=O one has to prescribe the axes a_1,a_2 of the system. The system is parabolic:

|XM|^2=|OM|^2\ , \quad |XN|^2=|ON|^2 with M on a_1 and N on a_2.

Straightedge and compass construction:

A system of orthogonal circles is determined uniquely by its poles P_1,P_2:

The axes (radical axes) are the lines \overline{P_1P_2} and the line segment bisector g_{12} of the poles.
The circles (green in the diagram) through P_1,P_2 have their centers on g_{12}. They can be drawn easily. For a point N the radius is \;r_N=|NP_1|\;.
In order to draw a circle of the second pencil (in diagram blue) with center M on \overline{P_1P_2}, one has to determine the radius r_M applying the theorem of Pythagoras: \; r_M^2=|OM|^2-|OP_1|^2\; (see diagram).

In case of P_1=P_2 the axes have to be chosen additionally. The system is parabolic and can be drawn easily.

Coaxal circles

Definition and properties:

Let c_1,c_2 be two circles and \Pi_1,\Pi_2 their power functions. Then for any \lambda\ne 1

\Pi_1(x,y)-\lambda\Pi_2(x,y)=0

is the equation of a circle c(\lambda) (see below). Such a class of circles is called the system of coaxal circles generated by the circles c_1,c_2. (In the case \lambda=1 the equation describes the radical axis of c_1,c_2.) ^[7]^[8]

The power function of c(\lambda) is

\ \Pi(\lambda,x,y)=\frac{\Pi_1(x,y)-\lambda\Pi_2(x,y)}{1-\lambda}.

The normed equation (the coefficients of x^2,y^2 are 1) of c(\lambda) is \ \Pi(\lambda,x,y)=0.

A simple calculation shows:

c(\lambda),c(\mu),\ \lambda\ne\mu\ , have the same radical axis as c_1 and c_2.

Allowing \lambda to move to infinity, one can see that c_1 and c_2 are members of the system of coaxal circles: c_1=c(0),\; c_2=c(\infty).

(E): If c_1,c_2 intersect at two points P_1,P_2, then P_1,P_2 are points of any circle c(\lambda), and the line \overline{P_1P_2} is their common radical axis. Such a system is called elliptic.
(P): If c_1 and c_2 are tangent at P, then any circle of the system is tangent to both c_1 and c_2 at point P. The common tangent is their common radical axis. Such a system is called parabolic.
(H): If c_1 and c_2 have no point in common, then the same is true for any pair of circles of the system. The radical axis of any pair of circles is the radical axis of c_1 and c_2. This system is called hyperbolic.

In detail:

Introducing coordinates such that

c_1: (x-d_1)^2+y^2=r_1^2

c_2: (x-d_2)^2+y^2= d_2^2+r_1^2-d_1^2,

then the y-axis is their radical axis (see above).

Calculating the power function \Pi(\lambda,x,y) gives the normed circle equation:

c(\lambda): \ x^2+y^2-2\tfrac{d_1-\lambda d_2}{1-\lambda}\; x +d_1^2-r_1^2=0\ .

Completing the square and substituting \delta_2=\tfrac{d_1-\lambda d_2}{1-\lambda} (x-coordinate of the center) yields the centered form of the equation

c(\lambda): \ (x-\delta_2)^2+y^2=\delta_2^2+r_1^2-d_1^2.

When r_1>d_1 the circles c_1,c_2,c(\lambda) intersect at the two points

P_1=\big(0,\sqrt{r_1^2-d_1^2}\big),\quad P_2=\big(0,-\sqrt{r_1^2-d_1^2}\big)

and the system of coaxal circles is elliptic.

If r_1=d_1, the circles c_1,c_2,c(\lambda) have the point P_0=(0,0) in common and the system is parabolic.

If r_1<d_1, the circles c_1,c_2,c(\lambda) have no point in common and the system is hyperbolic.

Alternative equations:
1) In the defining equation of a coaxal system of circles there can be used multiples of the power functions, too.
2) The equation of one of the circles can be replaced by the equation of the desired radical axis. The radical axis can be seen as a circle with an infinitely large radius. For example:

(x-x_1)^2+y^2-r^2_1\ - \ \lambda\; 2(x-x_2)\ =0\ \Leftrightarrow

(x-(x_1+\lambda))^2+y^2 =(x_1+\lambda)^2+r_1^2-x_1^2-2\lambda x_2,

describes all circles, which have with the first circle the line x=x_2 as radical axis.
3) In order to express the equal status of the two circles, the following form is often used:

\mu\Pi_1(x,y)+\nu\Pi_2(x,y)=0\; .

But in this case the representation of a circle by the parameters \mu,\nu is not unique.

Applications:
a) Circle inversions and Möbius transformations preserve angles and generalized circles. 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]

Radical center of three circles, construction of the radical axis

For three circles c_1,c_2,c_3, no two of which are concentric, there are three radical axes g_{12},g_{23},g_{31}. If the centers of these circles are not collinear, their radical axes intersect in a common point R, the radical center of the three circles. The circle which is orthogonal to the two of the circles with center in R is orthogonal to the third circle (radical circle).

Proof. The radical axis g_{ik} contains all points which have equal tangential distance to the circles c_i,c_k. The intersection point R of g_{12} and g_{23} has the same tangential distance to the all three circles. Hence, R is a point on the radical axis g_{31}, too.

This property allows one to construct the radical axis of two non intersecting circles c_1,c_2 with centers M_1,M_2: Draw a third circle c_3 with center not collinear to the given centers that intersects c_1,c_2. The radical axes g_{13},g_{23} can be drawn. Their intersection point is the radical center R of the three circles and lies on g_{12}. The line through R which is perpendicular to \overline{M_1M_2} is the radical axis g_{12}.

Additional construction method:

All points which have the same power to a given circle c lie on a circle concentric to c. Let us call it an equipower circle. This property can be used for an additional construction method of the radical axis of two circles:

For two non intersecting circles c_1,c_2, there can be drawn two equipower circles c'_1,c'_2, which have the same power with respect to c_1,c_2 (see diagram). In detail: \Pi_1(P_1)=\Pi_2(P_2). If the power is large enough, the circles c'_1,c'_2 have two points in common, which lie on the radical axis g_{12}.

Relation to bipolar coordinates

In general, any two disjoint, non-concentric circles can be aligned with the circles of a system of bipolar coordinates. In that case, the radical axis is simply the y-axis of this system of coordinates. Every circle on the axis that passes through the two foci of the coordinate system intersects the two circles orthogonally. 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.

Radical center in trilinear coordinates

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 which has sides with lengths a = |BC|, b = |CA|, c = |AB|, and represent the circles as follows:

(dx+ ey + fz)(ax + by + cz) + g(ayz + bzx + cxy) = 0,

(hx + iy + jz)(ax + by + cz) + k(ayz + bzx + cxy) = 0,

(lx + my + nz)(ax + by + cz) + p(ayz + bzx + cxy) = 0.

Then the radical center is the point

\begin{vmatrix}
g&k&p\\
e&i&m\\
f&j&n
\end{vmatrix}: 
\begin{vmatrix}
g&k&p\\
f&j&n\\
d&h&l
\end{vmatrix} : 
\begin{vmatrix}
g&k&p \\
d&h&l\\e&i&m\end{vmatrix}.

Radical plane and hyperplane

The radical plane of two non-concentric spheres in three dimensions is defined similarly: it is the locus of points from which tangents to the two spheres have same lengths.^[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 non-concentric hyperspheres.

Notes

^ Michel Chasles, C. H. Schnuse: Die Grundlehren der neuern Geometrie, erster Theil, Verlag Leibrock, Braunschweig, 1856, p. 312
^ Ph. Fischer: Lehrbuch der analytische Geometrie, Darmstadt 1851, Verlag Ernst Kern, p. 67
^ H. Schwarz: Die Elemente der analytischen Geometrie der Ebene, Verlag H. W. Schmidt, Halle, 1858, p. 218
^ Jakob Steiner: Einige geometrische Betrachtungen. In: Journal für die reine und angewandte Mathematik, Band 1, 1826, p. 165
^ A. Schoenfliess, R. Courant: Einführung in die Analytische Geometrie der Ebene und des Raumes, Springer-Verlag, 1931, p. 113
^ C. Carathéodory: Funktionentheorie, Birkhäuser-Verlag, Basel, 1961, ISBN 978-3-7643-0064-7, p. 46
^ Dan Pedoe: Circles: A Mathematical View, mathematical Association of America, 2020, ISBN 9781470457327, p. 16
^ R. Lachlan: An Elementary Treatise On Modern Pure Geometry, MacMillan&Co, New York,1893, p. 200
^ Carathéodory: Funktionentheorie, p. 47.
^ R. Sauer: Ingenieur-Mathematik: Zweiter Band: Differentialgleichungen und Funktionentheorie, Springer-Verlag, 1962, ISBN 978-3-642-53232-0, p. 105
^ Clemens Schaefer: Elektrodynamik und Optik, Verlag: De Gruyter, 1950, ISBN 978-3-11-230936-0, p. 358.
^ See Merriam–Webster online dictionary.

References

R. A. Johnson (1960). Advanced Euclidean Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. reprint of 1929 edition by Houghton Mifflin ed. New York: Dover Publications. pp. 31–43. ISBN 978-0-486-46237-0.

External links

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