Radical axis

Last updated
.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}
Two circles, centered at M1, M2
Radical axis, with sample point P
Tangential distances from both circles to P
The tangent lines must be equal in length for any point on the radical axis:
|
P
T
1
|
=
|
P
T
2
|
.
{\displaystyle |PT_{1}|=|PT_{2}|.}
If P, T1, T2 lie on a common tangent, then P is the midpoint of
T
1
T
2
-
.
{\displaystyle {\overline {T_{1}T_{2}}}.} Potenz-gerade-def.svg
  Two circles, centered at M1, M2
  Radical axis, with sample point P
  Tangential distances from both circles to P
The tangent lines must be equal in length for any point on the radical axis: If P, T1, T2 lie on a common tangent, then P is the midpoint of

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. For this reason the radical axis is also called the power line or power bisector of the two circles. In detail:

Contents

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 point P is outside the circles, P has equal tangential distance to both 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

On notations

The notation radical axis was used by the French mathematician M. Chasles as axe radical. [1]
J.V. Poncelet used 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 power line (Potenzgerade). [4]

Properties

Geometric shape and its position

Let be the position vectors of the points . Then the defining equation of the radical line can be written as:

Definition and calculation of
d
1
,
d
2
{\displaystyle d_{1},d_{2}} Potenz-gerade-ber-d1d2.svg
Definition and calculation of

From the right equation one gets

( is a normal vector to the radical axis !)

Dividing the equation by , one gets the Hessian normal form. Inserting the position vectors of the centers yields the distances of the centers to the radical axis:

,
with .

( may be negative if is not between .)

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

Radical axis: variations Potenz-gerade-var.svg
Radical axis: variations
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 equipower circles (see below).

Orthogonal circles

The touching points of the tangents through
P
{\displaystyle P}
lie on the orthogonal circle (green) Potenz-gerade-co.svg
The touching points of the tangents through lie on the orthogonal circle (green)
If is a point of the radical axis, then the four points lie on circle , which intersects 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 be two apart lying circles (as in the previous section), their centers and radii and their radical axis. Now, all circles will be determined with centers on line , which have together with line as radical axis, too. If is such a circle, whose center has distance to the center and radius . From the result in the previous section one gets the equation

, where are fixed.

With the equation can be rewritten as:

.
System of orthogonal circles: construction Kreise-orth-sys-e.svg
System of orthogonal circles: construction

If radius is given, from this equation one finds the distance to the (fixed) radical axis of the new center. In the diagram the color of the new circles is purple. Any green circle (see diagram) has its center on the radical axis and intersects the circles orthogonally and hence all new circles (purple), too. Choosing the (red) radical axis as y-axis and line as x-axis, the two pencils of circles have the equations:

purple:
green:

( is the center of a green circle.)

Properties:
a) Any two green circles intersect on the x-axis at the points , the poles of the orthogonal system of circles. That means, the x-axis is the radical line 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 .

Parabolic orthogonal system Kreis-sys-orth-pa.svg
Parabolic orthogonal system
Coaxal circles: types Kreis-buesch-typen.svg
Coaxal circles: types

Special cases:
a) In case of 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 to its center , i. e. , the equations turn into a more simple form and one gets .

Conclusion:
a) For any real the pencil of circles

has the property: The y-axis is the radical axis of .
In case of the circles intersect at points .
In case of they have no points in common.
In case of they touch at and the y-axis is their common tangent.

b) For any real the two pencils of circles

form a system of orthogonal circles. That means: any two circles intersect orthogonally.

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

Orthogonal system of circles to given poles
P
1
,
P
2
{\displaystyle P_{1},P_{2}} Kreise-orth-sys-p1p2.svg
Orthogonal system of circles to given poles
For the given points , their midpoint and their line segment bisector the two equations
with on , but not between , and on
describe the orthogonal system of circles uniquely determined by which are the poles of the system.
For one has to prescribe the axes of the system, too. The system is parabolic:
with on and on .

Straightedge and compass construction:

Orthogonal system of circles: straightedge and compass construction Kreise-os-konstr.svg
Orthogonal system of circles: straightedge and compass construction

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

  1. The axes (radical axes) are the lines and the Line segment bisector of the poles.
  2. The circles (green in the diagram) through have their centers on . They can be drawn easily. For a point the radius is .
  3. In order to draw a circle of the second pencil (in diagram blue) with center on , one determines the radius applying the theorem of Pythagoras: (see diagram).

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

Coaxal circles

Definition and properties:

Let be two circles and their power functions. Then for any

is the equation of a circle (see below). Such a system of circles is called coaxal circles generated by the circles . (In case of the equation describes the radical axis of .) [7] [8]

The power function of is

.

The normed equation (the coefficients of are ) of is .

A simple calculation shows:

Allowing to move to infinity, one recognizes, that are members of the system of coaxal circles: .

(E): If intersect at two points , any circle contains , too, and line is their common radical axis. Such a system is called elliptic.
(P): If are tangent at , any circle is tangent to at point , too. The common tangent is their common radical axis. Such a system is called parabolic.
(H): If have no point in common, then any pair of the system, too. The radical axis of any pair of circles is the radical axis of . The system is called hyperbolic.

In detail:

Introducing coordinates such that

,

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

Calculating the power function gives the normed circle equation:

Completing the square and the substitution (x-coordinate of the center) yields the centered form of the equation

.

In case of the circles have the two points

in common and the system of coaxal circles is elliptic.

In case of the circles have point in common and the system is parabolic.

In case of the circles 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:

,

describes all circles, which have with the first circle the line as radical axis.
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 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

Radical center of three circles
The green circle intersects the three circles orthogonally. Potenz-gerade-3k.svg
Radical center of three circles
The green circle intersects the three circles orthogonally.
Proof: the radical axis contains all points which have equal tangential distance to the circles . The intersection point of and has the same tangential distance to all three circles. Hence is a point of the radical axis , too.
This property allows one to construct the radical axis of two non intersecting circles with centers : Draw a third circle with center not collinear to the given centers that intersects . The radical axes can be drawn. Their intersection point is the radical center of the three circles and lies on . The line through which is perpendicular to is the radical axis .

Additional construction method:

Construction of the radical axis with circles
c
1
'
,
c
2
'
{\displaystyle c'_{1},c'_{2}}
of equal power. It is
P
1
(
P
1
)
=
P
2
(
P
2
)
{\displaystyle \Pi _{1}(P_{1})=\Pi _{2}(P_{2})}
. Potenz-gerade-konstr-e.svg
Construction of the radical axis with circles of equal power. It is .

All points which have the same power to a given circle lie on a circle concentric to . 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 , there can be drawn two equipower circles , which have the same power with respect to (see diagram). In detail: . If the power is large enough, the circles have two points in common, which lie on the radical axis .

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 -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 with sidelengths 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

Radical plane and hyperplane

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.

Notes

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

Related Research Articles

<span class="mw-page-title-main">Ellipse</span> Plane curve: conic section

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity , a number ranging from to .

<span class="mw-page-title-main">Hyperbola</span> Plane curve: conic section

In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

<span class="mw-page-title-main">Parabola</span> Plane curve: conic section

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

<span class="mw-page-title-main">Ellipsoid</span> Quadric surface that looks like a deformed sphere

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension D) in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D = 1 in the case of conic sections. When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a degenerate quadric or a reducible quadric.

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints. It is named after the mathematician Joseph-Louis Lagrange.

<span class="mw-page-title-main">Separation of variables</span> Technique for solving differential equations

In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.

<span class="mw-page-title-main">Pedal curve</span> Curve generated by the projections of a fixed point on the tangents of another curve

In mathematics, a pedal curve of a given curve results from the orthogonal projection of a fixed point on the tangent lines of this curve. More precisely, for a plane curve C and a given fixed pedal pointP, the pedal curve of C is the locus of points X so that the line PX is perpendicular to a tangent T to the curve passing through the point X. Conversely, at any point R on the curve C, let T be the tangent line at that point R; then there is a unique point X on the tangent T which forms with the pedal point P a line perpendicular to the tangent T – the pedal curve is the set of such points X, called the foot of the perpendicular to the tangent T from the fixed point P, as the variable point R ranges over the curve C.

<span class="mw-page-title-main">Cissoid</span> Plane curve constructed from two other curves and a fixed point

In geometry, a cissoid is a plane curve generated from two given curves C1, C2 and a point O. Let L be a variable line passing through O and intersecting C1 at P1 and C2 at P2. Let P be the point on L so that Then the locus of such points P is defined to be the cissoid of the curves C1, C2 relative to O.

<span class="mw-page-title-main">Line (geometry)</span> Straight figure with zero width and depth

In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word line may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points.

<span class="mw-page-title-main">Cassini oval</span> Class of quartic plane curves

In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points (foci) is constant. This may be contrasted with an ellipse, for which the sum of the distances is constant, rather than the product. Cassini ovals are the special case of polynomial lemniscates when the polynomial used has degree 2.

<span class="mw-page-title-main">Power of a point</span> Relative distance of a point from a circle

In elementary plane geometry, the power of a point is a real number that reflects the relative distance of a given point from a given circle. It was introduced by Jakob Steiner in 1826.

<span class="mw-page-title-main">Pencil (geometry)</span> Family of geometric objects with a common property

In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane.

Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces, the ellipsoidal coordinate system is based on confocal quadrics.

In geometry and linear algebra, a principal axis is a certain line in a Euclidean space associated with an ellipsoid or hyperboloid, generalizing the major and minor axes of an ellipse or hyperbola. The principal axis theorem states that the principal axes are perpendicular, and gives a constructive procedure for finding them.

<span class="mw-page-title-main">Conic section</span> Curve from a cone intersecting a plane

A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.

In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles.

<span class="mw-page-title-main">Intersection (geometry)</span> Shape formed from points common to other shapes

In geometry, an intersection is a point, line, or curve common to two or more objects. The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point or does not exist. Other types of geometric intersection include:

<span class="mw-page-title-main">Confocal conic sections</span> Conic sections with the same foci

In geometry, two conic sections are called confocal if they have the same foci.

<span class="mw-page-title-main">Dupin's theorem</span>

In differential geometry Dupin's theorem, named after the French mathematician Charles Dupin, is the statement:

References

Further reading