Carlyle Circle

Quadratic function with associated Carlyle circle

The Carlyle circle (also Lill circle ) is a special circle in the Cartesian coordinate system whose points of intersection with the x-axis coincide with the points of intersection of a normalized quadratic function and the x-axis. It can thus be used for the geometric construction of the zeros of a normalized quadratic function.

Definition and characteristics

For a normalized quadratic function , the Carlyle circle is defined as the circle that has the connecting line of the points and as the diameter . ${\displaystyle f(x)=x^{2}+px+q}$ ${\displaystyle AD}$ ${\displaystyle A(0|1)}$ ${\displaystyle D(-p|q)}$

The circle defined in this way has the center point , the radius and the x-coordinates of its intersection points with the x-axis are the zeros of the associated quadratic function. The latter can be seen by looking at the intersection points with the x-axis in the circular equation: ${\displaystyle M=\left({\tfrac {-p}{2}}|{\tfrac {q+1}{2}}\right)}$ ${\displaystyle r={\tfrac {\sqrt {p^{2}+(q-1)^{2}}}{2}}}$

${\displaystyle \left(x-{\tfrac {-p}{2}}\right)^{2}+\left(y-{\tfrac {q+1}{2}}\right)^{2}={\tfrac {p^{2}+(q-1)^{2}}{4}}}$

For the points of intersection with the x-axis it is now also true that their y-coordinate is 0. If you insert this into the circular equation and then solve the equation for x, you get the pq formula, which calculates the zeros of a normalized quadratic function:

${\displaystyle x_{1,2}=-{\tfrac {p}{2}}\pm {\sqrt {\left({\tfrac {p}{2}}\right)^{2}-q}}}$

Trapeze and Carlyle Circle

If you add the points and from the definition to the points and , you get a trapezoid whose side is the diameter of the Carlyle circle. This also intersects the side BC at points and . According to Thales's theorem , the angle is a right angle, which in turn means that the triangles and are similar. This then gives the following relationship equation: ${\displaystyle A(0|1)}$ ${\displaystyle D(-p|q)}$ ${\displaystyle B(0|0)}$ ${\displaystyle C(-p|0)}$ ${\displaystyle AD}$ ${\displaystyle S_{1}(x_{1}|0)}$ ${\displaystyle S_{2}(x_{2}|0)}$ ${\displaystyle \angle AS_{2}D}$ ${\displaystyle \triangle ABS_{2}}$ ${\displaystyle \triangle S_{2}CD}$

${\displaystyle {\frac {1}{x_{2}}}={\frac {x_{1}}{q}}\,\Rightarrow \,x_{1}x_{2}=q}$

Furthermore, and therefore due to the inversion of the root law of Vieta , it also applies that and are zeros of . ${\displaystyle x_{1}+x_{2}=-p}$ ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle f(x)=x^{2}+px+q}$

Complex zeros of a normalized parabola with the Carlyle circle:
${\displaystyle {\begin{aligned}\,&x^{2}+px+q=0\land {\tfrac {p^{2}}{4}}-q<0\\\Rightarrow \,&x_{1,2}=a\pm b\cdot i\end{aligned}}}$

This geometric proof of the properties of the Carlyle circle also shows that it can be understood as a modification of a special case of Lill's method , a graphical procedure for determining the zeros of a polynomial. For a given polynomial, starting from a common point, two polygons of lengths and are constructed. If their end points now also coincide, the tangent of the angle that the two polygons form at the common starting point is a zero of the polynomial. For a normalized quadratic function one obtains as polygons and and since these both end in the common point , there is a zero of the quadratic function. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n-1}$ ${\displaystyle ABCD}$ ${\displaystyle AS_{2}D}$ ${\displaystyle D}$ ${\displaystyle \tan(\angle BAS_{2})=x_{2}}$

The Carlyle cycle can also be used to construct the complex roots of a normalized quadratic function. In this case, it does not intersect the x-axis itself, but rather another circle that is constructed with its help. The identical real part of the two complex zeros corresponds to the signed distance of the vertical through the center of the Carlyle circle to the y-axis. Like the Carlyle circle, the center of the second circle lies on the vertical and its radius corresponds to the distance of the center of the Carlyle circle from the x-axis. Furthermore, both circles touch each other from the outside. The tangent from the intersection of the vertical with the axis x-axis at Carlyle's circle touches this in and the second circle intersects the x-axis in and . The amount of the imaginary part of the zero then corresponds to the lengths of the line , or . ${\displaystyle a}$ ${\displaystyle g}$ ${\displaystyle M}$ ${\displaystyle T}$ ${\displaystyle g}$ ${\displaystyle M}$ ${\displaystyle S(a|0)}$ ${\displaystyle g}$ ${\displaystyle R}$ ${\displaystyle U_{1}}$ ${\displaystyle U_{2}}$ ${\displaystyle b}$ ${\displaystyle SR}$ ${\displaystyle SU_{1}}$ ${\displaystyle SU_{2}}$

Polygon constructions

Roots of on the unit circle in the complex number plane ${\displaystyle z^{5}=1}$

Carlyle circles can be used for the compass-and-ruler construction of various regular polygons. It is used here that all points of the plane with rational coordinates can in principle be constructed with compasses and rulers; therefore it is possible to use a Carlyle circle with a point for without breaking the rules of construction. ${\displaystyle D(-p|q)}$ ${\displaystyle p,q\in \mathbb {Q} }$

The principle idea that enables the use of Carlyle circles is to understand the corner points of a regular polygon as the complex roots of the equation on the unit circle in the complex number plane. Two new numbers are then derived from several complex roots, which are the zeros of a normalized quadratic equation and can thus be constructed with the help of a Carlyle circle. The two numbers are also chosen in such a way that knowing them can be used to construct the complex roots used for them, i.e. the corner points of the polygon. This approach is described below using the example of the pentagon ; in addition, it can also be used in particular for the construction of the seventeenth , the 257 and the 65537 corners . ${\displaystyle z^{n}=1}$

Pentagon construction with the Carlyle circle (green)

The roots of are for . Now you choose the two numbers and . These satisfy the two equations and , so they are zeros of the normalized quadratic function . Since and and and conjugate roots it is also true that and correspond to twice the real part of the associated complex roots. ${\displaystyle z^{5}=1}$ ${\displaystyle Z_{i}=\exp({\tfrac {2\pi }{5}}\cdot i)}$ ${\displaystyle i=0,...,4}$ ${\displaystyle s_{1}=Z_{2}+Z_{3}}$ ${\displaystyle s_{2}=Z_{1}+Z_{4}}$ ${\displaystyle s_{1}\cdot s_{2}=-1}$ ${\displaystyle s_{1}+s_{2}=-1}$ ${\displaystyle f(x)=x^{2}+x-1}$ ${\displaystyle Z_{1}}$ ${\displaystyle Z_{4}}$ ${\displaystyle Z_{2}}$ ${\displaystyle Z_{3}}$ ${\displaystyle s_{1}}$ ${\displaystyle s_{2}}$

This results in the following construction method: First draw the unit circle and then the Carlyle circle that belongs to it . A circle with radius 1 is drawn around the intersection of the Carlyle circle with the x-axis. These two circles intersect the unit circle in and (construction of the perpendiculars of and ). is obtained as the intersection of the unit circle with the x-axis, thus all corner points of the pentagon are constructed. ${\displaystyle f(x)}$ ${\displaystyle D(-1|-1)}$ ${\displaystyle Z_{1},Z_{2},Z_{3}}$ ${\displaystyle Z_{4}}$ ${\displaystyle s_{1}}$ ${\displaystyle s_{2}}$ ${\displaystyle Z_{0}}$

history

Carlyle's solution to Leslie's problem. The black line is divided in such a way that its sections form a rectangle (green) that is the same area as a given rectangle (red).

According to Howard Eves (1911–2004), the mathematician John Leslie (1766–1832) described the construction of zeros with the help of the Carlyle circle in his book Elements of Geometry , where he noted that one of his students, Thomas Carlyle (1795– 1881), had been proposed. Leslie's representation contains an analog circular construction, but still without a Cartesian coordinate system, quadratic function or explicit quadratic equation, instead it is clad in an elementary geometric problem for the construction of rectangles of equal area. Carlyle used a circle and a trapezoid to solve the problem:

Divide a line so that its sections form the sides of a new rectangle that is the same area as a given rectangle (Proposition XVII in the third edition of John Leslie's Elements of Geometry ).

In 1867 the Austrian engineer and civil servant Eduard Lill published a graphic method for determining the zeros of a polynomial ( Lills method ); If one applies this to a normalized quadratic function, one obtains the trapezoid mentioned above with and that as the diameter of the Carlyle circle. In an article published in 1925, GA Miller describes that in the case of a normalized quadratic function, Lill's method can be used to derive a circular construction whose intersection points with the x-axis coincide with the zeros of the quadratic function, thus providing the modern definition of Carlyle -Circle. ${\displaystyle ABCD}$ ${\displaystyle |AB|=1}$ ${\displaystyle AD}$

Eves used the circle in the modern sense for an exercise in his book Introduction to the History of Mathematics (1953) and referred to Carlyle in a note. Later publications increasingly begin to use the designation Carlyle circle or Carlyle method. Duane W. DeTemple used the Carlyle circle (1989, 1991) to obtain the simplest possible compass-and-ruler constructions of certain regular polygons. In 1999 Ladislav Beran described how the Carlyle circle can be used to geometrically construct the complex zeros of a normalized quadratic function.

literature

• Rainer Kaenders (Hrsg.), Reinhard Schmidt (Hrsg.): Understanding more mathematics with GeoGebra . 2nd Edition. Springer Spectrum, 2014, ISBN 978-3-658-04222-6 , pp. 68-71
• Patricia R. Allaire, Robert E. Bradley: Geometric Approaches to Quadratic Equations from Other Times and Places . (PDF) In: The Mathematics Teacher , Vol. 94, No. 4, April 2001, pp. 308-313 ( JSTOR )
• Duane W. DeTemple: Simple Constructions for the Regular Pentagon and Heptadecagon . In: The Mathematics Teacher , Vol. 82, No. 5 (May 1989), pp. 361-365 ( JSTOR 27966269 )
• Duane W. DeTemple: Carlyle Circles and the Lemoine Simplicity of Polygon Constructions (PDF) In: The American Mathematical Monthly , Vol. 98, No. 2 (Feb. 1991), pp. 97-108 ( JSTOR 2323939 )
• E. John Hornsby, Jr .: Geometrical and Graphical Solutions of Quadratic Equations (PDF) In: The College Mathematics Journal , Vol. 21, No. 5 (Nov. 1990), pp. 362-369 ( JSTOR 2686901 )
• Walter M. Patterson III, Andre M. Lubecke: A Special Circle for Quadratic Equations . In: The Mathematics Teacher , Vol. 84, No. 2 (February 1991), pp. 125-127 ( JSTOR 27967040 )
• GA Miller: Geometric Solution of the Quadratic Equation . In: The Mathematical Gazette , Vol. 12, No. 179 (Dec., 1925), pp. 500-501 ( JSTOR 3602823 )
• Ladislav Beran: The Complex Roots of a Quadratic from a Circle . In: The Mathematical Gazette , Vol. 83, No. 497 (Jul., 1999), pp. 287-291 ( JSTOR 3619064 )

Web links

Commons : Carlyle circle  - collection of images, videos and audio files

• Eric W. Weisstein : Carlyle Circle . In: MathWorld (English).

Individual evidence

1. ^ ^{A b c} Duane W. DeTemple: Carlyle Circles and the Lemoine Simplicity of Polygon Constructions . ( Memento from August 11, 2011 in the Internet Archive ; PDF) In: The American Mathematical Monthly , Vol. 98, No. 2 (Feb. 1991), pp. 97-108 ( JSTOR 2323939 )
2. ↑ ^{a b c d e f} Rainer Kaenders (ed.), Reinhard Schmidt (ed.): Understanding more mathematics with GeoGebra . Springer Spectrum, 2nd edition, 2014, ISBN 978-3-658-04222-6 , pp. 68-71
3. ↑ ^{a b} See Kaenders / Schmidt, pp. 70–73. Please note that Lill's method does not actually provide the specified trapezoidal figure, but a figure that is congruent to it, rotated by 90 ° and mirrored on the horizontal. As a result, the intersection angle at the starting point of the congruent figure constructed according to Lill's method is negative and accordingly the tangent of the intersection angle is not a zero of the polynomial, but the negative tangent
4. ↑ ^{a b} Ladislav Beran: The Complex Roots of a Quadratic from a Circle . In: The Mathematical Gazette , Vol. 83, No. 497 (Jul., 1999), pp. 287-291 ( JSTOR 3619064 )
5. ↑ See z. B. Hornsby, DeTemple or Howard Eves: An Introduction into the History of Mathematics . 3rd edition. Holt, Reinhart and Winston, 1969, p. 73
6. ^ ^{A b} John Leslie: Elements of geometry and plane trigonometry: With an appendix, and copious notes and illustrations . 3rd edition. Archibald Constable & Co, 1817, p. 176 , p. 430 . It should be noted that the comment on Carlyle is not yet included in the earlier editions of the book (1809, 1811).
7. ^ GA Miller: Geometric Solution of the Quadratic Equation . In: The Mathematical Gazette , Vol. 12, No. 179 (Dec., 1925), pp. 500-501 ( JSTOR 3602823 )