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Generalised circle
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<h2 id="extended-complex-plane">Extended complex plane</h2> <p>The extended Euclidean plane can be identified with the <a href="/facts/Extended_complex_plane/4XgBzplC">extended complex plane</a>, so that equations of <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> can be used to describe lines, circles and inversions. </p> <h3>Bivariate linear equation</h3> <p>A <a href="/facts/Circle/9fy5ggKn">circle</a> Γ {\displaystyle \Gamma } is the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of <a href="/facts/Point_(geometry)/6YPAvX2a">points</a> z {\displaystyle z} in a plane that lie at <a href="/facts/Radius/9Gr9deYD">radius</a> r {\displaystyle r} from a center point γ . {\displaystyle \gamma .} </p> Γ ( γ , r ) = { z : the distance between z and γ is r } {\displaystyle \Gamma (\gamma ,r)=\{z:{\text{the distance between }}z{\text{ and }}\gamma {\text{ is }}r\}} <p>In the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>, γ {\displaystyle \gamma } is a complex number and Γ {\displaystyle \Gamma } is a set of complex numbers. Using the property that a complex number multiplied by its <a href="/facts/Complex_conjugate/7TEaoQDe">conjugate</a> is the square of its <a href="/facts/Absolute_value/dwJBpEs5">modulus</a> (its <a href="/facts/Euclidean_distance/9qDoQKQe">Euclidean distance</a> from the origin), an <a href="/facts/Implicit_equation/UJmQ2Cyk">implicit equation</a> for Γ {\displaystyle \Gamma } is: </p> r 2 = | z − γ | 2 = ( z − γ ) ( z − γ ) ¯ 0 = z z ¯ − γ ¯ z − γ z ¯ + ( γ γ ¯ − r 2 ) . {\displaystyle {\begin{aligned}r^{2}&=\left|z-\gamma \right|^{2}=(z-\gamma ){\overline {(z-\gamma )}}\\[5mu]0&=z{\bar {z}}-{\bar {\gamma }}z-\gamma {\bar {z}}+\left(\gamma {\bar {\gamma }}-r^{2}\right).\end{aligned}}} <p>This is a <a href="/facts/Homogeneous_equation/fk9CFdgp">homogeneous</a> <a href="/facts/Bivariate_polynomial/Lzak8VVx">bivariate</a> <a href="/facts/Linear_polynomial/Lzak8VVx">linear polynomial</a> equation in terms of the complex variable z {\displaystyle z} and its conjugate z ¯ , {\displaystyle {\bar {z}},} of the form </p> A z z ¯ + B z + C z ¯ + D = 0 , {\displaystyle Az{\bar {z}}+Bz+C{\bar {z}}+D=0,} <p>where <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a> A {\displaystyle A} and D {\displaystyle D} are <a href="/facts/Real_number/R02gw5Pb">real</a>, and B {\displaystyle B} and C {\displaystyle C} are <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugates</a>. </p><p>By dividing by A {\displaystyle A} and then reversing the steps above, the radius r {\displaystyle r} and center γ {\displaystyle \gamma } can be recovered from any equation of this form. The equation represents a generalized circle in the plane when r {\displaystyle r} is real, which occurs when A D < B C {\displaystyle AD<BC} so that the squared radius r 2 = ( B C − A D ) / A 2 {\displaystyle r^{2}=(BC-AD)/A^{2}} is positive. When A {\displaystyle A} is zero, the equation defines a straight line. </p> <h3>Complex reciprocal</h3> <p>That the reciprocal transformation z ↦ w = 1 / z {\displaystyle z\mapsto w=1/z} maps generalized circles to generalized circles is straight-forward to verify: </p> 0 = A z z ¯ + B z + C z ¯ + D = A w w ¯ + B w + C w ¯ + D = A + B w ¯ + C w + D w w ¯ = D w ¯ w + C w + B w ¯ + A . {\displaystyle {\begin{aligned}0&=Az{\bar {z}}+Bz+C{\bar {z}}+D\\[5mu]&={\frac {A}{w{\bar {w}}}}+{\frac {B}{w}}+{\frac {C}{\bar {w}}}+D\\[5mu]&=A+B{\bar {w}}+Cw+Dw{\bar {w}}\\[5mu]&=D{\bar {w}}w+Cw+B{\bar {w}}+A.\end{aligned}}} <p>Lines through the origin ( A = D = 0 {\displaystyle A=D=0} ) map to lines through the origin; lines not through the origin ( A = 0 , D ≠ 0 {\displaystyle A=0,D\neq 0} ) map to circles through the origin; circles through the origin ( A ≠ 0 , D = 0 {\displaystyle A\neq 0,D=0} ) map to lines not through the origin; and circles not through the origin ( A ≠ 0 , D ≠ 0 {\displaystyle A\neq 0,D\neq 0} ) map to circles not through the origin. </p> <h3>Complex matrix representation</h3> <p>The defining equation of a generalized circle </p> 0 = A z z ¯ + B z + C z ¯ + D {\displaystyle 0=Az{\bar {z}}+Bz+C{\bar {z}}+D} <p>can be written as a matrix equation </p> 0 = ( z 1 ) ( A B C D ) ( z ¯ 1 ) . {\displaystyle 0={\begin{pmatrix}z&1\end{pmatrix}}{\begin{pmatrix}A&B\\C&D\end{pmatrix}}{\begin{pmatrix}{\bar {z}}\\1\end{pmatrix}}.} <p>Symbolically, </p> 0 = z T C z ¯ , {\displaystyle 0=\mathbf {z} ^{\text{T}}{\mathfrak {C}}\,{\bar {\mathbf {z} }},} <p>with coefficients placed into an <a href="/facts/Invertible_matrix/fPqXk3V8">invertible</a> <a href="/facts/Hermitian_matrix/qArPblYo">hermitian matrix</a> C = C † {\displaystyle {\mathfrak {C}}={\mathfrak {C}}^{\dagger }} representing the circle, and z = ( z 1 ) T {\displaystyle \mathbf {z} ={\begin{pmatrix}z&1\end{pmatrix}}^{\text{T}}} a vector representing an extended complex number. </p><p>Two such matrices specify the same generalized circle <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> one is a <a href="/facts/Scalar_multiplication/jroFDfbM">scalar multiple</a> of the other. </p><p>To transform the generalized circle represented by C {\displaystyle {\mathfrak {C}}} by the <a href="/facts/M%C3%B6bius_transformation/YEkTTLfK">Möbius transformation</a> H , {\displaystyle {\mathfrak {H}},} apply the inverse of the Möbius transformation G = H − 1 {\displaystyle {\mathfrak {G}}={\mathfrak {H}}^{-1}} to the vector z {\displaystyle \mathbf {z} } in the implicit equation, </p> 0 = ( G z ) T C ( G z ) ¯ = z T ( G T C G ¯ ) z ¯ , {\displaystyle {\begin{aligned}0&=\left({\mathfrak {G}}\mathbf {z} \right)^{\text{T}}{\mathfrak {C}}\,{\overline {({\mathfrak {G}}\mathbf {z} )}}\\[5mu]&=\mathbf {z} ^{\text{T}}\left({\mathfrak {G}}^{\text{T}}{\mathfrak {C}}{\bar {\mathfrak {G}}}\right){\bar {\mathbf {z} }},\end{aligned}}} <p>so the new circle can be represented by the matrix G T C G ¯ . {\displaystyle {\mathfrak {G}}^{\text{T}}{\mathfrak {C}}{\bar {\mathfrak {G}}}.} </p> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Hans_Schwerdtfeger/xagkNoHK">Hans Schwerdtfeger</a>, <i><a href="/facts/Geometry_of_Complex_Numbers/Yx3Btn71">Geometry of Complex Numbers</a></i>, <a href="/facts/Courier_Dover_Publications/RwjvlSFv">Courier Dover Publications</a>, 1979</li> <li>Michael Henle, "Modern Geometry: Non-Euclidean, Projective, and Discrete", 2nd edition, <a href="/facts/Prentice_Hall/bynWAuat">Prentice Hall</a>, 2001</li> <li>David W. Lyons (2021) <a href="https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/03%3A_Geometries/3.02%3A_M%C3%B6bius_geometry">Möbius Geometry</a> from <a href="/facts/LibreTexts/8ackRcD8">LibreTexts</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Hitchman, Michael P. (2009). Geometry with an Introduction to Cosmic Topology. Jones & Bartlett. p. 43. <a href="https://books.google.com/books?id=yveG5B1few4C&pg=PA43" target="_blank">https://books.google.com/books?id=yveG5B1few4C&pg=PA43</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>