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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>Möbius Transformation</title>
<link rel="icon" href="images/riemann_sphere">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta name="keywords" content="mathematics, geometry, visualisation, mobius, symplectic, group, veronese, lagrangian">
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</head>
<body>
<header><a href="index.html"><input type="button" value="Home"></a> <a href="mobius-fr.html"><input type="button" value="Français"></a></header>
<h1>Möbius Transformation</h1>
<p>
A Möbius transformation \(M\) is a function \(M: \hat{\mathbb{C}}\rightarrow\hat{\mathbb{C}}\), where
\(\hat{\mathbb{C}} = \mathbb{C}\cup\{\infty\}\) and is called the <i>Riemann sphere</i>.<br>
More precisely, a Möbius transformation is a function defined by
\begin{align*}
M : \hat{\mathbb{C}} &\rightarrow \hat{\mathbb{C}}\\
z &\mapsto \frac{az+b}{cz+d}
\end{align*}
with \(a,b,c,d \in \mathbb{C}\) and \(ad-bc \neq 0\).
</p>
<p>
The following program uses the fact that
there exists a unique \(M\) such that for \(\{z_{1},z_{2},z_{3}\}\) and \(\{w_{1},w_{2},w_{3}\}\) two sets
of distinct elements in \(\hat{\mathbb{C}}\), we have
\(M(z_{i}) = w_{i}\) for \(i=1,2,3\).<br>
The two grey points represent \(z_{1} = w_{1}\) and \(z_{2} = w_{2}\), and by dragging your mouse,
you specify \(z_{3} \mapsto w_{3}\). The program then finds the corresponding \(M\) and applies the transformation on the Riemann sphere!
</p>
<p>
On the left, the visualisation uses domain coloring, and on the right, it uses discrete points \(a+b\mathrm{i}\)
with \(a,b \in \mathbb{Z}\).
</p>
<div class="picture_box">
<a href="visualisation/domaincoloring.html"><img src="images/picture_dom_col.png"></a>
<a href="visualisation/points.html"><img src="images/picture_points.png"></a>
</div>
</body>
</html>