Post by Gary on Nov 9, 2016 18:29:04 GMT
Vasco,
Regarding Subsection 10 The Poincaré Disc, it seems that enlightenment lurks just behind the hyperbolic plane. Part of my difficulty with this subsection (and others in the section) is what appears to be a change in the presentation format. Whereas previously, we had the $\mathbb{C}$ plane, the Riemann sphere, and projections between them, here we have Euclidian objects, "the map" of hyperbolic objects, and objects in the hyperbolic plane (e.g. pseudosphere) apparently depicted on the same diagram. I have the impression that objects from at least two different planes are appearing superimposed. This leads to confusion because when I see a circle, I wonder if I should think about it as Euclidean or hyperbolic. As a case in point here are three sentences from p. 317, $\mathbb{P} 5$:
$\hspace{5em}$1. Figure [36b] illustrates an infinitesimal disc of Euclidean radius 'ds' centred at $z = r e^{i\theta}$ in the $Poincar\acute{e}$ disc.
In 1, we have an Euclidean radius in a $Poincar\acute{e}$ disc, which I believe to be a kind of hyperbolic map.
$\hspace{5em}$2. Because the map is conformal, the h-length '$d\hat{s}$' of '$ds$' is independent of the direction of '$ds$', ...
Sentence 2 indicates that when a transformation is applied to an Euclidean object such as 'ds', the transformation from Euclidean space to the map is conformal. Since the map is described as 'conformal', it must be that all objects on the map result from conformal $M\ddot{o}bius$ transformations of Euclidean space. But sometimes, transformations are applied to hyperbolic objects, such as rotation of the h-line in $\mathbb{P}$ 4.
$\hspace{5em}$3. The equidistant curve '$e$' is then the illustrated arc of a Euclidean circle through the ends of '$l$'.
In 3, an equidistant curve is an object in "the map", so it is hyperbolic. But it is also an arc of a Euclidean circle. So it seems we are being asked to look at its Euclidean features and transform them into hyperbolic map features. Is this circle on the page an overloaded symbol?
On p. 301, $\mathbb{P}$, Needham wrote
$\hspace{5em}$To avoid confusion, let us use the prefix 'h-' to distinguish hyperbolic concepts from their Euclidean descriptions in the map. For example an 'h-line' will mean a 'hyperbolic line' (i.e. a geodesic), while a 'line' will refer to an ordinary straight line in the map.
I am wondering if the relation between the Euclidean and the hyperbolic parts of the map is broadly analogous to the relation F(z) on p. 163 that swaps $\xi_{\pm}$ to $0$ and $\infty$. F(z) is a $M\ddot{o}bius$ transformation and other $M\ddot{o}bius$ transformations can be applied to either the LHS of [29] or to its image. By analogy, certain transformations such as $d\hat{s} = ds/y$ transform Euclidean space on the $\mathbb{C}$ plane in the map to the hyperbolic space in the map (also on the $\mathbb{C}$ plane, I think). Other transformations, such as $\mathcal{I}_L$, can work on the hyperbolic part of the map or on the Euclidean part. Whether the transformations are Euclidean-to-hyperbolic or hyperbolic-to-hyperbolic, it seems they must work in the framework of the complex plane.
I wonder how you think about these relations and whether my thinking about them is off the mark.
Gary
Regarding Subsection 10 The Poincaré Disc, it seems that enlightenment lurks just behind the hyperbolic plane. Part of my difficulty with this subsection (and others in the section) is what appears to be a change in the presentation format. Whereas previously, we had the $\mathbb{C}$ plane, the Riemann sphere, and projections between them, here we have Euclidian objects, "the map" of hyperbolic objects, and objects in the hyperbolic plane (e.g. pseudosphere) apparently depicted on the same diagram. I have the impression that objects from at least two different planes are appearing superimposed. This leads to confusion because when I see a circle, I wonder if I should think about it as Euclidean or hyperbolic. As a case in point here are three sentences from p. 317, $\mathbb{P} 5$:
$\hspace{5em}$1. Figure [36b] illustrates an infinitesimal disc of Euclidean radius 'ds' centred at $z = r e^{i\theta}$ in the $Poincar\acute{e}$ disc.
In 1, we have an Euclidean radius in a $Poincar\acute{e}$ disc, which I believe to be a kind of hyperbolic map.
$\hspace{5em}$2. Because the map is conformal, the h-length '$d\hat{s}$' of '$ds$' is independent of the direction of '$ds$', ...
Sentence 2 indicates that when a transformation is applied to an Euclidean object such as 'ds', the transformation from Euclidean space to the map is conformal. Since the map is described as 'conformal', it must be that all objects on the map result from conformal $M\ddot{o}bius$ transformations of Euclidean space. But sometimes, transformations are applied to hyperbolic objects, such as rotation of the h-line in $\mathbb{P}$ 4.
$\hspace{5em}$3. The equidistant curve '$e$' is then the illustrated arc of a Euclidean circle through the ends of '$l$'.
In 3, an equidistant curve is an object in "the map", so it is hyperbolic. But it is also an arc of a Euclidean circle. So it seems we are being asked to look at its Euclidean features and transform them into hyperbolic map features. Is this circle on the page an overloaded symbol?
On p. 301, $\mathbb{P}$, Needham wrote
$\hspace{5em}$To avoid confusion, let us use the prefix 'h-' to distinguish hyperbolic concepts from their Euclidean descriptions in the map. For example an 'h-line' will mean a 'hyperbolic line' (i.e. a geodesic), while a 'line' will refer to an ordinary straight line in the map.
I am wondering if the relation between the Euclidean and the hyperbolic parts of the map is broadly analogous to the relation F(z) on p. 163 that swaps $\xi_{\pm}$ to $0$ and $\infty$. F(z) is a $M\ddot{o}bius$ transformation and other $M\ddot{o}bius$ transformations can be applied to either the LHS of [29] or to its image. By analogy, certain transformations such as $d\hat{s} = ds/y$ transform Euclidean space on the $\mathbb{C}$ plane in the map to the hyperbolic space in the map (also on the $\mathbb{C}$ plane, I think). Other transformations, such as $\mathcal{I}_L$, can work on the hyperbolic part of the map or on the Euclidean part. Whether the transformations are Euclidean-to-hyperbolic or hyperbolic-to-hyperbolic, it seems they must work in the framework of the complex plane.
I wonder how you think about these relations and whether my thinking about them is off the mark.
Gary
