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Chapter 4: Problem 16

Let \(\mathbf{x}: U \subset R^{2} \rightarrow R^{3}\), where $$ \begin{aligned} U &=\left\\{(\theta, \varphi) \in R^{2} ; 0<\theta<\pi, 0<\varphi<2 \pi\right\\}, \\ \mathbf{x}(\theta, \varphi) &=(\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta), \end{aligned} $$ be a parametrization of the unit sphere \(S^{2}\). Let $$ \log \tan \frac{1}{2} \theta=u, \quad \varphi=v, $$ and show that a new parametrization of the coordinate neighborhood \(\mathbf{x}(U)=V\) can be given by $$ \mathbf{y}(u, v)=(\operatorname{sech} u \cos v, \operatorname{sech} u \sin v, \tanh u) . $$ Prove that in the parametrization \(\mathbf{y}\) the coefficients of the first fundamental form are $$ E=G=\operatorname{sech}^{2} u, \quad F=0 $$ Thus, \(\mathbf{y}^{-1}: V \subset S^{2} \rightarrow R^{2}\) is a conformal map which takes the meridians and parallels of \(S^{2}\) into straight lines of the plane. This is called Mercator's projection.

Short Answer

Expert verified
The new parametrization is:\(\mathbf{y}(u, v) = (\operatorname{sech} u \cos v, \operatorname{sech} u \sin v, \tanh u)\).The coefficients of the first fundamental form are:\( E = G = \operatorname{sech}^2 u, F = 0 \).

Step by step solution

01

Define Variables

Given the parametrization \( \mathbf{x}(\theta, \varphi)=(\sin \theta \cos \varphi, \sin\theta \sin\varphi, \cos\theta) \), let \( u = \log \tan \frac{1}{2} \theta \) and \( v = \varphi \). Our goal is to reparametrize \( \mathbf{x}(\theta, \varphi) \) using the new variables \( (u, v) \).
02

Express \(\theta\) in terms of \(u\)

Since \( u = \log \tan \frac{1}{2} \theta \), solve for \( \theta \):1. Express \( \tan \frac{1}{2} \theta \) in terms of \( u \):\[ \tan \frac{1}{2} \theta = e^u \]2. Use the double-angle identity for tangent:\[ \tan \theta = \frac{2 \tan \frac{1}{2} \theta}{1 - \tan^2 \frac{1}{2} \theta} = \frac{2e^u}{1 - e^{2u}}. \]This allows us to find \( \sin \theta \) and \( \cos \theta \).
03

Find \( \sin \theta \) and \( \cos \theta \) in terms of \(u\)

Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{2e^u}{1 - e^{2u}} \). We have:1. Using \( \sin \theta = 2 \sin \frac{1}{2} \theta \cos \frac{1}{2} \theta \):2. With \( \sin \frac{1}{2} \theta = \frac{e^u}{\sqrt{1+e^{2u}}} \) and \( \cos \frac{1}{2} \theta = \frac{1}{\sqrt{1+e^{2u}}} \):\[ \sin \theta = \frac{2e^u}{1 + e^{2u}} \]3. Similarly for \( \cos \theta \):\[ \cos \theta = \frac{1 - e^{2u}}{1 + e^{2u}} \]
04

Express \( \mathbf{x}(\theta, \varphi) \) in terms of \(u\) and \(v\)

1. Using the expressions for \( \sin \theta \) and \( \cos \theta \):\[ \mathbf{y}(u, v) = (\sin \theta \cos v, \sin \theta \sin v, \cos \theta) \]2. Substitute \( \sin \theta = \frac{2e^u}{1 + e^{2u}} \) and \( \cos \theta = \frac{1 - e^{2u}}{1 + e^{2u}} \) into the parametrization:\[ \mathbf{y}(u, v) = \left( \frac{2e^u}{1 + e^{2u}} \cos v, \frac{2e^u}{1 + e^{2u}} \sin v, \frac{1 - e^{2u}}{1 + e^{2u}} \right) \]
05

Simplify the Expression for \( \mathbf{y}(u, v) \)

Using the hyperbolic identities \( \operatorname{sech} u = \frac{2 e^u}{1+e^{2u}} \) and \( \tanh u = \frac{1-e^{2u}}{1+e^{2u}} \), we get\[ \mathbf{y}(u, v) = (\operatorname{sech} u \cos v, \operatorname{sech} u \sin v, \tanh u) \]
06

Calculate the First Fundamental Form

1. The first fundamental form coefficients are: \( E = \mathbf{E} \cdot \mathbf{E} \), \( F = \mathbf{E} \cdot \mathbf{F} \), and \( G = \mathbf{F} \cdot \mathbf{F} \).2. Compute the partial derivatives of \( \mathbf{y}(u, v) \):\[ \frac{\partial \mathbf{y}}{\partial u} = (-\operatorname{sech} u \tanh u \cos v, -\operatorname{sech} u \tanh u \sin v, \operatorname{sech}^2 u) \]\[ \frac{\partial \mathbf{y}}{\partial v} = (-\operatorname{sech} u \sin v, \operatorname{sech} u \cos v, 0) \]3. Calculate \( E, F, G \):\[ E = (\operatorname{sech} u)^2 \]\[ F = 0 \]\[ G = (\operatorname{sech} u)^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

unit sphere parametrization
Understanding the parametrization of a unit sphere is crucial in differential geometry. The unit sphere, denoted as \( S^2 \), can be parameterized in several ways. One effective way is using spherical coordinates, where a point on the sphere is represented by \( (\theta, \varphi) \). In this context, \( \theta \) is the polar angle (measured from the positive z-axis), and \( \varphi \) is the azimuthal angle (measured from the positive x-axis in the xy-plane).

The given parametrization is \( \mathbf{x}(\theta, \varphi) = (\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta) \). This describes how a point on the unit sphere relates to the angles \( \theta \) and \( \varphi \). As \( \theta \) varies from 0 to \( \pi \) and \( \varphi \) varies from 0 to 2\( \pi \), every point on the surface of the sphere is covered.

Reparametrizing these coordinates can simplify various calculations. For example, switching to coordinates \( (u, v) \), where \( u = \log \tan \frac{1}{2} \theta \) and \( v = \varphi \), transforms the original parametrization into a more manageable form, \( \mathbf{y}(u, v) = (\operatorname{sech} u \cos v, \operatorname{sech} u \sin v, \tanh u) \). This new parametrization has useful properties for differential geometry and mapping tasks.
Mercator's projection
Mercator's projection is an essential concept in cartography and differential geometry. It's a type of cylindrical map projection where the meridians and parallels of the sphere map onto straight lines on a plane, making it particularly useful for navigation.

In the given exercise, the parametrization \( \mathbf{y}(u, v) = (\operatorname{sech} u \cos v, \operatorname{sech} u \sin v, \tanh u) \) is shown to convert the coordinates of the sphere into straight lines in the plane. This transformation is conformal, meaning it preserves angles. Specifically, on Mercator's projection, meridians (lines of constant longitude) appear as vertical lines, while parallels (lines of constant latitude) appear as horizontal lines. This characteristic is particularly advantageous for plotting a course in navigation, as the angle represented on the map corresponds directly to the actual course angle.

The conformality is proven by showing that the first fundamental form coefficients \( E \) and \( G \) are equal (both \( \operatorname{sech}^2 u \)), and \( F = 0 \). This implies that the map's grid is scaled uniformly in every direction at any point, preserving the angle between intersecting lines.
first fundamental form
The first fundamental form is a fundamental concept in differential geometry, representing the metric structure of a surface. It encodes essential information about the surface's geometry, such as distances and angles.

For a parametrized surface given by \( \mathbf{y}(u, v) \), the first fundamental form is expressed via the coefficients E, F, and G:
  • \( E = \left( \frac{\partial \mathbf{y}}{\partial u} \cdot \frac{\partial \mathbf{y}}{\partial u} \right) \)
  • \( F = \left( \frac{\partial \mathbf{y}}{\partial u} \cdot \frac{\partial \mathbf{y}}{\partial v} \right) \)
  • \( G = \left( \frac{\partial \mathbf{y}}{\partial v} \cdot \frac{\partial \mathbf{y}}{\partial v} \right) \)
The derivatives of \( \mathbf{y}(u, v) \) with respect to \( u \) and \( v \) can be calculated as:

\( \frac{\partial \mathbf{y}}{\partial u} = (-\operatorname{sech} u \tanh u \cos v, -\operatorname{sech} u \tanh u \sin v, \operatorname{sech}^2 u) \)
\( \frac{\partial \mathbf{y}}{\partial v} = (-\operatorname{sech} u \sin v, \operatorname{sech} u \cos v, 0) \)

These lead to:
  • \( E = (\operatorname{sech} u)^2 \)
  • \( F = 0 \)
  • \( G = (\operatorname{sech} u)^2 \)
Since \( E = G \) and \( F = 0 \), this configuration exemplifies a conformal map, maintaining angles and thus suitable for projections like Mercator's projection.

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Most popular questions from this chapter

A diffeomorphism \(\varphi: S_{1} \rightarrow S_{2}\) is said to be a geodesic mapping if for every geodesic \(C \subset S_{1}\) of \(S_{1}\), the regular curve \(\varphi(C) \subset S_{2}\) is a geodesic of \(S_{2}\). If \(U\) is a neighborhood of \(p \in S_{1}\), then \(\varphi: U \rightarrow S_{2}\) is said to be a local geodesic mapping in \(p\) if there exists a neighborhood \(V\) of \(\varphi(p)\) in \(S_{2}\) such that \(\varphi: U \rightarrow V\) is a geodesic mapping. a. Show that if \(\varphi: S_{1} \rightarrow S_{2}\) is both a geodesic and a conformal mapping, then \(\varphi\) is a similarity; that is, $$ \langle v, w\rangle_{p}=\lambda\left\langle d \varphi_{p}(v), d \varphi_{p}(w)\right\rangle_{\varphi(p)}, \quad p \in S_{1}, v, w \in T_{p}\left(S_{1}\right), $$ where \(\lambda\) is constant. b. Let \(S^{2}=\left\\{(x, y, z) \in R^{3} ; x^{2}+y^{2}+z^{2}=1\right\\}\) be the unit sphere, \(S^{-}=\) \(\left\\{(x, y, z) \in S^{2} ; z<0\right\\}\) be its lower hemisphere, and \(P\) be the plane \(z=-1\). Prove that the map (central projection) \(\varphi: S^{-} \rightarrow P\) which takes a point \(p \in S^{-}\)to the intersection of \(P\) with the line that connects \(p\) to the center of \(S^{2}\) is a geodesic mapping. c. Show that a surface of constant curvature admits a local geodesic mapping into the plane for every \(p \in S\).

Let \(S\) be a connected surface and let \(\varphi, \psi: S \rightarrow S\) be two isometries of \(S .\) Assume that there exists a point \(p \in S\) such that \(\varphi(p)=\psi(p)\) and \(d \varphi_{p}(v)=d \psi_{p}(v)\) for all \(v \in T_{p}(S)\). Prove that \(\varphi(q)=\psi(q)\) for all \(q \in S\).

Surfaces of Liouville are those surfaces for which it is possible to obtain a system of local coordinates \(\mathbf{x}(u, v)\) such that the coefficients of the first fundamental form are written in the form $$ E=G=U+V, \quad F=0 $$ where \(U=U(u)\) is a function of \(u\) alone and \(V=V(v)\) is a function of \(v\) alone. Observe that the surfaces of Liouville generalize the surfaces of revolution and prove that (cf. Example 5) a. The geodesics of a surface of Liouville may be obtained by integration in the form $$ \int \frac{d u}{\sqrt{U-c}}=\pm \int \frac{d v}{\sqrt{V+c}}+c_{1} $$ where \(c\) and \(c_{1}\) are constants that depend on the initial conditions. b. If \(\theta, 0 \leq \theta \leq \pi / 2\), is the angle which a geodesic makes with the curve \(v=\) const., then $$ U \sin ^{2} \theta-V \cos ^{2} \theta=\mathrm{const} $$ (Notice that this is the analogue of Clairaut's relation for the surfaces of Liouville.)

A diffeomorphism \(\varphi: S \rightarrow \bar{S}\) is said to be area- preserving if the area of any region \(R \subset S\) is equal to the area of \(\varphi(R)\). Prove that if \(\varphi\) is area-preserving and conformal, then \(\varphi\) is an isometry.

(Gauss Theorem on the Sum of the Internal Angles of a "Small" Geodesic Triangle.) Let \(\Delta\) be a geodesic triangle (that is, its sides are segments of geodesics) on a surface \(S\). Assume that \(\Delta\) is sufficiently small to be contained in a normal neighborhood of some of its vertices. Prove directly (i.e., without using the Gauss-Bonnet theorem) that $$ \iint_{\Delta} K d A=\left(\sum_{i=1}^{3} \alpha_{i}\right)-\pi $$ where \(K\) is the Gaussian curvature of \(S\), and \(0<\alpha_{i}<\pi, i=1,2,3\), are the internal angles of the triangle \(\Delta\).
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