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Chapter 3: Problem 8

Describe the region of the unit sphere covered by the image of the Gauss map of the following surfaces: a. Paraboloid of revolution \(z=x^{2}+y^{2}\). b. Hyperboloid of revolution \(x^{2}+y^{2}-z^{2}=1\). c. Catenoid \(x^{2}+y^{2}=\cosh ^{2} z\).

Short Answer

Expert verified
a. Upper hemisphere of the unit sphere. b. Entire unit sphere. c. Entire unit sphere.

Step by step solution

01

Understand the Gauss Map

The Gauss map sends a point on a surface to the corresponding point on the unit sphere, which is given by the normal vector at that point.
02

Paraboloid of Revolution

For the paraboloid of revolution given by the equation \(z = x^2 + y^2\), compute the normal vector at any point and then find the Gauss map image. The normal vector is \ = (-2x, -2y, 1)//// and the corresponding unit normal vector is \[\frac{(-2x, -2y, 1)}{\sqrt{4x^2 + 4y^2 + 1}}\]. Notice that as \(x\) and \(y \) vary over all real numbers, the unit normal vectors cover the upper hemisphere of the unit sphere.
03

Hyperboloid of Revolution

For the hyperboloid of revolution given by the equation \(x^2 + y^2 - z^2=1\), compute the normal vector at any point and then find the Gauss map image. The normal vector is \( = (2x, 2y, -2z)\) and the corresponding unit normal vector is \[\frac{(2x, 2y, -2z)}{\sqrt{4x^2 + 4y^2 + 4z^2}} = \frac{(x, y, -z)}{\sqrt{x^2 + y^2 + z^2 }} \]. Since \(x and y\) can take any values and \(z\) is a parameter based on \(x\) and \(y\), this normal vector maps to the entire unit sphere.
04

Catenoid

For the catenoid given by the equation \(x^2 + y^2 = \cosh^2 z\), compute the normal vector at any point and then find the Gauss map image. The normal vector is \( = \left( \frac{x}{\cosh z}, \frac{y}{\cosh z}, -\sinh z \right)\) and the corresponding unit normal vector is \[\left( \frac{x}{\cosh z}, \frac{y}{\cosh z}, -\sinh z \right) = (a \cos \theta , a\sin \theta, -a \tanh z) \]. As \(x and y\) vary and the hyperbolic functions cover their respective range, the normal vectors cover the entire unit sphere.

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Key Concepts

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

unit sphere
The term 'unit sphere' refers to a sphere with a radius of one unit. It is centered at the origin in three-dimensional Cartesian coordinates (0, 0, 0). This sphere is a fundamental surface in differential geometry.
The unit sphere is often used as a reference to map normal vectors from other surfaces. Its equation can be given as:
\[ x^2 + y^2 + z^2 = 1 \]
In simpler terms, every point on the unit sphere is exactly one unit away from the origin.
Key properties of the unit sphere:
  • All points on the surface are equidistant from the origin.
  • Any normal vector on the unit sphere is aligned with the radius at any given point.
The Gauss map, which connects points from other surfaces to points on the unit sphere, involves these normal vectors.
normal vector
A normal vector is a vector that is perpendicular to the surface at a given point. For any given surface, the normal vector plays a crucial role in defining its orientation.
Computing a normal vector is necessary in differential geometry for understanding the behavior of surfaces.
For a surface defined by a function \( z = f(x, y) \), the normal vector \( \textbf{N} \) can often be obtained from the gradient:
  • For the paraboloid of revolution \( z = x^2 + y^2 \), the normal vector is:
    \[ \textbf{N} = (-2x, -2y, 1) \]
  • For the hyperboloid of revolution \( x^2 + y^2 - z^2 = 1 \), the normal vector is:
    \[ \textbf{N} = (2x, 2y, -2z) \]
  • For the catenoid \( x^2 + y^2 = \text{cosh}^2 z \), the normal vector is:
    \[ \textbf{N} = \frac{(x, y, -\text{sinh} z)}{\text{cosh} z} \]
The unit normal vector is then computed by normalizing the normal vector, ensuring it has a length of one.
paraboloid of revolution
The paraboloid of revolution is a quadratic surface generated by rotating a parabola around its axis of symmetry.
Its standard form is given by \( z = x^2 + y^2 \). This surface is used commonly in optics and radio telescopes due to its reflective properties.
  • The normal vector at a point \((x, y, z)\) on this surface is calculated as \( \textbf{N} = (-2x, -2y, 1) \).
  • The corresponding unit normal vector then becomes:
    \[\frac{(-2x, -2y, 1)}{\text{sqrt}(4x^2 + 4y^2 + 1)}\].
  • This map's image covers the upper hemisphere of the unit sphere due to the nature of the normal vectors' directions.
hyperboloid of revolution
The hyperboloid of revolution is a surface generated by rotating a hyperbola around its central axis.This surface has two types: one-sheeted and two-sheeted. For our purposes, we use a one-sheeted hyperboloid given by the equation:
  • \[ x^2 + y^2 - z^2 = 1 \]

Key aspects include:
  • Normal vector at a point \((x, y, z)\) is computed as:\[ \textbf{N} = (2x, 2y, -2z) \]
  • The unit normal vector simplifies to:
    \[\frac{(x, y, -z)}{\text{sqrt}(x^2 + y^2 + z^2)}\].
  • This map covers the entire unit sphere because \(x, y\), and \(z\) span their full ranges, giving a complete picture of the surface's normal vectors.
catenoid
The catenoid is a distinctive surface generated by rotating a catenary curve around a horizontal axis.
Its mathematical description is given as:
  • \[ x^2 + y^2 = \text{cosh}^2 z \]
This shape is commonly associated with minimal surfaces, as it represents a surface with minimal area for a given boundary.
Here's how we handle the normal vector:
  • For a point on the catenoid, the normal vector is:
    \[\textbf{N} = \frac{(x, y, -\text{sinh} z)}{\text{cosh} z}\].
  • This leads to the unit normal vector of:
    \[ (\text{a cos }\theta, \text{a sin }\theta, -a \text{tanh } z) \], where \( a \) covers the full range.
  • As a result, the image under the Gauss map from the catenoid covers the entire unit sphere.

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Most popular questions from this chapter

(The Hessian.) Let \(h: S \rightarrow R\) be a differentiable function on a surface \(S\), and let \(p \in S\) be a critical point of \(h\) (i.e., \(d h_{p}=0\) ). Let \(w \in T_{p}(S)\) and let $$ \alpha:(-\epsilon, \epsilon) \rightarrow S $$ be a parametrized curve with \(\alpha(0)=p, \alpha^{\prime}(0)=w\). Set $$ H_{p} h(w)=\left.\frac{d^{2}(h \circ \alpha)}{d t^{2}}\right|_{t=0} $$ a. Let \(\mathbf{x}: U \rightarrow S\) be a parametrization of \(S\) at \(p\), and show that (the fact that \(p\) is a critical point of \(h\) is essential here) $$ H_{p} h\left(u^{\prime} \mathbf{x}_{u}+v^{\prime} \mathbf{x}_{v}\right)=h_{u u}(p)\left(u^{\prime}\right)^{2}+2 h_{u v}(p) u^{\prime} v^{\prime}+h_{v v}(p)\left(v^{\prime}\right)^{2} . $$ Conclude that \(H_{p} h: T_{p}(S) \rightarrow R\) is a well-defined (i.e., it does not depend on the choice of \(\mathbf{x}\) ) quadratic form on \(T_{p}(S) . H_{p} h\) is called the Hessian of \(h\) at \(p\). b. Let \(h: S \rightarrow R\) be the height function of \(S\) relative to \(T_{p}(S)\); that is, \(h(q)=\langle q-p, N(p)\rangle, q \in S\). Verify that \(p\) is a critical point of \(h\) and thus that the Hessian \(H_{p} h\) is well defined. Show that if \(w \in T_{p}(S)\), \(|w|=1\), then \(H_{p} h(w)=\) normal curvature at \(p\) in the direction of \(w .\) Conclude that the Hessian at \(\mathrm{p}\) of the height function relative to \(\mathrm{T}_{\mathrm{p}}(\mathrm{S})\) is the second fundamental form of \(\mathrm{S}\) at \(\mathrm{p}\).

Consider the parametrized surface \(\mathbf{x}(u, v)=\left(\sin u \cos v, \sin u \sin v, \cos u+\log \tan \frac{u}{2}+\varphi(v)\right)\), where \(\varphi\) is a differentiable function. Prove that a. The curves \(v=\) const. are contained in planes which pass through the \(z\) axis and intersect the surface under a constant angle \(\theta\) given by $$ \cos \theta=\frac{\varphi^{\prime}}{\sqrt{1+\left(\varphi^{\prime}\right)^{2}}} $$ Conclude that the curves \(v=\) const. are lines of curvature of the surface. b. The length of the segment of a tangent line to a curve \(v=\) const., determined by its point of tangency and the \(z\) axis, is constantly equal to 1 . Conclude that the curves \(v=\) const. are tractrices (cf. Exercise 6).

Let \(\lambda_{1}, \ldots, \lambda_{m}\) be the normal curvatures at \(p \in S\) along directions making angles \(0,2 \pi / m, \ldots,(m-1) 2 \pi / m\) with a principal direction, \(m>2\). Prove that $$ \lambda_{1}+\cdots+\lambda_{m}=m H, $$ where \(H\) is the mean curvature at \(p\).

Let \(p\) be an elliptic point of a surface \(S\), and let \(r\) and \(r^{\prime}\) be conjugate directions at \(p\). Let \(r\) vary in \(T_{p}(S)\) and show that the minimum of the angle of \(r\) with \(r^{\prime}\) is reached at a unique pair of directions in \(T_{p}(S)\) that are symmetric with respect to the principal directions.

Determine the asymptotic curves of the catenoid $$ \mathbf{x}(u, v)=(\cosh v \cos u, \cosh v \sin u, v) . $$
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