Problem 34 - Thomas Calculus

Question

Hyperboloid of one sheet a. Find a parametrization for the hyperboloid of one sheet \(x^{2}+y^{2}-z^{2}=1\) in terms of the angle \(\theta\) associated with the circle \(x^{2}+y^{2}=r^{2}\) and the hyperbolic parameter \(u\) associated with the hyperbolic function \(r^{2}-z^{2}=1 .\) Hint: \(\cosh ^{2} u-\sinh ^{2} u=1 . )\) b. Generalize the result in part (a) to the hyperboloid \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)-\left(z^{2} / c^{2}\right)=1\)

Step-by-Step Solution

Answer

Parametrize with \(x = \cosh u \cos \theta, y = \cosh u \sin \theta, z = \sinh u\); generalize with coefficients for each variable.

Step 1: Identify Parameters and Equation Transformations

The given hyperboloid equation is \(x^{2}+y^{2}-z^{2}=1\). Recognize that this relates to two sub-expressions: the circle \(x^{2}+y^{2}=r^{2}\) with parameter \(\theta\), and the hyperbolic expression \(r^{2}-z^{2}=1\) with parameter \(u\).

Step 2: Parametrize the Circle with \(\theta\)

The circle \(x^{2}+y^{2}=r^{2}\) is parameterized in terms of \(\theta\) as: \(x = r\cos\theta\) and \(y = r\sin\theta\).

Step 3: Introduce Hyperbolic Trigonometric Functions

Relate the parameter \(u\) to the hyperbolic functions \(\cosh u\) and \(\sinh u\) because \(\cosh^2 u - \sinh^2 u = 1\). This leads to defining \(r = \cosh u\) and \(z = \sinh u\).

Step 4: Substitute to Complete the Parametrization

Substitute the polar and hyperbolic parameterizations into the hyperboloid equation: \(x = \cosh u \cdot \cos \theta\), \(y = \cosh u \cdot \sin \theta\), and \(z = \sinh u\).

Step 5: Generalize to a General Hyperboloid

In the general equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\), use similar substitutions: \(x = a\cosh u \cdot \cos \theta\), \(y = b\cosh u \cdot \sin \theta\), and \(z = c\sinh u\).

Key Concepts

Parametrization

Parametrization is a mathematical concept used to express geometric shapes and surfaces using parameters. These parameters allow us to represent the coordinates of points on a surface using simpler expressions. In the context of the hyperboloid of one sheet, we utilize two parameters:

• \( \theta \) - associated with the parametric representation of a circle, controls the angle for circular motion in the \( xy \)-plane.
• \( u \) - related to hyperbolic functions, defines the hyperbolic aspect of the structure in terms of height or depth.

This method provides an elegant way to handle complex surfaces, reducing them to functions of simpler variables. In this case, the parametrization allows us to connect trigonometric and hyperbolic components in the hyperboloid's equation, making the relationship between \( x, y, \) and \( z \) more comprehensible. It simplifies the manipulation and visualization of such surfaces in further mathematical analysis.

Hyperbolic functions

Hyperbolic functions are analogues to the trigonometric functions but for hyperbolas, not circles. The key hyperbolic functions are hyperbolic sine (\( \sinh \)) and hyperbolic cosine (\( \cosh \)). They are defined as:

• \( \sinh u = \frac{e^u - e^{-u}}{2} \)
• \( \cosh u = \frac{e^u + e^{-u}}{2} \)

The identity \( \cosh^2 u - \sinh^2 u = 1 \) mirrors the trigonometric identity \( \cos^2 \theta + \sin^2 \theta = 1 \). These functions are most useful when dealing with hyperbolic geometry or scenarios involving rapid growth or decay, typical in relativity and engineering applications. In our hyperboloid example, these functions determine the extent and ancillarily direct the formation of the shape, especially affecting the radial portion orthogonal to the circular base.

Trigonometry

Trigonometry primarily involves the relationships between angles and lengths in triangles. It's cornerstone in geometry revolving around circles. The specific use of trigonometric functions like sine and cosine are pivotal for parametric equations involving rotations or angles.In our hyperboloid demonstration:

• The circle component is parameterized as \( x = r \cos \theta \) and \( y = r \sin \theta \) where \( r \) is the radius (here \( \cosh u \)).
• This characterizes the circular aspect on the \( xy \)-plane.

These equations assist in converting planar circle concepts cylindrically across three-dimensional spaces. They systematically explain why structures rotate or stretch within any given plane, incorporating angular dimensions directly into algebraic expressions linked with hyperbolas.

Conic sections

Conic sections are curves obtained from intersecting a cone with a plane. Depending on the angle of intersection, these shapes manifest as ellipses, parabolas, or hyperbolas. Hyperboloids are related to hyperbolas from conic sections, especially as they extend these concepts into three dimensions. Key aspects of conic sections:

• Ellipses represent closed and bounded paths within circles and ovals.
• Parabolas are reflective, depicting paths like trajectories.
• Hyperbolas describe double curves diverging in opposite directions.

The hyperboloid involves these conic perspectives, creating a surface encapsulating two infinite sheets. By incorporating hyperbolic patterns vertically and circular ones horizontally using parametrizations, the hyperboloid embodies a dynamic intersection pattern. This three-dimensional adaptation of conic sections greatly enhances our capacity to analyze and visualize complex geometrical and physical structures.