Question

Describe geometrically Beltrami's parametrization of the sphere of radius \(k\) given by $$ \begin{aligned} &x=\frac{u k}{\sqrt{a^{2}+u^{2}+v^{2}}}, \quad y=\frac{v k}{\sqrt{a^{2}+u^{2}+v^{2}}} \\ &z=\frac{a k}{\sqrt{a^{2}+u^{2}+v^{2}}} \end{aligned} $$

Short answer

Question: Describe geometrically Beltrami's parametrization of the sphere of radius \(k\). Answer: Geometrically, Beltrami's parametrization represents points on a sphere with radius \(k\) using two parameters, \(u\) and \(v\). By examining the sum of the squares of each parametrization equation, we can see that it is equal to \(k^2\), confirming that the parametrization indeed describes a sphere of radius \(k\).

Step-by-step solution

Understand Beltrami's Parametrization of the Sphere

Beltrami's parametrization represents the coordinates (x, y, z) of points on a sphere of radius \(k\) in terms of two parameters, \(u\) and \(v\). This parametrization has the form: $$ \begin{aligned} &x=\frac{u k}{\sqrt{a^{2}+u^{2}+v^{2}}}, \quad y=\frac{v k}{\sqrt{a^{2}+u^{2}+v^{2}}} \\ &z=\frac{a k}{\sqrt{a^{2}+u^{2}+v^{2}}} \end{aligned} $$ Let us analyze the parametrization to understand its geometric properties and relation to a sphere of radius \(k\).

Examine the square of the parametrization equations

To understand further the geometric properties of Beltrami's parametrization, we can examine the sum of the squares of each parametrization equation. This will give us insight into the relationship between \(u\), \(v\), \(x\), \(y\), and \(z\): $$ x^2+y^2+z^2 = \left(\frac{u k}{\sqrt{a^{2}+u^{2}+v^{2}}}\right)^2 + \left(\frac{v k}{\sqrt{a^{2}+u^{2}+v^{2}}}\right)^2 + \left(\frac{a k}{\sqrt{a^{2}+u^{2}+v^{2}}}\right)^2 $$ $$= \frac{u^2 k^2 + v^2 k^2 + a^2 k^2}{a^2 + u^2 + v^2} $$ Now we want to simplify and understand what this expression means geometrically.

Simplify the sum of the squares and interpret geometrically

Now let's simplify the expression obtained above: $$ \frac{u^2 k^2 + v^2 k^2 + a^2 k^2}{a^2 + u^2 + v^2} = k^2 \frac{u^2 + v^2 + a^2}{a^2 + u^2 + v^2} = k^2 $$ We see that the sum of the squares of the three coordinates (x, y, z) is equal to the square of the radius \(k\). Geometrically, this means that the parametrization describes points on a sphere with radius \(k\). Thus, Beltrami's parametrization of the sphere represents points on a sphere with radius \(k\) in terms of two parameters \(u\) and \(v\).

Key concepts

Geometric Parametrization

Geometric parametrization is a mathematical approach used for representing the points of a geometric figure using parameters. In the context of Beltrami's parametrization, it takes the form of two variables, u and v, to describe all points on the surface of a sphere of radius k.

The advantage of this method is that it simplifies the dealing with complex shapes, like spheres, by reducing the dimensional complexity. Imagine trying to specify each point on a sphere with precise coordinates; it would be a daunting task. Parametrization makes it much simpler by providing a formulaic approach. Each pair of (u, v) corresponds to exactly one point on the sphere, thereby mapping a 2D space of parameters to the 3D curved surface of the sphere. This is a powerful tool in mathematical geometry and is widely used in fields such as computer graphics, where it aids in texture mapping and 3D modeling.

Spherical Coordinates

Spherical coordinates provide a way to express the position of points in three-dimensional space using three values: the radius r, the inclination theta, and the azimuthal angle phi. While Beltrami's parametrization might not look exactly like traditional spherical coordinates, it is closely related.

How do these relate to Beltrami's parametrization?

In the case of Beltrami, the coordinates are replaced by a more generalized pair of parameters u and v. By manipulating these two, we can traverse the entire surface of the sphere. This is akin to spherical coordinates, where varying theta and phi sweeps across the sphere's surface. It's crucial to understand that while the formulation differs, the underlying principle of using a fixed radius and two varying parameters to outline the surface remains consistent.

Mathematical Geometry

When talking about mathematical geometry, we're referring to the abstract language of shapes, sizes, and the relative position of figures. It serves as the basis for parametrizations and thoughtful simplifications like Beltrami's. Through algebraic formulas and geometric reasoning, we can capture the very essence of multidimensional objects on a piece of paper or a computer screen.

Mathematical geometry ties beautifully with the sphere's parametrization: We use algebra identified with geometrical meaning to express the complex relationship of points on the sphere. By evaluating the sum of squares of x, y, and z from Beltrami’s parametrization and seeing that they simplify to the sphere's radius squared, we marry algebra and geometry, resulting in a deep understanding of the sphere's geometry.