Question

The transformation from Cartesian rhoϕθ-space to Cartesian xyz-space is a mathematical mapping that allows us to convert from one coordinate system to another. It's important to note that in this context, rho, ϕ, and θ are spherical coordinates, while x, y, and z are Cartesian coordinates.

The transformation equations from spherical coordinates rho, ϕ, and θ to Cartesian coordinates x, y, and z are as follows:
1. rho represents the radial distance from the origin to a point in 3D space.
2. ϕ represents the polar angle, which is the angle between the positive z-axis and the line connecting the origin to the point.
3. θ represents the azimuthal angle, which is the angle between the positive x-axis and the projection of the line connecting the origin to the point onto the xy-plane.

Answer 1

The transformation from spherical coordinates to Cartesian coordinates allows the conversion of a point's location in 3D space from the spherical system (rho, phi, theta) to Cartesian (x, y, z) using trigonometric relationships.

Step-by-step explanation:

The transformation from spherical coordinates (rho (ρ), phi (ϕ), and theta (θ)) to Cartesian coordinates (x, y, z) is a mathematical process used to convert the representation of a point in three-dimensional space from one coordinate system to another. In spherical coordinates, ρ represents the radial distance from the origin to the point, ϕ is the polar angle measured from the positive z-axis, and θ is the azimuthal angle measured in the xy-plane from the positive x-axis.

The equations to transform spherical to Cartesian coordinates are given by:
x = ρ sin(ϕ) cos(θ)
y = ρ sin(ϕ) sin(θ)
z = ρ cos(ϕ)

These conversions are derived from the fundamental principles of trigonometry and the Pythagorean theorem, highlighting the relationship between the lengths and angles within a triangle. The conversion is particularly useful in fields like physics and engineering, where symmetrical systems or rotational dynamics are studied.