TechTorch

Location:HOME > Technology > content

Technology

Converting Between Cartesian and Spherical Coordinates in 3D

March 23, 2025Technology2467
Converting Between Cartesian and Spherical Coordinates in 3D Understan

Converting Between Cartesian and Spherical Coordinates in 3D

Understanding how to convert between Cartesian and spherical coordinates in three-dimensional space is a fundamental skill in many scientific and engineering fields. This article will guide you through the process, including important distinctions between coordinates and components.

Introduction to 3D Coordinate Systems

In three-dimensional space, Cartesian coordinates (also known as rectangular coordinates) define a position using x, y, and z. Spherical coordinates, on the other hand, define a position using r, θ, and φ. The r coordinate is the radial distance from the origin, θ is the polar angle (or zenith angle) from the positive z-axis, and φ is the azimuthal angle from the positive x-axis in the x-y plane.

Converting from Cartesian to Spherical Coordinates

To convert from Cartesian coordinates to spherical coordinates, we start with the Cartesian coordinates x, y, and z. Here are the step-by-step transformations:

Evaluating the Radial Distance

The radial distance r can be found using the Pythagorean theorem:

$$ r sqrt{x^2 y^2 z^2} $$

This measures the straight-line distance from the origin to the point in 3D space.

Calculating the Polar Angle θ

The polar angle θ is the angle between the positive z-axis and the line connecting the origin to the point. This can be calculated using:

$$ vartheta arccosleft(frac{z}{r}right) $$

The function arccos ensures that θ is in the range [0, π].

Determining the Azimuthal Angle φ

The azimuthal angle φ is the angle measured in the x-y plane from the positive x-axis. Unlike θ, φ can extend from -π to π or 0 to 2π. When x is zero, additional care is required:

If x 0 and y 0, then φ π/2. If x 0 and y 0, then φ -π/2. If x 0 and y 0, φ can be any value, and typically it is set to 0.

For the general case where x ≠ 0, use the two-argument arctangent function:

$$ varphi text{atan2}(y, x) $$

This function takes into account the quadrant of the point (x, y) and provides the correct angle.

Converting from Spherical to Cartesian Coordinates

To convert from spherical coordinates to Cartesian coordinates, use the following formulas:

$$ x r sinvartheta cosvarphi $$ $$ y r sinvartheta sinvarphi $$ $$ z r cosvartheta $$

These formulas can be used to check the accuracy of your transformations.

Components vs. Coordinates

It is crucial to understand the difference between coordinates and vector components. Coordinates, as discussed, are quantities that specify a position in a coordinate system. In contrast, components are values of a vector along specific directions in the coordinate system. For instance, the position vector always has only one component in the radial direction, while other vectors can have components in all three directions.

Conclusion

Mastery of coordinate conversions and understanding the subtle differences between coordinates and components is essential for anyone working with spatial data in 3D. Applying these concepts correctly ensures accurate and meaningful interpretations of geometric and physical phenomena.