Vector Coordinate System Converter

How the Three Systems Relate

Cartesian (x, y, z) is the everyday grid system. Cylindrical (r, theta, z) keeps the z-axis but describes the x-y position as a distance from the origin (r) and an angle (theta) - useful for anything with rotational symmetry around one axis, like a cylinder or a spinning object. Spherical (r, theta, phi) drops Cartesian z entirely and describes position with a distance from the origin (r) and two angles: theta (azimuthal, around the z-axis) and phi (polar, down from the z-axis).

Worked Example: Cartesian to Spherical

Convert the point (x=3, y=4, z=12) to spherical coordinates. Distance from origin: r = square root of (x squared+y squared+z squared) = square root of (9+16+144) = square root of 169 = 13. Azimuthal angle: theta = atan2(4, 3) = 0.9273 radians (53.13 degrees). Polar angle: phi = acos(z/r) = acos(12/13) = acos(0.9231) = 0.3948 radians (22.62 degrees).

You can verify this reverses correctly: z = r x cos(phi) = 13 x cos(0.3948) = 13 x 0.9231 = 12.00, and x = r x sin(phi) x cos(theta) = 13 x 0.3846 x 0.6 = 3.00.

Worked Example: Cartesian to Cylindrical

The same point (3, 4, 12) in cylindrical form: r = square root of (x squared+y squared) = square root of (9+16) = square root of 25 = 5.00, theta = atan2(4,3) = 0.9273 radians (53.13 degrees - the same azimuthal angle as the spherical case, since both measure the angle in the x-y plane), and z stays 12.00 unchanged.

Why Angles Are in Radians

The underlying JavaScript trig functions (sin, cos, atan2) all operate in radians natively, so the tool keeps radians as its primary unit and adds a degree conversion alongside it for readability, rather than converting back and forth internally.

Where r Can't Be Negative

In both cylindrical and spherical systems, r represents a physical distance, which can't be negative by definition - entering a negative r value is rejected rather than silently reflected to the opposite side of the origin.