Interactive model

QA: Coordinate Mapper

Explore Cartesian and spherical coordinates with a live 3D mapper.

This interactive page studies one point or vector in ordinary three-dimensional space. Enter Cartesian values $(x, y, z)$ to compare the coordinate tuple with the spherical description $(r, θ, φ)$ .

The mapper reports radius, polar angle, azimuth angle, and the local spherical basis vectors written in Cartesian components. The goal is to make the change of coordinates visible before using the same language in quantum-state diagrams.

Interactive model

3D coordinate mapper

x coordinate y coordinate z coordinate

Radius r5.385

Polar theta42.031 deg

Azimuth phi56.31 deg

Local spherical basis in Cartesian components

e_r     = [0.371, 0.557, 0.743]
e_theta = [0.412, 0.618, -0.67]
e_phi   = [-0.832, 0.555, 0]

Coordinate Maps Are Not New Spaces

A coordinate map changes how the same geometric object is described. The point is not replaced by the numbers; the numbers are labels after a basis and coordinate convention have been chosen.

For the convention used here,

r = \sqrt x 2 + y 2 + z 2

θ = arccos(z r)

φ = atan2(y, x)

The polar angle $θ$ is measured down from the positive z-axis, and the azimuth angle $φ$ is measured in the x-y plane from the positive x-axis.

QA: Coordinate Mapper

3D coordinate mapper

Coordinate Maps Are Not New Spaces

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