BIZOMAT MAKEOS · Geometry
Cylindrical and Spherical Coordinates
What are the rectangular, cylindrical or spherical coordinates of this point?
Convert a point between rectangular (x, y, z), cylindrical (r, θ, z) and spherical (ρ, θ, φ) coordinates, for five-axis work, probing and layout.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Convert from | choice | The coordinate system of the entered point. |
| x, or r, or ρ | mm | x (rectangular), r (cylindrical radius) or ρ (spherical distance from origin). |
| y, or θ (angle in XY plane from +X) | mm or ° | y (rectangular) or θ in degrees (cylindrical and spherical), measured in the XY plane from +X. |
| z, or φ (angle from +Z axis) | mm or ° | z (rectangular and cylindrical) or φ in degrees from the +Z axis (spherical). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Convert from
- What it is
- The coordinate system your point is written in: rectangular (x, y, z), cylindrical (radius, angle, height) or spherical (distance, two angles). It sets the meaning of the three inputs below.
- Units and format
- Choose one of 3 options from the list: Rectangular (x, y, z); Cylindrical (r, θ, z); Spherical (ρ, θ, φ).
- Example
- The form opens with “Cylindrical (r, θ, z)” selected as a worked example. Choose the option that matches your case.
- Get it right
- Angles are taken in degrees in both cylindrical and spherical form.
What this means: x, or r, or ρ
- What it is
- The first coordinate: x for a rectangular point, the radius r from the Z axis for a cylindrical point, or the straight-line distance ρ from the origin for a spherical point.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000000 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Cylindrical r is measured from the Z axis in the XY plane; spherical ρ is measured from the origin itself.
What this means: y, or θ (angle in XY plane from +X)
- What it is
- The second coordinate: y for a rectangular point, or the angle θ in the XY plane measured from the +X axis for a cylindrical or spherical point.
- Units and format
- Enter a number in mm or °: mm or °. The form accepts -10000000 mm or ° to 10000000 mm or °. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 mm or ° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- In rectangular mode this is a length in mm; in the other modes it is an angle in degrees, positive anticlockwise from +X.
What this means: z, or φ (angle from +Z axis)
- What it is
- The third coordinate: z for a rectangular or cylindrical point, or the angle φ measured down from the +Z axis for a spherical point.
- Units and format
- Enter a number in mm or °: mm or °. The form accepts -10000000 mm or ° to 10000000 mm or °. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 mm or ° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Spherical φ is from the +Z axis, not up from the XY plane; some software uses elevation from the plane instead.
Method and formulas
- Cylindrical to rectangular: x = r cos θ, y = r sin θ, z = z.
- Spherical to rectangular: x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ.
- Back: r = √(x² + y²), ρ = √(x² + y² + z²), θ = atan2(y, x), φ = arccos(z/ρ).
- Example: cylindrical (50, 30°, 20) is x 43.301, y 25, z 20.
Limitations
- Mathematical convention (φ from +Z); some CAD systems measure elevation from the XY plane instead.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Program points given in cylindrical or spherical form
- Convert probe data
- Lay out holes on cylinders and domes
Who it is for
- CNC programmers
- Quality inspectors
- Design engineers
What you get
- Rectangular x, y, z
- Cylindrical r, θ, z
- Spherical ρ, θ, φ
Interpreting the result
- Check the angle conventions of your machine or software before using the values.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 43–45. Topic reference; no source tables or recommended settings are reproduced.
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