🌐 3D Spatial Geometry & Vector Analysis
3D Spherical, Cylindrical & Cartesian Converter
Convert 3D space coordinates between Cartesian $(x,y,z)$, Cylindrical $(\rho, \phi, z)$, and Spherical $(r, \theta, \phi)$ systems in degrees and radians.
Cartesian Input $(x, y, z)$
Spherical Coordinates (r, θ, φ)
r = 7.071, θ = 45.0°, φ = 53.1°
Radial Radius ($r$)
7.071
Polar Angle ($\theta$)
45.00° (0.785 rad)
Azimuth ($\phi$)
53.13° (0.927 rad)
💡 Transformation Formula:
$$r = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{50} \approx 7.071, \quad \theta = \arccos(5 / 7.071) = 45.0^\circ$$
3D Coordinate Transformations
Standard ISO physics convention for 3D coordinate transformations:
- Spherical Radius: $$r = \sqrt{x^2 + y^2 + z^2}$$
- Polar / Inclination Angle ($\theta$ from $+Z$ axis): $$\theta = \arccos\left(\frac{z}{r}\right)$$
- Azimuthal Angle ($\phi$ from $+X$ axis in $XY$ plane): $$\phi = \operatorname{atan2}(y, x)$$
- Cylindrical Radius ($\rho$ in $XY$ plane): $$\rho = \sqrt{x^2 + y^2}$$