CALCULATOR
Spherical Polar Angle from Positive-z Cartesian Point Calculator
Spherical Polar Angle from Positive-z Cartesian Point. Example: X: 3 m, Y: 4 m, Z: 5 m gives 45 degrees. Positive z, polar angle only; whole-sphere coordinates deferred. Original useful product scope; historical catalog title does not grant missing modes or source parity. Version2 precision contract: Positive z. Small-angle approximation has relative analytic error≤t²/3≤3.34e-17 for t=ρ/z≤1e-8, b
Positive z, polar angle only; whole-sphere coordinates deferred. Original useful product scope; historical catalog title does not grant missing modes or source parity. Version2 precision contract: Positive z. Small-angle approximation has relative analytic error≤t²/3≤3.34e-17 for t=ρ/z≤1e-8, below binary64 precision. Degree conversion precedes multiplication by tiny m to preserve representable subnormal-degree results. Ordinary rounding/underflow of final output remains possible. Derivation: Compute scaled radial norm. For radial/z≤1e-8 evaluate the small-angle result in degrees before multiplying tiny scale; otherwise use atan2.
Accepted inputs: X: -1,000 to 1,000 m; Y: -1,000 to 1,000 m; Z: 0.1 to 1,000 m.
How the calculation works
m=max(abs(x),abs(y)); s=max(m,MIN_VALUE); u=sqrt((x/s)²+(y/s)²); ρ=m*u; if ρ≤1e-8 z: degrees=m*((u*(180/π))/z); else degrees=atan2(ρ,z)*180/π.
Worked example
With x 3 m, y 4 m, z 5 m, spherical polar angle from positive-z cartesian point is 45 degrees.
Assumptions and limits
Positive z, polar angle only; whole-sphere coordinates deferred. Original useful product scope; historical catalog title does not grant missing modes or source parity. Version2 precision contract: Positive z. Small-angle approximation has relative analytic error≤t²/3≤3.34e-17 for t=ρ/z≤1e-8, below binary64 precision. Degree conversion precedes multiplication by tiny m to preserve representable subnormal-degree results. Ordinary rounding/underflow of final output remains possible. Derivation: Compute scaled radial norm. For radial/z≤1e-8 evaluate the small-angle result in degrees before multiplying tiny scale; otherwise use atan2.