Spherical Harmonic Calculator

1) What the calculator evaluates

Spherical harmonics form an orthogonal basis on the unit sphere. This tool evaluates Y_l^m(θ, φ) from an associated Legendre term and an azimuthal phase, reporting real/imag parts, magnitude, and phase. It also outputs a real-harmonic value for visualization workflows quickly.

2) Input ranges and angle meaning

Degree l is supported from 0 to 60; order m must satisfy −l ≤ m ≤ l. Theta θ is colatitude: θ=0 along +z, θ=π/2 at the equator. Phi φ is azimuth around the z-axis, typically 0…2π in many physics problems used here too.

3) Associated Legendre computation

Angular dependence enters through P_l^|m|(cosθ). The calculator uses upward recurrences rather than symbolic differentiation, which is efficient for interactive use. Recursion reduces cancellation from direct closed forms and supports repeated evaluation for sweeps without heavy overhead across l and m up to the tool limits.

4) Condon–Shortley phase control

Many physics references include the Condon–Shortley factor (−1)^m within P_l^m. Some engineering and graphics conventions omit it, so the tool provides a toggle. Changing it mainly flips the sign for odd m, affecting table comparisons and fitted coefficients in downstream spherical expansions.

5) Normalization choices and scaling

Normalization changes amplitude, not angular shape. Orthonormal N_lm enforces ∫|Y_l^m|²dΩ = 1 on the unit sphere. Schmidt normalization multiplies m≠0 terms by √2, common in some geomagnetism and gravity models. Unnormalized output helps match legacy notes and quick derivations when coefficients were defined without N_lm.

6) Complex and real output conventions

The complex form uses Y_l^m(θ, φ)=N_lmP_l^m(cosθ) e^imφ. Negative m values follow the standard conjugation relation with a parity factor. The displayed real harmonic combines √2·cos(mφ) or √2·sin(|m|φ) with P_l^|m| to produce real basis functions for maps and finite elements.

7) Sweep mode for angular scans

Phi-sweep and theta-sweep modes generate a table over a start–end range. Choose 2 to 721 samples for coarse previews or dense scans. The page shows 30 rows for readability, while CSV exports all rows, ready for plotting lobes or checking symmetries.

8) Export, validation, and practical use

CSV export supports downstream analysis, fitting, and visualization. The PDF export creates a portable summary with parameter metadata and a preview of up to 20 rows. For validation, compute the same (l, m) under different normalizations and confirm only amplitude changes. For speed, keep l modest and tune samples to match resolution when scanning ranges of θ or φ.

1) Is θ latitude or colatitude?

Here θ is colatitude: 0 at +z, π/2 on the equator, π at −z. If you have geographic latitude, convert using θ = π/2 − latitude.

2) Why can m be negative?

Negative orders encode the same angular frequencies with a defined symmetry. The calculator uses the standard physics relation linking Y_l^−m to the complex conjugate of Y_l^m, with a parity factor.

3) Which normalization should I choose?

Use orthonormal when you want unit-power basis functions on the sphere. Use Schmidt if your reference uses √2 scaling for m ≠ 0 terms. Choose unnormalized only when matching legacy formulas that omit N_lm.

4) What does the phase toggle change?

It controls whether the Condon–Shortley factor (−1)^m is included in P_l^m. This can flip signs for odd m and will affect comparisons against sources that use a different convention.

5) Why do values look unstable for large l?

Large l and |m| amplify numerical sensitivity because P_l^m spans very large and very small scales across θ. Reduce l, adjust normalization, or use fewer sweep samples if you see noisy phase or abrupt jumps.

6) How do sweeps interpret start and end angles?

Start and end are read in your selected unit. Phi-sweep varies φ while holding θ fixed, and theta-sweep varies θ while holding φ fixed. The tool interpolates linearly across the requested number of samples.

7) Can I reconstruct a function on the sphere?

Yes. Expand f(θ, φ) as a sum of coefficients times Y_l^m. This calculator provides point values that help you test coefficients, validate conventions, and generate angular grids for fitting or simulation pipelines.

The complex spherical harmonic (physics convention) is:

The orthonormal normalization used in many texts is:

A common real-harmonic construction is:

This calculator computes P_l^m via stable recursion and reports Re, Im, magnitude, and phase.