Spherical Harmonic Surface
Spherical Harmonic Surface is a spectral surface, given by a parametrisation, immersed.
Open Spherical Harmonic Surface in the interactive viewer →
aperiodic def-parametric immersed implemented spectral
Properties
- Family
- spectral
- Given by
- a parametrisation
- Curvature
- no condition imposed
- Periodicity
- not periodic
- Ends
- 0
- Embedding
- immersed
- Fidelity
- exact
Definition
No chart is stored: r = f(Y_l^m) needs the associated Legendre functions, which the expression language does not have. The shipped implementation is authoritative; an unverified transcription would silently define a different surface.
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- Spherical harmonics: P. S. Laplace, "Theorie des attractions des spheroides et de la figure des planetes", Memoires de l'Academie royale des Sciences, 1785; A.-M. Legendre, "Recherches sur l'attraction des spheroides homogenes", Memoires de Mathematique et de Physique, 1785.
- Real form, normalisation and the Condon-Shortley phase: E. U. Condon and G. H. Shortley, "The Theory of Atomic Spectra", Cambridge University Press, 1935.
- The associated Legendre recurrences used here: M. Abramowitz and
- I. A. Stegun, "Handbook of Mathematical Functions", Dover, 1965, chapter 8.
- The eight-parameter sculptural family r = sin(m0 phi)^m1 + cos(m2 phi)^m3 + sin(m4 theta)^m5 + cos(m6 theta)^m7 is Paul Bourke's "Spherical Harmonics" form (February 1990), http://paulbourke.net/geometry/sphericalh/ . The parameter sets offered as presets below are project-chosen, not his.