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原文摘要

Every real spherical harmonic of even degree is congruent to its mirror image. We prove that a real spherical harmonic of odd degree $n\ge3$ with distinct Maxwell axes is congruent to its mirror image if and only if the hyperelliptic curve of genus $n-1$ defined by its binary form has a reduced automorphism of even order whose lift acts on the form by $-1$. We then determine the locus of these curves: it is the locus of curves with an extra involution if and only if $n-1$ is a power of two, and otherwise it also contains, for each even divisor $k$ of $n-1$ with odd cofactor $d\ge3$, an irreducible family $y^2=x K(x^{2k})$, $°K=d$, with generic reduced automorphism group $\Z/2k$. For $n=3$ the condition is the vanishing of the skew invariant of the binary sextic. As an application, the invariants of the truncated spherical harmonic expansion of a genus-zero surface, in a conformal parameterization normalized by its conformal barycenter, give a descriptor that is exactly invariant under parameterization, position and scale, records handedness, and is stable in the orbit distance.