Spherical harmonics are a complete set of orthogonal basis functions defined on the surface of a sphere, used to represent functions of direction compactly as a weighted sum of coefficients. Analogous to a Fourier series on the sphere, they allow smooth angular functions — such as incoming light or a directional colour — to be approximated with a small number of low-order coefficients. In computer graphics they underpin precomputed radiance transfer, irradiance environment lighting, and, more recently, view-dependent colour in Gaussian splatting and neural rendering.
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- Spherical harmonics solve a recurring problem in graphics: how to represent a function defined over all directions — the light arriving at a point, the colour seen from each viewing angle — without storing a dense table of samples. Just as a Fourier series expresses a periodic signal as a sum of sinusoids, spherical harmonics express a function on the sphere as a sum of basis functions ordered by angular frequency, so that a few low-order terms capture the smooth, dominant structure.
- The power of the representation comes from orthogonality and rotational properties. Because the basis functions are orthogonal, projecting a function onto them yields independent coefficients computed by integration, and reconstruction is a simple weighted sum. Crucially, integrals of products of functions reduce to dot products of their coefficient vectors, turning expensive directional integrals — exactly what lighting calculations require — into cheap vector operations.
- This is why spherical harmonics became central to real-time global illumination. Precomputed radiance transfer projects how a scene responds to distant lighting into spherical-harmonic coefficients offline, so that at runtime, relighting under a changing environment reduces to a dot product. Irradiance environment maps use just nine coefficients to represent diffuse lighting from an entire environment with visually negligible error, an efficiency that made soft, realistic ambient lighting feasible in interactive applications.
- The technique has found renewed prominence in modern neural and point-based rendering. In 3D Gaussian splatting, each primitive stores low-order spherical-harmonic coefficients to encode how its colour changes with viewing direction, reproducing specular highlights and view-dependent effects without per-frame shading networks. This pairing of a classical compact angular basis with learned scene representations exemplifies how spherical harmonics remain a foundational tool wherever directional functions must be represented efficiently.