Spherical functions appear throughout computer graphics, from spherical harmonic lighting and precomputed radiance transfer to neural radiance fields and procedural planet rendering. Efficient evaluation is critical for real-time applications, yet existing approaches face a quality-performance trade-off: bilinear LUT sampling is fast but produces faceting, while bicubic filtering requires 16 texture samples. Most implementations use finite differences for normals, requiring extra samples and introducing noise. This paper presents Spherical Hermite Maps, a derivative-augmented LUT representation that resolves this trade-off. By storing function values alongside scaled partial derivatives at each texel of a padded cubemap, bicubic-Hermite reconstruction is enabled from only four texture samples (a 2x2 footprint) while providing continuous gradients from the same samples. The key insight is that Hermite interpolation reconstructs smooth derivatives as a byproduct of value reconstruction, making surface normals effectively free. In controlled experiments, Spherical Hermite Maps improve PSNR by 8-41 dB over bilinear interpolation and match 16-tap bicubic quality at one-quarter the cost. Analytic normals reduce mean angular error by 9-13% on complex surfaces while yielding stable specular highlights. Three applications demonstrate versatility: spherical harmonic glyph visualization, radial depth-map impostors for mesh level-of-detail, and procedural planet/asteroid rendering with spherical heightfields.

翻译：球面函数在计算机图形学中广泛存在，从球谐光照与预计算辐射传输到神经辐射场和程序化行星渲染。高效求值对实时应用至关重要，但现有方法面临质量与性能的权衡：双线性查找表采样速度快但会产生面片化伪影，而双三次滤波需要16次纹理采样。大多数实现使用有限差分计算法线，这需要额外采样并引入噪声。本文提出球面埃尔米特映射，一种基于导数增强的查找表表示方法，解决了这一权衡问题。通过在填充立方体贴图的每个纹素处存储函数值及缩放偏导数，仅需四次纹理采样（2x2采样区域）即可实现双三次-埃尔米特重构，并同时从相同采样中提供连续梯度。核心洞见在于：埃尔米特插值在重构函数值的同时，能自然重构平滑导数，使得表面法线计算实质上无需额外开销。在受控实验中，球面埃尔米特映射相较于双线性插值将峰值信噪比提升了8-41 dB，并以四分之一成本达到了16次采样双三次滤波的质量。解析法线在复杂表面上将平均角度误差降低了9-13%，同时产生稳定的高光反射。三个应用展示了其多功能性：球谐函数符号可视化、用于网格细节层次的径向深度图代理，以及基于球面高度场的程序化行星/小行星渲染。