We present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. We also construct orthogonal polynomials on the surface of quadratic surfaces of revolution, generalizing spherical harmonics to the surface of a cone, hyperboloid, and paraboloid. We use this construction to develop cubature and fast approximation methods.

中文翻译：

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旋转二次曲面内和上的正交多项式

我们提出了二次旋转体内正交多项式的显式构造，包括圆锥、双曲面和抛物面。我们还在二次旋转曲面的表面上构造正交多项式，将球谐函数推广到圆锥、双曲面和抛物面的表面。我们使用这种结构来开发体积和快速逼近方法。