In real-time computer graphics, approximations are often used to allow for interactive virtual scene rendering. Concerning the on-line lighting of such scenes, an approach getting increased recognition is to approximate the light in every direction of the hemisphere of a surface point using suitable mathematical distribution functions, such as the well-known Spherical Gaussian. A drawback of this distribution is that current methods using it are inaccurate and do not reflect the correct lighting integral over the surface hemisphere. We show new and more accurate convolution of a Spherical Gaussian with a clamped cosine distribution. In short, we propose a closed form approximation of the hemispherical integral of such a distribution in an arbitrary hemisphere. While our use case is the approximation of the hemispherical lighting situation, we believe that our general formulation of the hemispherical integral of a Spherical Gaussian can also be useful in other areas.
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