Gaussian Over Parallelogram

Integration of a Gaussian over a parallelogram

This app performs the integration of a Gaussian (normal distribution) over a parallelogram or a rectangle. For most of the inputs the 15 decimals shown in the output are exact. This app is implemented using JavaScript and is explained in:

  • R. García-Pelayo, J. L. Gonzalo and C. Bombardelli, "Rate and collision probability of tethers and sails against debris or spacecraft," Journal of Guidance, Control, and Dynamics, vol. 6, no. 6, pp. 1330-1342, 2019.
It is based on an algorithm due to Genz:
  • A. Genz, "Numerical computation of rectangular bivariate and trivariate normal and t probabilities," Statistics and Computing, vol. 14, pp. 251-260. 2004.


The parallelogram is defined by three consecutive vertices, either in clockwise or counter-clockwise order. The last vertex is computed automatically.

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Parameters of the Gaussian

The diagonal elements of the covariance matrix are σx2 and σy2, ρxyσxσy is its non diagonal element with ρxy ∈ [-1,1]