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For the general form of the equation the coefficient ''A'' is the height of the peak and {{math|(''x''<sub>0</sub>, ''y''<sub>0</sub>)}} is the center of the blob.
If we set
<math display="block"> \begin{align}
a &= \frac{\cos^2\theta}{2\sigma_X^2} + \frac{\sin^2\theta}{2\sigma_Y^2}, \\
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\end{align}
</math>then we rotate the blob by a positive, counter-clockwise angle <math>\theta</math> (for negative, clockwise rotation, invert the signs in the ''b'' coefficient).<ref>{{cite web |last1=Nawri |first1=Nikolai |title=Berechnung von Kovarianzellipsen |url=http://imkbemu.physik.uni-karlsruhe.de/~eisatlas/covariance_ellipses.pdf |access-date=14 August 2019 |url-status=dead |archive-url=https://web.archive.org/web/20190814081830/http://imkbemu.physik.uni-karlsruhe.de/~eisatlas/covariance_ellipses.pdf |archive-date=2019-08-14}}</ref>
To get back the coefficients <math>\theta</math>, <math>\sigma_X</math> and <math>\sigma_Y</math> from <math>a</math>, <math>b</math> and <math>c</math> use
<math display="block">\begin{align}
\theta &= \frac{1}{2}\arctan\left(\frac{2b}{a-c}\right), \quad \theta \in [-45, 45], \\
\sigma_X^2 &= \frac{1}{2 (a \cdot \cos^2\theta + 2 b \cdot \cos\theta\sin\theta + c \cdot \sin^2\theta)}, \\
\sigma_Y^2 &= \frac{1}{2 (a \cdot \sin^2\theta - 2 b \cdot \cos\theta\sin\theta + c \cdot \cos^2\theta)}.
\end{align}</math>
Example rotations of Gaussian blobs can be seen in the following examples:
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Also see [[multivariate normal distribution]].
=== Higher-order Gaussian or super-Gaussian function or generalized Gaussian function ===
A more general formulation of a Gaussian function with a flat-top and Gaussian fall-off can be taken by raising the content of the exponent to a power <math>P</math>:
<math display="block">f(x) = A \exp\left(-\left(\frac{(x - x_0)^2}{2\sigma_X^2}\right)^P\right).</math>
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