Complex analytic geometry

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In mathematics, complex analytic geometry sometimes denotes the application of complex numbers to plane geometry.

Rather than represent a point in the plane as a pair of Cartesian coordinates, it can be represented as a single complex number, which can be written at will in either rectangular or polar form.

Complex analytical geometry deals with solving geometrical problems, especially those involving angles, by means of complex algebra. Using Euler's formula,

$e^{i\theta}=\cos\theta+i\sin\theta,\quad$

it is possible to represent any rotation as a complex number.

Complex numbers can therefore be used to advantage to solve problems in elementary geometry. This method is sometimes used in the advanced high-school mathematics, since it allows schoolchildren to write down angle relations in terms of simple formulas.