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2D Coordinates Systems

Tool to achieve coordinates system changes in the 2d-plane (cartesian, polar, etc.). These are mathematical operations representing the same elements but in different referentials.

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2D Coordinates Systems -

Tag(s) : Geometry

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# 2D Coordinates Systems

## Change of 2D Coordinates (plane)

### How to convert cartesian coordinates to polar?

The base / referential change using cartesian coordinates $(x, y)$ to another referential using polar coordinates $(r, \theta)$ obey the equations: $$r = \sqrt{x^2 + y^2} \\ \theta = 2\arctan\left(\frac y{x+ \sqrt{x^2+y^2}} \right)$$ with $\arctan$ the reciprocal of the function $\tan$ (tangent).

NB: the value of $\theta$ calculated here is included in the inverval $] -\pi, \pi]$ (to have it in the interval $] 0, 2\pi]$ add $2 \pi$ if the value of the angle is negative)

If $r = 0$ then the angle can be defined by any real number

Example: The point of the plane in position $(1,1)$ in Cartesian coordinates is defined by the polar coordinates $r = \sqrt {2}$ and $\theta = \pi/4$

### How to convert polar coordinates to cartesian?

The base / referential change from polar coordinates $(r, \theta)$ to another referential using cartesian coordinates $(x, y)$ follows the equations: $$x = r \cos (\theta) \\ y = r \sin (\theta)$$

with $r$ a positive real number and $\theta$ an angle defined between $] -\pi, \pi]$

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