Two languages for the same point

Remark — Two languages for the same point

Cartesian and polar are two different languages speaking of the same geometric object (a point of the plane): the translation is one-to-one (except for the pole, an “untranslatable word”). A circle centred at the origin has equation $x^2+y^2=a^2$ in one language and $\rho=a$ in the other: the second is ten times shorter. Being able to change language is a key mathematical skill: as with natural languages, some things are easier to say in one language rather than the other (Esty).

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Topics: Trigonometry
Concepts: Cartesian coordinates · Polar coordinates · Cartesian plane