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I've spent the day learning elementary vector math and I'm curious: how can a circle be represented through vector notation? My textbook doesn't mention it, and Google doesn't seem to help.


Edit: in case my question doesn't make sense, what I mean by vector math is, e.g. representing a line using notation such as $$r = (a,b,c) + \alpha(d,e,f)$$

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up vote 7 down vote accepted

If $m$ is the vector of the center of the circle and $r$ its radius, we can define the vectors $x$ of the points of the circle by:

$$|x-m|=r $$

which just says that the distance between $x$ and $m$ is $r$.

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Isn't this a sphere? – N. McA. Mar 23 '15 at 12:06
If x and m are 3D vectors, yes. If x and m are 2D vectors, it's a circle. – mach Apr 12 '15 at 22:45

Based on your example, I assume you are looking for a parametric equation of a circle in 3D. You need to supply a "coordinate system" $(c, u, v)$ in which $c=(c_x, c_y, c_z)$ is the center of the circle, and $u=(u_x, u_y, u_z)$ and $v=(v_x, v_y, v_z)$ are two orthogonal unit vectors spanning the plane that contains the circle.

For a parameter $t$ in $[0, 2 \pi]$ you can now find the points on the circle by

$$\mathbf{x} = \mathbf{c} + r \cos t\, \mathbf{u} + r \sin t\, \mathbf{v}$$

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