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Consider a sine wave in Cartesian coordinate system, with positive part in XY-plane and the negative part in XZ-plane:

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The two planes here are orthogonal, but they can have an angle $\theta$ , in general. How this wave is related to an ordinary sine wave with all two parts on the same axis?

Michael Medvinsky
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Osh
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  • I am not sure what function you are describing, could you make its form more explicit? – Justpassingby Dec 28 '15 at 22:14
  • "This wave" is "related to" the "ordinary sine wave with all two parts" in the same plane by the fact its negative part is rotated around the $x$ axis by $\theta$. – dbanet Dec 28 '15 at 22:34

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You could turn a regular sine wave into such a function if you were to define sine as a vector-valued position function: $$\vec{r} = \begin{bmatrix}cos(\frac{-\pi}{2}*g(t)) & 0 & sin(\frac{-\pi}{2}*g(t))\\0 & 1 & 0\\ -sin(\frac{-\pi}{2}*g(t)) & 0 & cos(\frac{-\pi}{2}*g(t))\end{bmatrix} \begin{bmatrix}0\hat{i} \\ t\hat{j} \\ sin(t)\hat{k}\end{bmatrix}$$

where g(t) is defined as: $$ g(t) = \left\{\begin{aligned} &0, \ \ where&& \lfloor t \rfloor \ne n\\ &1, \ \ where&& \lfloor t \rfloor = n \end{aligned} \right.$$

$$where \ n \in \mathbb{Z}: \frac{n}{2} \notin \mathbb{Z} \ (n \ is \ an \ odd \ integer \ only)$$

That would make this the identity matrix when it's not an odd integer multiple of $\pi$, and it rotates $90^\circ$ clockwise for that second half of the period

Jeff Strom
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