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How do I change the coordinate system from two dimensional cartesian to two dimensional curve linear and polar by using Matrix?

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  • $\begingroup$ Welcome to Engineering! This question seems like a bad fit for this SE. Feel free to ask again at Mathematics. $\endgroup$
    – Wasabi
    Nov 21 '19 at 0:13
  • $\begingroup$ @Wasabi Matrix transformation of cartesian to polar is indeed a pure mathematical question and beyond the scope of the this community, however cartesian to curvilinear transformation, is related to dynamic. The mathematical approach, is systematic, you follow the rules, you'll find the answer. The physical interpretation here, provides OP more insight than pure mathematical approach to the problem, which is almost abstract. $\endgroup$ Nov 21 '19 at 11:37
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The coordinate transformation of polar to cartesian is given by: $$\begin{pmatrix} r \\ \theta \end{pmatrix} = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$$

For a point mass, moving along a path in space, we can define an arbitrary point, and call it origine. Notice that the polar and the cartesian coordinate systems, have the same origine. And the $(x, y)$ or $(r, \theta)$ represent the position of the the point mass relative to that origine.

The tangent/normal (n-t) or curve linear coordinate system, moves with the point mass along the same path, notice that the origine of the new coordinate system now moves with the point mass.

Now, you want to express the position of the point mass in the curve linear coordinate in function of the old (cartesian) coordinates, but the position of the point mass coincide with the origine, so there is no explicit transformation here.

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