Given a family of curves $\mathcal{F}_1$ and an orthogonal family $\mathcal{F_2}$ in rectangular coordinates, how does one create a coordinate system and basis based on these two families.
For example, if the first family is a set of concentric circles centered at the origin and the second family is every straight line passing through the origin, then we can construct a coordinate system based on these families (polar coordinates).
I figured it out. If one defines both families in terms of an arbitrary constant and then treats the two families as a system of equations and solves for x and y, a new coordinate transformation can be defined and an orthogonal basis will easily follow. For example.
Given circles defined by $x^2+y^2=A^2$ and the orthogonal family, $y=Bx$, x and y can be solved for to return the coordinate transformation: $y=A\sqrt{\frac{B^2}{1+B^2}}$ and $x=A\sqrt{\frac{1}{1+B^2}}$ which can be transformed into the classic polar coordinate transformation of $A=r$ and $B=\tan(\theta)$.