How do I calculate a change of coordinates given two lines as new axis?
Solution 1
Let $\bf p$ be the intersection point of the lines, and let $\bf u$ and $\bf v$ be unit direction vectors associated to the two lines. The first part of the transformation should translate $\bf p$ to $\bf 0$; so wlog assume $\bf p=0$.
We want to write $\mathbf{x}=a\mathbf{u}+b\mathbf{v}$; to do this, form the matrix $\mathrm{U}=[\bf u~v]$ (we assume vectors are column vectors), so we can write $\mathbf{x}=\mathrm{U}[a~b]^T$ and hence we solve for $a,b$ via $[a~b]^T=\mathrm{U}^{-1}\bf x$. Then $\bf x$ in the old coordinates corresponds to $[a~b]^T$ in the new coordinates.
Two changes-of-coordinates of the sort we want are related by a third coordinate change taking place in the second coordinates. This new coordinate change preserves the origin and two axes, and therefore is precisely a rescaling of the $a$- and $b$-axes individually.
Solution 2
Let $P$ be the point $(x_3,y_3)$. Draw lines parallel to both $y=m_1x+b_1$ and $y=m_2x+b_2$ through $P$ and suppose those parallel lines meets $y=m_1x+b_1$ and $y=m_2x+b_2$ at $A$ and $B$ respectively. If $O$ is the intersection point of $y=m_1x+b_1$ and $y=m_2x+b_2$ then $(w,z)=(|OB|,|OA|)$
Solution 3
There is not enough information to convert coordinates. We are given the location of the axes, but not the coordinate scale on those axes. The $z$ and $w$ coordinates can be given by $$ \begin{align} z&=c_1(\color{#C00000}{m_1x+b_1-y})\\ w&=c_2(\color{#C00000}{m_2x+b_2-y}) \end{align}\tag{1} $$ for any $c_1,c_2\not=0$. Given the $(x,y)$ and $(z,w)$ for some point not on the $z$ or $w$ axes (so that we can divide by the red terms in $(1)$), we can compute $c_1$ and $c_2$, which determines the conversion $(1)$.
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Matt Groff
Updated on August 01, 2022Comments
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Matt Groff 5 months
Suppose we have an old coordinate system using the variables $x$ and $y$. We are given two equations for lines to form the axis for new coordinates. e.g. The line $z=0$ is given by the equation $y = m_1 x + b_1$, and the line $w=0$ is given by the equation $y = m_2 x + b_2$. Now suppose we are given a point $(x_3, y_3)$. What are the $w$ and $z$ coordinates of this point?
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Matt Groff over 10 yearsI think that I was making this out to be harder than it is. I only have to calculate the distance from one line, i.e. $z=0$, to get the $w$ coordinate, and vice versa. I guess that's as easy as it will get.
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anon over 10 yearsRe: your comment, Are the two axes necessarily perpendicular?
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Matt Groff over 10 years@anon: No. I guess that distance won't work. I'm hoping for the simplest possible method.
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anon over 10 yearsNote that there is actually more than one such coordinate transformation: for any such transformation, we can rescale the two coordinates afterwards, and composing the two procedures together is a different coordinate transformation. There is an obvious geometric case though: projecting points directly onto the lines, and then measuring the distance from the intersection along each line and making those two quantities the new coordinates.
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rschwieb over 10 years"Change of basis" should also probably be switched for "change of coordinates", because what's described here is an alias change of coordinates in the plane. (It includes a translation, and doesn't refer to the basis of the vector space directly.)
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Matt Groff over 10 yearsHmmm... If there is more than one possible solution/method, I'd like to know all possibilites. Maybe this should be a community wiki...
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anon over 10 years@MattGroff I don't think it should be CW; the possibilities are easy to describe (in fact my comment provides the way to obtain all possibilities from just one possibility (assuming of course you want rectilinear coordinates)).
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J. M. ain't a mathematician over 10 yearsHave you seen this?
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