Anyone know a good derivation of the linear interpolation:
$$\frac{y-y_0}{x-x_0}=\frac{y_1-y_0}{x_1-x_0}$$
Wikipedia gives one, which I don't understand.
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$\begingroup$Since it is a linear interpolation, just consider a straight line $y=a+ bx$ which goes through two points $(x_0,y_0)$ and $(x_1,y_1)$. So $$y_0=a+b x_0$$ $$y_1=a+b x_1$$ Solve for $a,b$.
$\endgroup$ $\begingroup$For any two points: $(x_0 ,y_0 )$ and $(x_1 ,y_1 )$
$y_0 =mx_0 +b$ and $y_1 =mx_1 +b$
Solve for $b$ for each equation...
Then set the two equal to each other...
Next solve for $m$
you get: $m=(y_0 -y_1 )/(x_0 -x_1 )$
Since this is true for any two points on the line, its also true for the first point and another point on the line where you may know the $y$ but not the $x$: say $(x,y_2 )$
so then:$$m= \frac{y_0 -y_1 }{x_0 -x_1 }=\frac{y_0 -y_2 }{x_0 -x}.$$
There it is!
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