I know that the formula for a dilation in the real plane $\mathbb{R}^2$ with center the origin $(0,0)$ is $(cx,cy)$, with $c \neq 0$. What is the formula for a dilation in the real plane with center an arbitrary point $(a,b)$?
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We can combine three movements.
- Translate the plane so that the point goes to the origin. $$(x,y)\to (x-a, y-b)$$
- Do the dilation $$(x-a, y-b)\to (c(x-a), c(y-b))$$
- Translate the plane so that the origin goes to the point $$(c(x-a), c(y-b)\to (c(x-a)+a, c(y-b)+b)$$
Combining that we get $$D_{(a,b)}(x,y)=(c(x-a)+a, c(y-b)+b)$$
John Douma
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An application like that is an affine map called homothety. Given the center $c$, the homothety $f$ of ratio $\lambda$ can be calculated as $f(p)=c+\lambda \vec{cp}$, where $p$ is a point of the affine space.
If you use the usual reference system and the center has coordinates $(a,b)$, the homothety of ratio $\lambda$ could be expressed as $f(x,y)=(a+\lambda(x-a),b+\lambda(y-b))$
Valere
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