I am learning about polar graphs for the first time, and I ran into some trouble understanding the transformations of polar graphs.
From https://brilliant.org/wiki/polar-curves/, the following linear transformations can be performed on a polar curve:
Rotations about the pole can be performed in polar form by replacing the parameter $θ$ with $(θ−ϕ)$; this will rotate the curve counterclockwise by $ϕ$ radians.
A reflection about a line $θ=ϕ$ can be performed by replacing the parameter $θ$ with $(2ϕ−θ)$.
A reflection about the pole can be performed by replacing the parameter θ with $(θ−π)$.
Where did each of these replacement parameters come from (especially that of a reflection about a line)/why do they work to transform the polar graphs in that way? Another point of confusion I have is that a rotation about the pole seems to have the same form as a reflection about the pole, with the only difference being that the latter specifies $ϕ=π$. How is the form for reflection and rotation around the pole the same, yet it produces different results?