Logaritmic spiral

[BobCat]

How to prove that scaled logaritmic spiral is same spiral just rotated (rotational invariance)?

 

[HallsofIvy]

First, what is the formula for the logarithmic spiral? Second, if you define \(\displaystyle x'= x cos(\theta)+ ysin(\theta)\), \(\displaystyle y'= - x sin(\theta)+ y'cos(\theta)\), where x and y satisfy that formula, do x' and y' also satisfy it?

 

[BobCat]

I found solution

equation of logaritam spiral is r(alpha)=A*e^(B*alpha) in polar coordinats where A,B are constants
or alpha(r)=(1/B)*ln(1/r) which is f(r)
let lambda be scaling factor, than to scale function f(r) we need to substitute r by r*lambda
alpha(r*lambda)=(1/B)*ln(1/r*lambda)
and that is alpha(r*lambda)=-(1/B)*ln(r)-(1/B)*ln(lambda)
which is alpha(r*lambda)=-f(r)-(1/B)*ln(lambda)
alpha(r*lambda)=-f(r)-C
And that is same function of logaritmic spiral rotated by constant angle C