Scaling a circle with cosine

Question

Take a simple circle:

Scale it down (2 methods):

-or-

Now, with that same simple circle, add a cosine variant:

How can the cosine variant be scaled without distorting the original shape? That is, how can the x-values from the original shape be preserved by scaling/translating them down to overlay on the smaller simple circle?

Answer 1

There is a general answer.

We need two geometric transformations: translation and homothecy.

Given a curve with equation $F(x,y)=0$, you may translate the curve by vector $\vec u$: replace the equation by $F(x-u_x,y-u_y)=0$.

Likewise, given a curve with equation $F(x,y)=0$, you may apply a homothecy of center $O$ (the origin of coordinates) and ratio $\lambda$: replace the equation by $F(x/\lambda,y/\lambda)=0$.

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Here, we would like a homothecy of ration $1/2$, but with center at $(1,2)$. We will apply a translation to move the center of homothecy to $O$, then apply the homothecy, then translate back.

Step by step:

1. Initial curve: $(x-1)^2+(y-2)^2=4+\cos(\pi x)$.
2. After the first translation: $x^2+y^2=4+\cos(\pi x+\pi)=4-\cos(\pi x)$.
3. After the homothecy: $4x^2+4y^2=4-\cos(2\pi x)$.
4. After the second translation: $4(x-1)^2+4(y-2)^2=4-\cos(2\pi x)$.

The equation of the scaled curve is thus

$$4(x-1)^2+4(y-2)^2=4-\cos(2\pi x)$$

Illustration: