What does it mean for a curve to be homothetic to another?

Question

What does it mean for one curve to be homothetic (< Greek θετικός, < τιθέναι to place) to another?

Answer 1

Hadamard's Lessons in Geometry: I. Plane Geometry (bk. 3, ch. 5 "Homothecy & Similarity", p. 145) defines "homothecy":

140. Definition. If we choose a point S as the center of homothecy, and a number $k$ called the ratio of homothecy or ratio of similarity, the homothetic image of an arbitrary point M is the point M' obtained by joining SM and taking a segment SM on this line or on its extension, starting from S (Fig. 136), such that $\frac{SM'}{SM}=k$.

The homothecy is said to be direct if SM' is taken in the sense of SM (fig. 136.) and inverse if these segments are in opposite senses (Fig. 137). The homothetic image of a figure F is the figure formed by the set of all points M', homothetic to the points which constitute the figure F.

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Lo Bello's Origins of Mathematical Words p. 164 says:

homothetic See the entry homo- above. The word homothetic is a modern invention created on the analogy of adjectives like synthetic from the verb τίθημι, to put or place. The associated adjective is θετικός, which means fit for placing, apposite, positive. A homothetic function of the plane is a dilation or contraction followed by a translation, that is, a function that takes $(x,y)$ to $(u,v)$ where $u = h + λx$ and $v = k + λy$, where $h$, $k$, and $λ$ are arbitrary real numbers. The idea intended is that geometric figures are placed in the same position relative to one another after the application of the homothetic function as they were before. However, the adjective θετικός never meant placed, which was τιθείς.

Answer 2

A homothety is a map $f:\>{\mathbb R}^n\to{\mathbb R}^n$ that can be described in the form $$f:\quad{\bf x}\to {\bf a}+\lambda{\bf x}$$ for some ${\bf a}\in{\mathbb R}^n$ and $\lambda\ne0$. The set ${\cal H}$ of these maps does not depend on a chosen basis, nor on a scalar product (euclidean metric) which might be present in ${\mathbb R}^n$.

It follows that two curves $\gamma$, $\gamma'\subset{\mathbb R}^2$ are homothetic if they are similar under a similarity $f:\>{\mathbb R}^2\to{\mathbb R}^2$ leaving directions invariant.