ancient principle of mathematics: figure = varying element

Question

On page 83 in the book Conceptual mathematics by Lawvere et al. it says:

An ancient principle of mathematics holds that a figure is the locus of a varying element.

What does this quote mean? In particular, what is the "locus of a varying element" and in which sense is it the same as a figure?

Answer 1

An example:

Let's $P(\alpha)$ be the point $e^{i\alpha}$ of the complex plan.

The locus of $P$ when $\alpha\in\mathbb{R}$ is the circle $C$ of radius $1$ whose center is the origin.

$P$ is the varying element, $C$ is the figure.

Answer 2

I am studying the same book (Perhaps different edition). Here is what I think author intended. Consider category Sets. Suppose $f: A \rightarrow B$. Now consider point $1 \rightarrow A$. A varying element V has following properties:

$$V(1 \rightarrow A)=1 \rightarrow B$$ (It appears that $V$ is a map from set of points of $A$ to set of points of $B$.) This can be shown by composing points with $f$ as follows: $$f∘a=b$$ Where $a: 1 \rightarrow A$ and $b:1 \rightarrow B$.

So evaluating $V$ at different points of $A$ gives a family of points of B. For every point of $A$, $f$ assigns a point of $B$. This is one perspective for a map. Now, we can replace 1 by D, and say that a map $f:A \rightarrow B$ is a family of D-elements of B.

What I do not understand though, is last sentence of that paragraph, which is: " For example, we can take D = A, and the identity $D \rightarrow A$, thus revealing that (c) the varying element, as a single thing, is a single figure or element itself. "

Any ideas on this?