When I read the Wiki about Legendre transformation, there is a statement
The Legendre transformation is an application of the duality relationship between points and lines.
What's the meaning of this statement?
When I read the Wiki about Legendre transformation, there is a statement
The Legendre transformation is an application of the duality relationship between points and lines.
What's the meaning of this statement?
For a convex function you can do the following:
For each point on the graph of the function, draw the line tangent to the function at that point. That point can now be identified by its original $x$ and $y=f(x)$ coordinates, or by specifying the slope of that tangent line and its corresponding y-intercept. Each point maps to one and only one line, and vice versa. For convex functions, the mapping is one-to-one. There is no ambiguity. Draw a sketch and you will soon be convinced.
The Legendre transformation gives you the value of the y-intercept if you give it the slope. So the Legendre transform is a plot of $b(m)$ vs $m$ (y-intercept as a function of slope) rather than $f(x)$ vs $x$. Either function represents the same data or concept. In a sense they contain the same information.
update Thanks to @EmilioPisanty for improved wording. See comments.