The Legendre transform f*(p) = sup_x [px − f(x)] converts a convex function f(x) into its "slope-space" dual, where p is the conjugate variable (slope). Geometrically, f*(p) is the y-intercept of the tangent line with slope p, negated. The transform is its own inverse for strictly convex functions: f** = f. In thermodynamics it connects energy representations: the Helmholtz free energy A(T,V) is the Legendre transform of internal energy U(S,V) with p = T, replacing entropy S with temperature T as the natural variable. In mechanics, the Hamiltonian H(q,p) is the Legendre transform of the Lagrangian L(q,q̇).