Try to see this as a 3-dimensional picture.
The black ellipse is an object, depicted in the z=1 plane
(the 'object plane').
The white curve is its slope transform (Legendre transform),
depicted in the y=-1 plane (the 'legendre plane').
The red point runs along the x-axis (in the object plane), its value indicated in yellow; it indexes a point on the black ellipse, to which a tangent is drawn in white. The slope of the tangent is in green, and the value of its Legendre transform is the intercept of the y-axis (in the object plane), in red.
The slope is then used as an index along the x-axis (in the legendre plane) for a the legendre transform in the z-direction. Its slope is minus the x-index (in yellow).
Created with Cinderella20010518