Are there shapes defined by 3 (or more) generative parameters whose mapping to psychological similarity space is known? I am trying to generate 4 shapes that are equidistant in psychological similarity space - meaning that they are all equally discriminable from one another - which differ in 3 parameters, such that those parameters have a known mapping between objective and psychological space.

A simple example may help.  We might define an object in terms of its width, height, and depth.  The four objects A, B, C and D would have values as follows:

    height   width    depth

A.    2        2        1

B.    1        1        1

C.    2        1        2

D.    1        2        2


These points are equidistant in 3d euclidean space (they are actually the vertices of a tetrahedron).  However, an increment of 1 in height may not be perceived psychologically as distinct as an increment of 1 in depth (in principle).  More problematically, going from 1 to 1.5 may not be psychologically the same as going from 1.5 to 2 on any dimension, meaning that while these values are continuous, they are not linear in psychological space. In other words, objects equidistant in the metric of the parameters I'm using to generate them may not actually be equidistant in the eye/mind of an observer.

This is particularly true for this example, as point B is clearly more visually distinct from the others - it's the only cube!

I am therefore looking for 3 shape parameters where the mapping from objective space to psychological space is known, and in which there is not a loss of dimensionality in this mapping to below 3 dimensions in psychological similarity space.

I suspect this is not known, but I would like to avoid reinventing the wheel (not to mention running the complex pilot experiment this would require - it's not even my question of primary interest!)
Consider the colour visual system. Take 3 monochromatic (in the physics sense) light sources of wavelength $420 \; nm$, $534 \; nm$, and $564 \; nm$; i.e. the peaks of spectral sensitivity for cones. Your 3 physical parameters are then 3 knobs $b$, $g$, $r$ that control the intensity of each light source as they shine on the same white surface. If you want more physical parameters, you can just add more knobs for other monochromatic lightsources, however 3 is sufficient. Colour perception is extremely well studied, so this comes as close as I can think of to a known mapping between between objective and psychological space. Unfortunately, you will run into some culture and language specific issues for judging the similarity of perceived colours.