"Shape is what is left when the influence of
position,
size
and rotation
has been removed."
↳ what remains is a fixed point on S²
— D. G. Kendall, 1984
Human vision and shape representations.
When you recognise a triangle, your visual system encounters it at many different positions, sizes and orientations — rotated, scaled, shifted across the retina.
Each of these is a different representation of the same shape in the image plane.
Kendall's shape space makes this precise: all those representations belong to a single equivalence class.
The brain, like Kendall's quotient, must be doing something that is invariant to position, size and rotation — discarding those degrees of freedom to recover the underlying shape.
This visualisation shows what that invariance looks like geometrically: infinitely many rotated triangles in ℝ², all mapping to one point on the shape sphere S².
Configuration Space — ℝ²
Each frame is a different representation of the same triangle — rotated by k/n of a full turn. In human vision these would each activate different orientation-tuned neurons, yet all depict the same shape.
Kendall Shape Space — S²
Shape = (z₃−z₁)/(z₂−z₁) ∈ ℂP¹ ≅ S²
Once position, size and rotation are quotiented out, all representations collapse to a single point. The shape point does not move.