Abstract: In this paper a new approach is derived in the context of shape analysis. The so called polynomial Eulerian shape theory solves an open problem proposed in [27] about the construction of certain shape density involving Euler hypergeometric functions of matrix arguments. The associated distribution is obtained by a connection between the required shape invariants and a known result on canonical correlations published in 1963. As usual in matrix variate statistical shape theory, the densities are expressed in terms of infinite series of zonal polynomials. However, if we consider certain parametric subspace for the parity of the number of landmarks, the computational problem can be solved analytically by deriving the Eulerian matrix relation of two matrix arguments. Under that restriction, the analysis of classical landmark data is based on polynomial distribution with small degree. Finally, a methodology to compare Eulerian shape and landmark discrimination under equivalent classes is proposed and applied in machine vision. © Allerton Press, Inc. 2024.