© 2018 Indian Statistical Institute A new model of statistical shape theory under elliptical models is proposed by using the polar decomposition. This work completes the group of SVD and QR shape densities obtained from the transpose of the square root of a non singular Wishart matrix. The associated non isotropic and non central polar shape distributions are set in the context of consistent computable series of zonal polynomials. Then the inference procedures with elliptical assumptions can be performed at the same computational cost of the published routines based on Gaussian models. As an example of the technique, a classical application in Biology is studied under three models, the usual Gaussian and two Kotz type models; then the best model is selected by a modified BIC∗criterion, and a test for equality in polar shapes is performed. The published results for this landmark data under isotropic Gaussian models and procrustes theory are also discussed.