  Airfoil shape design is a classical problem in engineering and manufacturing.
In this work, we combine principled physics-based considerations for the shape
design problem with modern computational techniques using a data-driven
approach. Modern and traditional analyses of 2D and 3D aerodynamic shapes
reveal a flow-based sensitivity to specific deformations that can be
represented generally by affine transformations (rotation, scaling, shearing,
translation). We present a novel representation of shapes that decouples
affine-style deformations over a submanifold and a product submanifold
principally of the Grassmannian. As an analytic generative model, the separable
representation, informed by a database of physically relevant airfoils, offers
(i) a rich set of novel 2D airfoil deformations not previously captured in the
data, (ii) an improved low-dimensional parameter domain for inferential
statistics informing design/manufacturing, and (iii) consistent 3D blade
representation and perturbation over a sequence of nominal 2D shapes.
