Statistical shape models using a principal-component analysis are inadequate for studying shapes that are in non-linear manifolds. Principal tangent components use a matrix Lie group that maps a non-linear manifold to a corresponding linear tangent space. Computations that are performed on the tangent space of the manifold use linear statistics to analyze non-linear shape spaces. The method was tested on bone surface from proximal femurs. Using only three components, the new model recovered 94% of the medical dataset, whereas a conventional method that used linear principal components needed 24 components to achieve the same reconstruction accuracy.