Figure 1
From: Ricci-flow based conformal mapping of the proximal femur to identify exercise loading effects
An illustrated description of the parametrising procedure developed for (a) proximal femur triangular surface meshes M N (V, E, F); where V = set of nodes; E = set of edges; F = set of faces (N = 111, in this study). (b) In the first parametrisation step, the surface is conformally mapped to its topological equivalent: disk. The single boundary (∂1M) at the distal end of the proximal femur (shaft) is mapped to the edge of the disk under a free boundary condition, where the metric on the boundary nodes is left unchanged (colour map: conformal factor). (c) The parametrised disk along with the conformal factor at the nodes as a height map. The femoral head (FH) and greater trochanter (GT) features are detected as the peaks (inset). The straight line between these features is used to introduce a second boundary (∂2M) by slitting the mesh along the line. (d) In the first parametrisation step, the surface is conformally mapped to its topological equivalent: annulus. The map is embedded in the complex plane by introducing a cut graph between the GT node and ∂1M. The embedded meshes are then transformed such that the ∂2M boundary lies on the imaginary axis scaled within [0, 2π]. An exponential map consequently results in the annulus. (e) Parametrised meshes in the a common coordinate frame. The boundary edges and feature points are colour coded consistently across all images.
