Section: New Results

Mechanical rods

Reduced-coordinate models for rods such as the articulated rigid body model or the super-helix model  [50] are able to capture the bending and twisting deformations of thin elastic rods while strictly and robustly avoiding stretching deformations. In this work we are exploring new reduced-coordinate models based on a higher-order geometry. Typically, elements are defined by a polynomial curvature function of the arc length, of degree $d\ge 1$. The main difficulty compared to the super-helix model (where $d=0$) is that the kinematics has no longer a closed form. Last year, in R. Casati's PhD's thesis, we extended this result to the full 3D case. The key idea was to integrate the rod's kinematics using power series expansion, and to design an accurate and efficient computational algorithm adapted to floating point arithmetics. Our method nicely propagates to the computation of the full dynamic of a linked chain of 3d clothoid. This year we thoroughly compared our methods against other rod models from the literature, in terms of both accuracy and computational efficiency. Our results demonstrate that our model is competitive compared to former models, and yields a better trade-off in the case of highly curly rods. All these results were published and presented this year at SIGGRAPH [25] . The source code is also freely distributed under a GPLv.3 license (see Section  5.3 ).

Controlling the input shape of slender structures such as rods is desirable in many design applications (such as hairstyling, reverse engineering, etc.), but solving the corresponding inverse problem is not straightforward. In  [54] , [55] we noted that reduced-coordinates models such as the super-helix are well-suited for static inversion in presence of gravity.

We are facing two main difficulties: 1/ the geometrical fitting of a piecewise helix to an arbitrary input curve and 2/ the inversion a super-helix subject to gravity and contacting forces.

In A. Derouet-Jourdan's PhD's thesis (co-supervised by Joëlle Thollot, EPI Maverick), we solved this problem by extending to 3d the floating tangents algorithm introduced in 2d in  [54] . In this new method, only tangents are strictly interpolated while points are displaced in an optimal way so as to lie in a feasible configuration, i.e., a configuration that is compatible with the interpolation by a helix. Our approach relies upon the co-helicity condition found by Ghosh  [56] , which was however only partially proved in  [56] . To ensure the existence of the helix and prove its uniqueness in the general case, we complete the proof which serves as the basis for our reconstruction algorithm.

Our method proves to be efficient and robust as it can successfully handle large and complex datasets from real curve aquisitions, such as the capture of hair fibers or the magnetic field of a star. We also compared our method against a standard nonlinear least-squares methods. Unlike the optimization approach which often fail to converge in the case of frizzy input curves, our method remains extremely fast regardless the complexity of the input curves. The set of these results was published this year at Computer-Aided Geometric Design [28] . This work has been transferred to L'Oréal in December 2013. Some source code is also freely released for academics under the GPLv.3 license (see Section  5.3 ).

In A. Derouet-Jourdan's PhD's thesis (co-supervised by Joëlle Thollot, EPI Maverick), we bring a first solution to the challenging problem consisting in identifying the intrinsic geometry of a fiber assembly under gravity and (unknown) frictional external and mutual contacts, from a single configuration geometry (a set of geometric curves). Taking an arbitrary fiber assembly geometry (such as hair) as input together with corresponding interacting meshes (such as the body mesh), we interpret the fiber assembly shape as a static equilibrium configuration of a fiber assembly simulator, in the presence of gravity as well as fiber-mesh and fiber-fiber frictional contacts. Assuming fibers parameters are homogeneous and lie in a plausible range of physical values, we show that this large, underdetermined inverse problem can be formulated as a well-posed constrained optimization problem (second-order cone quadratic program), which can be solved robustly and efficiently by leveraging the frictional contact solver of our direct simulator for fiber assemblies [8] . Our method was successfully applied to the animation of various hair geometries, ranging from synthetic hairstyles manually designed by an artist to the most recent human hair data reconstructed from capture. These results were published this year at SIGGRAPH Asia [27] .