International audienceThis paper presents a new physically-based method for predicting natural hairstyles in the presence of gravity and collisions. The method is based upon a mechanically accurate model for static elastic rods (Kirchhoff model), which accounts for the natural curliness of hair, as well as for hair ellipticity. The equilibrium shape is computed in a stable and easy way by energy minimization. This yields various typical hair configurations that can be observed in the real world, such as ringlets. As our results show, the method can generate different hair types with a very few input parameters, and perform virtual hairdressing operations such as wetting, cutting and drying hair

Audoly, Basile

Bertails, Florence

Cani, Marie-Paule

Leroy, Frédéric

Lévêque, Jean-Luc

Querleux, Bernard

Hal - Université Grenoble Alpes

Predicting Natural Hair Shapes by Solving the Statics of Flexible Rods

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HAL Id: inria-00402611https://hal.inria.fr/inria-00402611Submitted on 25 Apr 2012HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Predicting Natural Hair Shapes by Solving the Statics ofFlexible RodsFlorence Bertails, Basile Audoly, Bernard Querleux, Frédéric Leroy, Jean-LucLévêque, Marie-Paule CaniTo cite this version:Florence Bertails, Basile Audoly, Bernard Querleux, Frédéric Leroy, Jean-Luc Lévêque, et al.. Pre-dicting Natural Hair Shapes by Solving the Statics of Flexible Rods. Eurographics Short Papers, Aug2005, Dublin, Ireland. ￿inria-00402611￿EUROGRAPHICS 2005 / J. Dingliana and F. Ganovelli Short PresentationsPredicting Natural Hair Shapesby Solving the Statics of Flexible RodsFlorence Bertails1, Basile Audoly2 , Bernard Querleux3, Frédéric Leroy3, Jean-Luc Lévêque3 and Marie-Paule Cani11 GRAVIR-IMAG/INRIA, Grenoble, France 2 LMM-CNRS/Univ. Paris 6, Paris, France 3 L’Oréal Recherche, FranceFigure 1: Virtual versus real hair ringletsAbstractThis paper presents a new physically-based method for predicting natural hairstyles in the presence of gravityand collisions. The method is based upon a mechanically accurate model for static elastic rods (Kirchhoff model),which accounts for the natural curliness of hair, as well as for hair ellipticity. The equilibrium shape is computed ina stable and easy way by energy minimization. This yields various typical hair configurations that can be observedin the real world, such as ringlets. As our results show, the method can generate different hair types with a veryfew input parameters, and perform virtual hairdressing operations such as wetting, cutting and drying hair.Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Physically based modeling1. Introduction and Previous WorkDue to the relevance of hair in the overall appearance of acharacter, many recent works have focused on the challeng-ing issue of virtual hairstyling. However, there is currentlyno hairstyling method able to account for the structural prop-erties of the different hair types existing in the world - Asian,African and Caucasoid hair - nor to predict how a given hairwould look as it grows or gets wet. Such a model wouldprove very useful for performing virtual hairdressing opera-tions on any given hair type.The geometry of human hair results from complex biolog-ical and mechanical processes. Most CG hairstyling meth-ods reproduce this result using procedural techniques andmanual editing [KN02, CK05]. This gives a high controlla-bility, but also requires some significant input and model-ing skills from the user. Several attempts have been madeto use physically-based modelling in the design of hairstyles[AUK92, HMT00]. In these approaches, the finest geome-tric hair details, such as curls or waves, are still added ina procedural way. Image-based reconstruction techniqueshave recently proved useful in capturing the geometry of realhair [PBS04]. However, as they only model the visible partof hair, these methods can hardly capture the geometry ofcurly hair because of the occlusion issue.Our method is physically-based, and relies on the Kirch-hoff equations for static Cosserat rods. This accurate me-chanical model, which accounts for the curvatures and twistdeformation of elastic rods, was first introduced to the CGcommunity by Pai [Pai02]. Our specific contributions are:1. An improvement of Pai’s model, namely a new formu-lation of the Kirchhoff equations for an elastic rod in thestatic case, based on energy minimization. This allows forhandling external forces such as response to collisions,while still providing efficiency and robustness.2. The extension of Pai’s model to handle elliptic strands.This is mandatory for dealing with hair from differentc The Eurographics Association 2005.Bertails et al. / Predicting Natural Hair Shapes by Solving the Statics of Flexible Rodsethnies, since both the eccentricity1 and the natural curli-ness play a great role in the overall hair shape.3. A very simple handling of basic hairstyling operationssuch as wetting, cutting and drying, since only a few in-tuitive physical parameters need to be changed (length,mean radius and Young modulus). This opens the wayfor the design of future virtual prototyping systems to beused by hairdressers.2. Static Simulation of a Hair StrandBuilding an accurate hair strand model first requires a goodunderstanding of its geometric and mechanical properties.A hair strand is a very thin and light elliptic tube that de-forms in an anisotropic way. Whereas it can easily bend andsometimes twist, it strongly resists stretching and shearing.A hair strand also has elastic properties as it tends to recoverits original shape after the stress applied on it has been re-moved. Lastly, a real hair strand can be naturally straight,wavy or curly.As suggested by Audoly and Pomeau [AP05], we repre-sent a hair strand as a Cosserat rod obeying the Kirchhoffequations. These equations describe elastic rods subject toarbitrary external forces such as gravity.2.1. Cosserat Model for RodsWe define a hair strand as an inextensible, unshearable rod,with a boundary condition on the position and the orientationat the root (imposed by the hair clamping on the scalp). Inthis section, the strand is submitted to the gravity field only.In the Cosserat model, the configuration of a rod is de-scribed by its centerline, a space curve r  s  (where s denotesthe curvilinear abscissa of the rod), and a material frame  s  n1  s  n2  s 	 t  s  attached to each point on thiscurve (see Figure 1, left). Usually, vector t is the local tan-gent of the centerline r  s  , and vectors n1 and n2 are lyingin the plane of the local cross section of the rod. In our case,n1 and n2 are the principal axis of the elliptic cross section.A Cosserat rod can locally bend in both directions n1 andn2, or twist onto itself. The amount of bending around n1 andn2, and the amount of twisting are respectively characterizedby (material) curvatures κ1 and κ2, and torsion τ, through thefollowing kinematics equations:∂ui∂s Ω  ui (1)where Ω  κ1 n1  κ2 n2  τ t is the rotation vector of the rodand ui is n1, n2, t for i  1  2  3 respectively.As previously mentioned, hair can be naturally curly. Thisintrinsic curliness can be expressed by natural curvatures andtorsion κ01, κ02 and τ0, which describe the state of the rod1 We use the mathematical definition of the eccentricity e of an el-lipse. Thus, e  0 for a circle and e increases as the ellipse flattens.with no applied force. In the case of hair, it is reasonable toassume that natural curvatures and torsion are roughly con-stant along the hair shaft. As a result, the configuration of ahair strand in the absence of gravity is a helix (this is easy toshow, knowing that rotation vector Ω is a constant vector).2.2. Potential EnergyWe are looking for stable equilibrium configurations of astrand in the gravity field. One solution would consist ofsolving the Kirchhoff’s equations for static rods [Pai02].However, the nonlinearities in these equations and theboundary conditions that need to be imposed at both the freeand the clamped ends prevent the equations from being inte-grated in a single pass, and one is forced to iterate the inte-gration. Instead, our new approach is based on energy min-imization. It offers the advantage of being both robust andcomputationally cheaper, which is essential when handlinga large number of hair strands. Our method can also accountfor collision forces, as demonstrated in section 3.1.Finding static configurations of a physical system isequivalent to the search for its minimal potential energy. Thepotential energy of a rod of length L formulates as:hair   g   e (2)wheree is the internal elastic energy of the rod andg theenergy of the rod accounting for gravity. Assuming hair tobe a rod of elliptic cross section and obeying Hooke’s lawfor elasticity, the elastic energye can be written as:e  L0  E I12  κ1  s  κ01  2  E I22  κ2  s  κ02  2  µJ2  τ  s  τ0  2  ds (3)where E is the Young modulus, µ the shearing modulus2 , I1(resp. I2) the momentum of inertia of the rod’s cross sectionwith respect to n1 (resp. to n2) and J the axial momentum ofinertia. Momenta of inertia depend on the principal radii ofthe cross section, and thus on hair’s eccentricity e.Potential gravitational energyg can be written as:g  ρSg  L0z  s  ds (4)where ρ is the volumic mass of the rod, S the area of itscross section, g the gravity field value and z  s  the verticalcoordinate of element ds at curvilinear abscissa s.Note that the minimum forhair results from a balance oftwo antagonistic effects: the tendency to recover a naturallyhelical shape, represented bye, and the downward pull ofgravity, represented byg.2.3. Numerical SolvingWe first divide the rod into n slices si of equal length ds, forthe sake of simplicity. Along each (small) slice si, curvaturesand torsion are assumed to be constant. We note Cn the vec-tor of size 3  n composed of the n curvatures κi1, κi2 and2 The shearing modulus of the rod is related to the Young modulusE and to the Poisson ratio σ by the following formula: µ  E2  1  σ c The Eurographics Association 2005.Bertails et al. / Predicting Natural Hair Shapes by Solving the Statics of Flexible Rodstorsions τi. Given the fact that the initial material frameis imposed by the clamping into the scalp, vector Cn definesa unique configuration for the rod.Our aim is first to find the vector Cn that minimizes thediscrete energyhair, and then to compute the correspondingconfiguration of the rod. Our algorithm is as follows: We first initialize energy  hair and vector Cn; Then, until energy  hair stops decreasing, we iterativelyproceed the following steps:1. Compute the elastic energye using Equation (3);2. Calculate formally the configuration (r  s  ,   s  ) ofthe rod, each slice being a helix;3. Compute the potential energyg, which requires theintegration of the kinematic relations (1), in order todetermine the function z  s  . Indeed, z  s  r  s  z under the condition z  0   0, where z is the verticalnormalized axis. Using the relation t  ∂r∂s , we get thefollowing expression forg:g  ρSg  L0 L ! s "#$ t  s "%	 z  ds " (5)which can be accurately evaluated in practice.4. Perform a minimization step forhair   g   e, usinga fast gradient descent approach [FP63]. Knowing the vector Cn that minimizes energy  , we com-pute the final geometric configuration (r,) of the rod.Some typical strand configurations obtained by ourmethod are shown in Figures 1 and 2 (left). Our method isfast enough to handle several hair strands (sampled into 15points) in real-time, and a hundred strands within a few sec-onds. This enables to use the method within an interactivehairstyling system.Figure 2: Left: simulating a curly hair strand with differ-ent eccentricity values; note that increasing the eccentric-ity increases the regularity of the curls along the strand.Right: a curly hair strand growing on a sphere.3. Full Hair ModelingSimilarly to the approach of Choe et al. [CK05], we modelhair as a set of wisps where each wisp is composed of amaster strand and of numerous other strands that are proce-durally generated within a generalized cylinder surroundingthe master strand. The shape of the master strand is physi-cally computed by energy minimization, as explained above,whereas the shape of the other strands is generated using astochastic process similar to the one used in [CK05].3.1. CollisionsFor creating realistic hairstyles, it is necessary to account forboth hair-body collisions and hair self-collisions. The latterare essential for giving an adequate volume to hair.In our model, each wisp is modelled by a skeleton com-posed of the sample points of the master strand, and a thick-ness riw that is computed according to several factors such asthe number of hair strands within the wisp and the level ofhair curliness. The body is approximated by a set of spheresfor collision detection and response. We compute the colli-sion response using elastic penalty forces Fc. The advantageis that such forces derive from a potential energyc. As ourstatic model is based on energy minimization, accountingfor this kind of force simply amounts to minimizing the newenergyhair defined by:hair   g   e   cIn practice,c is computed using the amount x of penetra-tion:c  12 k x2, where k is a stiffness parameter. As illus-trated in Figure 2 (right), this method properly accounts forthe contacts between hair and a sphere.Hair self-collisions are computed using the multiple layerhulls method developed by Lee and Ko [LK01]. Resultsshow that this method is satisfying in the static case, andproperly accounts for the hair volume.3.2. Hairstyling Editing ToolsThis section briefly presents the virtual hairstyling tools thatare provided to the user for creating natural hairstyles. Thegreat advantage of our approach in comparison with exist-ing ones is that usual hairstyling operations such as curling,wetting, cutting or drying hair become straightforward.Hair wetting has a direct impact on the hair shape,as its stiffness coefficient is roughly divided by a factor10 [Rob02]. Moreover, as a hair strand absorbs water, itsradius increases of around 13%. Ward et al. have proposeda hair model accounting for these properties [WGL04]. But,as this model is not characterized by adequate physical pa-rameters, they need to control multiple structures to applythe physical changes to the hair strands. In contrast, withinour accurate physically-based modelling system, the wettingof any hair strand is simply effected by modifying two rele-vant parameters: the Young’s modulus and the mean radius.Additionally, when hair is wetted, the hair volume is proce-durally reduced through the hull layers. We simply processhair drying as the inverse process of hair wetting.Haircutting consists of removing the hair part located un-der a given z position. This is simply done by changing eachc The Eurographics Association 2005.Bertails et al. / Predicting Natural Hair Shapes by Solving the Statics of Flexible Rodshair length lprev into lprev ! lcut where lcut is the length of theremoved part. As a result, hair becomes lighter after cutting,and thus its overall shape is affected, as in the real world.4. Results and Concluding RemarksGenerated hairstyles are rendered using the accurate scatter-ing model of Marschner et al. [MJC & 03]. In particular, thismodel accounts for the visual effect due to the elliptic crosssection of a hair fiber. Thus, eccentricity e represents an uni-fied parameter for both our mechanical model and rendering.This parameter varies according to the hair type: for Asianhair, e is nearly equal to 0, whereas for Caucasoid hair, itranges from 0 to 0.1 and for African hair, it ranges from 0to 0.2. Other hair physical parameters E, µ and ρ are takenequal to their actual value, given by physical measurements:E  10GPa, µ  0 ' 3, ρ  1 ' 3g ( cm3 [Rob02].Figure 1 and 3 show various hairstyles obtained withour method for different hair types and haircuts. Pleasealso visit our website and watch our videos at http://www-evasion.imag.fr/Publications/2005/BAQLLC05/. Note that ourmethod is able to capture realistic natural hair shapes.Figure 3: Right: real pictures of (a) smooth, (b) curly and(c) African hair. Left: corresponding synthetic results, gen-erated with adequate values for e and κ0i . Each synthetichairstyle was created in less than half an hour.The main limitation of our method is the computationaltime required to calculate the equilibrium configuration ofhair (6 seconds in average for the hairstyles illustrated inFigure 3). This prevents us from using more than a hundredmaster strands within our interactive hairstyling software.However, results are very satisfactory with this limited num-ber of master strands, thanks to our procedural wisp model.We are currently working on the extension of our physicalmodel to the dynamic simulation of hair motion.Acknoledgments We would like to thank Steve Marschnerfor letting us use his original rendering code.References[AP05] AUDOLY B., POMEAU Y.: Elasticity and Geometry:from hair curls to the nonlinear response of shells. Ox-ford University Press, To Appear in 2005.[AUK92] ANJYO K., USAMI Y., KURIHARA T.: A simplemethod for extracting the natural beauty of hair. In Pro-ceedings of ACM SIGGRAPH’92 (August 1992), Com-puter Graphics Proceedings, Annual Conference Series,pp. 111–120.[CK05] CHOE B., KO H.-S.: A statiscal wisp model and pseu-dophysical approcahes for interactive hairstyle genera-tion. IEEE Transactions on Visualization and ComputerGraphics 11, 2 (March 2005).[FP63] FLETCHER R., POWELL M. J. D.: A rapidly conver-gent descent method for minimization. The ComputerJournal 6, 2 (1963), 163–168.[HMT00] HADAP S., MAGNENAT-THALMANN N.: Interactivehair styler based on fluid flow. In Computer Animationand Simulation ’00 (Aug. 2000), pp. 87–100.[KN02] KIM T.-Y., NEUMANN U.: Interactive multiresolutionhair modeling and editing. ACM Transactions on Graph-ics 21, 3 (July 2002), 620–629. Proceedings of ACMSIGGRAPH’02.[LK01] LEE D.-W., KO H.-S.: Natural hairstyle modeling andanimation. Graphical Models 63, 2 (March 2001), 67–85.[MJC ) 03] MARSCHNER S., JENSEN H., CAMMARANO M.,WORLEY S., HANRAHAN P.: Light scattering fromhuman hair fibers. ACM Transactions on Graphics(Proceedings of the SIGGRAPH conference) 22, 3 (July2003), 281–290.[Pai02] PAI D.: Strands: Interactive simulation of thin solidsusing cosserat models. Computer Graphics Forum 21, 3(2002), 347–352. Proceedings of Eurographics’02.[PBS04] PARIS S., BRICEÑO H., SILLION F.: Capture of hairgeometry from multiple images. ACM Transactions onGraphics (Proceedings of the SIGGRAPH conference)(2004).[Rob02] ROBBINS C.: Chemical and Physical Behavior of Hu-man Hair. 4th ed. Springer, 2002.[WGL04] WARD K., GALOPPO N., LIN M. C.: Modeling hair in-fluenced by water and styling products. In InternationalConference on Computer Animation and Social Agents(CASA) (May 2004).c The Eurographics Association 2005.

Predicting Natural Hair Shapes by Solving the Statics of Flexible Rods

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