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<!DOCTYPE HTML>
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<title>Filter Line Search for Non-Inversion - Physics-Based Simulation</title>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="preface.html">Preface</a></li><li class="chapter-item expanded affix "><li class="part-title">Simulation with Optimization</li><li class="chapter-item expanded "><a href="lec1-discrete_space_time.html"><strong aria-hidden="true">1.</strong> Discrete Space and Time</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec1.1-solid_rep.html"><strong aria-hidden="true">1.1.</strong> Representations of a Solid Geometry</a></li><li class="chapter-item expanded "><a href="lec1.2-newton_2nd_law.html"><strong aria-hidden="true">1.2.</strong> Newton's Second Law</a></li><li class="chapter-item expanded "><a href="lec1.3-time_integration.html"><strong aria-hidden="true">1.3.</strong> Time Integration</a></li><li class="chapter-item expanded "><a href="lec1.4-explicit_time_integration.html"><strong aria-hidden="true">1.4.</strong> Explicit Time Integration</a></li><li class="chapter-item expanded "><a href="lec1.5-implicit_time_integration.html"><strong aria-hidden="true">1.5.</strong> Implicit Time integration</a></li><li class="chapter-item expanded "><a href="lec1.6-summary.html"><strong aria-hidden="true">1.6.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec2-opt_framework.html"><strong aria-hidden="true">2.</strong> Optimization Framework</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec2.1-opt_time_integration.html"><strong aria-hidden="true">2.1.</strong> Optimization Time Integrator</a></li><li class="chapter-item expanded "><a href="lec2.2-dirichlet_BC.html"><strong aria-hidden="true">2.2.</strong> Dirichlet Boundary Conditions</a></li><li class="chapter-item expanded "><a href="lec2.3-contact.html"><strong aria-hidden="true">2.3.</strong> Contact</a></li><li class="chapter-item expanded "><a href="lec2.4-friction.html"><strong aria-hidden="true">2.4.</strong> Friction</a></li><li class="chapter-item expanded "><a href="lec2.5-summary.html"><strong aria-hidden="true">2.5.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec3-projected_Newton.html"><strong aria-hidden="true">3.</strong> Projected Newton</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec3.1-conv_issue_Newton.html"><strong aria-hidden="true">3.1.</strong> Convergence of Newton's Method</a></li><li class="chapter-item expanded "><a href="lec3.2-line_search.html"><strong aria-hidden="true">3.2.</strong> Line Search</a></li><li class="chapter-item expanded "><a href="lec3.3-grad_based_opt.html"><strong aria-hidden="true">3.3.</strong> Gradient-Based Optimization</a></li><li class="chapter-item expanded "><a href="lec3.4-summary.html"><strong aria-hidden="true">3.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec4-2d_mass_spring.html"><strong aria-hidden="true">4.</strong> Case Study: 2D Mass-Spring*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec4.1-discretizations.html"><strong aria-hidden="true">4.1.</strong> Spatial and Temporal Discretizations</a></li><li class="chapter-item expanded "><a href="lec4.2-inertia.html"><strong aria-hidden="true">4.2.</strong> Inertia Term</a></li><li class="chapter-item expanded "><a href="lec4.3-mass_spring_energy.html"><strong aria-hidden="true">4.3.</strong> Mass-Spring Potential Energy</a></li><li class="chapter-item expanded "><a href="lec4.4-opt_time_integrator.html"><strong aria-hidden="true">4.4.</strong> Optimization Time Integrator</a></li><li class="chapter-item expanded "><a href="lec4.5-sim_with_vis.html"><strong aria-hidden="true">4.5.</strong> Simulation with Visualization</a></li><li class="chapter-item expanded "><a href="lec4.6-summary.html"><strong aria-hidden="true">4.6.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><li class="part-title">Boundary Treatments</li><li class="chapter-item expanded "><a href="lec5-dirichlet_BC_solve.html"><strong aria-hidden="true">5.</strong> Dirichlet Boundary Conditions*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec5.1-equality_constraints.html"><strong aria-hidden="true">5.1.</strong> Equality Constraint Formulation</a></li><li class="chapter-item expanded "><a href="lec5.2-DOF_elimin.html"><strong aria-hidden="true">5.2.</strong> DOF Elimination Method</a></li><li class="chapter-item expanded "><a href="lec5.3-hanging_square.html"><strong aria-hidden="true">5.3.</strong> Case Study: Hanging Sqaure*</a></li><li class="chapter-item expanded "><a href="lec5.4-summary.html"><strong aria-hidden="true">5.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec6-slip_DBC.html"><strong aria-hidden="true">6.</strong> Slip Dirichlet Boundary Conditions</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec6.1-axis_aligned.html"><strong aria-hidden="true">6.1.</strong> Axis-Aligned Slip DBC</a></li><li class="chapter-item expanded "><a href="lec6.2-change_of_vars.html"><strong aria-hidden="true">6.2.</strong> Change of Variables</a></li><li class="chapter-item expanded "><a href="lec6.3-general_slip_DBC.html"><strong aria-hidden="true">6.3.</strong> General Slip DBC</a></li><li class="chapter-item expanded "><a href="lec6.4-summary.html"><strong aria-hidden="true">6.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec7-dist_barrier.html"><strong aria-hidden="true">7.</strong> Distance Barrier for Nonpenetration</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec7.1-signed_dists.html"><strong aria-hidden="true">7.1.</strong> Signed Distances</a></li><li class="chapter-item expanded "><a href="lec7.2-dist_barrier_formulation.html"><strong aria-hidden="true">7.2.</strong> Distance Barrier</a></li><li class="chapter-item expanded "><a href="lec7.3-sol_accuracy.html"><strong aria-hidden="true">7.3.</strong> Solution Accuracy</a></li><li class="chapter-item expanded "><a href="lec7.4-summary.html"><strong aria-hidden="true">7.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec8-filter_line_search.html"><strong aria-hidden="true">8.</strong> Filter Line Search*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec8.1-tunneling.html"><strong aria-hidden="true">8.1.</strong> Tunneling Issue</a></li><li class="chapter-item expanded "><a href="lec8.2-nonpenetration_traj.html"><strong aria-hidden="true">8.2.</strong> Penetration-free Trajectory</a></li><li class="chapter-item expanded "><a href="lec8.3-square_drop.html"><strong aria-hidden="true">8.3.</strong> Case Study: Square Drop*</a></li><li class="chapter-item expanded "><a href="lec8.4-summary.html"><strong aria-hidden="true">8.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec9-friction.html"><strong aria-hidden="true">9.</strong> Frictional Contact</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec9.1-smooth_fric.html"><strong aria-hidden="true">9.1.</strong> Smooth Dynamic-Static Transition</a></li><li class="chapter-item expanded "><a href="lec9.2-semi_imp_fric.html"><strong aria-hidden="true">9.2.</strong> Semi-Implicit Discretization</a></li><li class="chapter-item expanded "><a href="lec9.3-fixed_point_iter.html"><strong aria-hidden="true">9.3.</strong> Fixed-Point Iteration</a></li><li class="chapter-item expanded "><a href="lec9.4-summary.html"><strong aria-hidden="true">9.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec10-square_on_slope.html"><strong aria-hidden="true">10.</strong> Case Study: Square On Slope*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec10.1-ground_to_slope.html"><strong aria-hidden="true">10.1.</strong> From Ground To Slope</a></li><li class="chapter-item expanded "><a href="lec10.2-slope_fric.html"><strong aria-hidden="true">10.2.</strong> Slope Friction</a></li><li class="chapter-item expanded "><a href="lec10.3-summary.html"><strong aria-hidden="true">10.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec11-mov_DBC.html"><strong aria-hidden="true">11.</strong> Moving Boundary Conditions*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec11.1-penalty_method.html"><strong aria-hidden="true">11.1.</strong> Penalty Method</a></li><li class="chapter-item expanded "><a href="lec11.2-compress_square.html"><strong aria-hidden="true">11.2.</strong> Case Study: Compressing Square*</a></li><li class="chapter-item expanded "><a href="lec11.3-summary.html"><strong aria-hidden="true">11.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><li class="part-title">Hyperelasticity</li><li class="chapter-item expanded "><a href="lec12-kinematics.html"><strong aria-hidden="true">12.</strong> Kinematics Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec12.1-continuum_motion.html"><strong aria-hidden="true">12.1.</strong> Continuum Motion</a></li><li class="chapter-item expanded "><a href="lec12.2-deformation.html"><strong aria-hidden="true">12.2.</strong> Deformation</a></li><li class="chapter-item expanded "><a href="lec12.3-summary.html"><strong aria-hidden="true">12.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec13-strain_energy.html"><strong aria-hidden="true">13.</strong> Strain Energy</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec13.1-rigid_null_rot_inv.html"><strong aria-hidden="true">13.1.</strong> Rigid Null Space and Rotation Invariance</a></li><li class="chapter-item expanded "><a href="lec13.2-polar_svd.html"><strong aria-hidden="true">13.2.</strong> Polar Singular Value Decomposition</a></li><li class="chapter-item expanded "><a href="lec13.3-simp_model_inversion.html"><strong aria-hidden="true">13.3.</strong> Simplified Models and Invertibility</a></li><li class="chapter-item expanded "><a href="lec13.4-summary.html"><strong aria-hidden="true">13.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec14-stress_and_derivatives.html"><strong aria-hidden="true">14.</strong> Stress and Its Derivatives</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec14.1-stress.html"><strong aria-hidden="true">14.1.</strong> Stress</a></li><li class="chapter-item expanded "><a href="lec14.2-compute_P.html"><strong aria-hidden="true">14.2.</strong> Computing Stress</a></li><li class="chapter-item expanded "><a href="lec14.3-compute_stress_deriv.html"><strong aria-hidden="true">14.3.</strong> Computing Stress Derivatives</a></li><li class="chapter-item expanded "><a href="lec14.4-summary.html"><strong aria-hidden="true">14.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec15-inv_free_elasticity.html"><strong aria-hidden="true">15.</strong> Case Study: Inversion-free Elasticity*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec15.1-linear_tri_elem.html"><strong aria-hidden="true">15.1.</strong> Linear Triangle Elements</a></li><li class="chapter-item expanded "><a href="lec15.2-energy_grad_hess.html"><strong aria-hidden="true">15.2.</strong> Computing Energy, Gradient, and Hessian</a></li><li class="chapter-item expanded "><a href="lec15.3-filter_line_search.html" class="active"><strong aria-hidden="true">15.3.</strong> Filter Line Search for Non-Inversion</a></li><li class="chapter-item expanded "><a href="lec15.4-summary.html"><strong aria-hidden="true">15.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><li class="part-title">Governing Equations</li><li class="chapter-item expanded "><a href="lec16-strong_and_weak_forms.html"><strong aria-hidden="true">16.</strong> Strong and Weak Forms</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec16.1-mass_conserv.html"><strong aria-hidden="true">16.1.</strong> Conservation of Mass</a></li><li class="chapter-item expanded "><a href="lec16.2-momentum_conserv.html"><strong aria-hidden="true">16.2.</strong> Conservation of Momentum</a></li><li class="chapter-item expanded "><a href="lec16.3-weak_form.html"><strong aria-hidden="true">16.3.</strong> Weak Form</a></li><li class="chapter-item expanded "><a href="lec16.4-summary.html"><strong aria-hidden="true">16.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec17-disc_weak_form.html"><strong aria-hidden="true">17.</strong> Discretization of Weak Forms</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec17.1-discrete_space.html"><strong aria-hidden="true">17.1.</strong> Discrete Space</a></li><li class="chapter-item expanded "><a href="lec17.2-discrete_time.html"><strong aria-hidden="true">17.2.</strong> Discrete Time</a></li><li class="chapter-item expanded "><a href="lec17.3-summary.html"><strong aria-hidden="true">17.3.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec18-BC_and_fric.html"><strong aria-hidden="true">18.</strong> Boundary Conditions and Frictional Contact</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec18.1-incorporate_BC.html"><strong aria-hidden="true">18.1.</strong> Incorporating Boundary Conditions</a></li><li class="chapter-item expanded "><a href="lec18.2-normal_contact.html"><strong aria-hidden="true">18.2.</strong> Normal Contact for Nonpenetration</a></li><li class="chapter-item expanded "><a href="lec18.3-barrier_potential.html"><strong aria-hidden="true">18.3.</strong> Barrier Potential</a></li><li class="chapter-item expanded "><a href="lec18.4-friction_force.html"><strong aria-hidden="true">18.4.</strong> Friction Force</a></li><li class="chapter-item expanded "><a href="lec18.5-summary.html"><strong aria-hidden="true">18.5.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><li class="part-title">Finite Element Method</li><li class="chapter-item expanded "><a href="lec19-linear_FEM.html"><strong aria-hidden="true">19.</strong> Linear Finite Elements</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec19.1-linear_disp_field.html"><strong aria-hidden="true">19.1.</strong> Piecewise Linear Displacement Field</a></li><li class="chapter-item expanded "><a href="lec19.2-mass_matrix.html"><strong aria-hidden="true">19.2.</strong> Mass Matrix and Lumping</a></li><li class="chapter-item expanded "><a href="lec19.3-elasticity_term.html"><strong aria-hidden="true">19.3.</strong> Elasticity Term</a></li><li class="chapter-item expanded "><a href="lec19.4-summary.html"><strong aria-hidden="true">19.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec20-pw_linear_boundary.html"><strong aria-hidden="true">20.</strong> Piecewise Linear Boundaries</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec20.1-boundary_conditions.html"><strong aria-hidden="true">20.1.</strong> Boundary Conditions</a></li><li class="chapter-item expanded "><a href="lec20.2-obstacle_contact.html"><strong aria-hidden="true">20.2.</strong> Solid-Obstacle Contact</a></li><li class="chapter-item expanded "><a href="lec20.3-self_contact.html"><strong aria-hidden="true">20.3.</strong> Self-Contact</a></li><li class="chapter-item expanded "><a href="lec20.4-summary.html"><strong aria-hidden="true">20.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec21-2d_self_contact.html"><strong aria-hidden="true">21.</strong> Case Study: 2D Self-Contact*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec21.1-scene_setup.html"><strong aria-hidden="true">21.1.</strong> Scene Setup and Boundary Element Collection</a></li><li class="chapter-item expanded "><a href="lec21.2-point_edge_dist.html"><strong aria-hidden="true">21.2.</strong> Point-Edge Distance</a></li><li class="chapter-item expanded "><a href="lec21.3-barrier_and_derivatives.html"><strong aria-hidden="true">21.3.</strong> Barrier Energy and Its Derivatives</a></li><li class="chapter-item expanded "><a href="lec21.4-ccd.html"><strong aria-hidden="true">21.4.</strong> Continuous Collision Detection</a></li><li class="chapter-item expanded "><a href="lec21.5-summary.html"><strong aria-hidden="true">21.5.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec22-2d_self_fric.html"><strong aria-hidden="true">22.</strong> 2D Frictional Self-Contact*</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec22.1-disc_and_approx.html"><strong aria-hidden="true">22.1.</strong> Discretization and Approximation</a></li><li class="chapter-item expanded "><a href="lec22.2-precompute.html"><strong aria-hidden="true">22.2.</strong> Precomputing Normal and Tangent Information</a></li><li class="chapter-item expanded "><a href="lec22.3-fric_and_derivatives.html"><strong aria-hidden="true">22.3.</strong> Friction Energy and Its Derivatives</a></li><li class="chapter-item expanded "><a href="lec22.4-summary.html"><strong aria-hidden="true">22.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec23-3d_elastodynamics.html"><strong aria-hidden="true">23.</strong> 3D Elastodynamics</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec23.1-kinematics.html"><strong aria-hidden="true">23.1.</strong> Kinematics</a></li><li class="chapter-item expanded "><a href="lec23.2-mass_matrix.html"><strong aria-hidden="true">23.2.</strong> Mass Matrix</a></li><li class="chapter-item expanded "><a href="lec23.3-elasticity.html"><strong aria-hidden="true">23.3.</strong> Elasticity</a></li><li class="chapter-item expanded "><a href="lec23.4-summary.html"><strong aria-hidden="true">23.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="lec24-3d_fric_self_contact.html"><strong aria-hidden="true">24.</strong> 3D Frictional Self-Contact</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="lec24.1-barrier_and_dist.html"><strong aria-hidden="true">24.1.</strong> Barrier and Distances</a></li><li class="chapter-item expanded "><a href="lec24.2-collision_detection.html"><strong aria-hidden="true">24.2.</strong> Collision Detection</a></li><li class="chapter-item expanded "><a href="lec24.3-friction.html"><strong aria-hidden="true">24.3.</strong> Friction</a></li><li class="chapter-item expanded "><a href="lec24.4-summary.html"><strong aria-hidden="true">24.4.</strong> Summary</a></li></ol></li><li class="chapter-item expanded "><a href="bibliography.html">Bibliography</a></li></ol>
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<h1 class="menu-title">Physics-Based Simulation</h1>
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<h2 id="filter-line-search-for-non-inversion"><a class="header" href="#filter-line-search-for-non-inversion">Filter Line Search for Non-Inversion</a></h2>
<p>To guarantee non-inversion just like non-interpenetration (see <a href="lec8-filter_line_search.html">Filter Line Search</a>) during the simulation, we can similarly filter the line search initial step size with a critical step size <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span></span></span></span></span></span></span> that first brings the volume of any triangles to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. This can be obtained by solving a 1D equation per triangle:
<span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1413em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="enclosing" id="eq:lec15:V_eq_0"></span></span><span class="tag"><span class="strut" style="height:1.1413em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">15.3.1</span></span><span class="mord">)</span></span></span></span></span></span>
and taking the minimum of the solved step sizes. Here <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the search direction of node <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>, and in 2D, Equation <a href="#eq:lec15:V_eq_0">(15.3.1)</a> is equivalent to:
<span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">])</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0975em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0975em;vertical-align:-0.3831em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3831em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="enclosing" id="eq:lec15:det_tri_equals_0"></span></span><span class="tag"><span class="strut" style="height:1.1331em;vertical-align:-0.3831em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">15.3.2</span></span><span class="mord">)</span></span></span></span></span></span>
with <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0592em;vertical-align:-0.3948em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3948em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1274em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7305em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7305em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7305em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7305em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>. Expanding Equation <a href="#eq:lec15:det_tri_equals_0">(15.3.2)</a> we obtain:
<span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1774em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1774em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1774em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span><span class="mpunct mtight">,</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1774em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span></span></span></span></span>
which can be reorganized as a quadratic equation of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span></span></span></span></span></span></span>:
<span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1413em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">])</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">])</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1413em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]))</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">])</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.</span></span></span></span></span>
Here, note that <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">])</span></span></span></span> can be very tiny when the nodes do not move much or when their movement barely changes to triangle area in the current timestep, thus the equation can be degenerated into a linear one. To robustly detect this degenerate case, we cannot directly check whether <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">])</span></span></span></span> is <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> due to numerical errors. In fact, checking whether <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">])</span></span></span></span> is below an epsilon is still tricky, because the scale of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">])</span></span></span></span> heavily depends on the scene dimension and nodal velocity during the simulation. Therefore, we use <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">])</span></span></span></span> as a scaling and obtain a scaled but equivalent equation:
<span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">])</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">])</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">])</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">])</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop">det</span><span class="mopen">([</span><span class="mord"><span class="mord mathbf">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">21</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">31</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">])</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="enclosing" id="eq:lec15:scaled_quad_eq_of_alphaI"></span></span><span class="tag"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">15.3.3</span></span><span class="mord">)</span></span></span></span></span></span>
where magnitude checks can be safely performed on any coefficients with unitless thresholds.</p>
<p>In practice, we also need to allow some slackness so that the step size to be taken will not lead to an exactly <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> volume. Thus, we solve <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">I</span></span></span></span></span></span></span></span></span></span></span> such that it first decreases the volume of any triangles by <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">90%</span></span></span></span>, which can be realized by modifying the constant term coefficient in Equation <a href="#eq:lec15:scaled_quad_eq_of_alphaI">(15.3.3)</a> from <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.9</span></span></span></span>:</p>
<p><a name="imp:lec15:filter_line_search"></a>
<strong>Implementation 15.3.1 (Filter line search, NeoHookeanEnergy.py).</strong></p>
<pre><code class="language-python">def init_step_size(x, e, p):
alpha = 1
for i in range(0, len(e)):
x21 = x[e[i][1]] - x[e[i][0]]
x31 = x[e[i][2]] - x[e[i][0]]
p21 = p[e[i][1]] - p[e[i][0]]
p31 = p[e[i][2]] - p[e[i][0]]
detT = np.linalg.det(np.transpose([x21, x31]))
a = np.linalg.det(np.transpose([p21, p31])) / detT
b = (np.linalg.det(np.transpose([x21, p31])) + np.linalg.det(np.transpose([p21, x31]))) / detT
c = 0.9 # solve for alpha that first brings the new volume to 0.1x the old volume for slackness
critical_alpha = utils.smallest_positive_real_root_quad(a, b, c)
if critical_alpha > 0:
alpha = min(alpha, critical_alpha)
return alpha
</code></pre>
<p>Here, if the equation does not have a positive real root, that means for this specific triangle, the step size can be taken arbitrarily large and it will not trigger inversion.</p>
<p>The quadratic equation can be solved as</p>
<p><a name="imp:lec15:find_positive_real_root"></a>
<strong>Implementation 15.3.2 (Solve quadratic equation, utils.py).</strong></p>
<pre><code class="language-python">def smallest_positive_real_root_quad(a, b, c, tol = 1e-6):
# return negative value if no positive real root is found
t = 0
if abs(a) <= tol:
if abs(b) <= tol: # f(x) = c > 0 for all x
t = -1
else:
t = -c / b
else:
desc = b * b - 4 * a * c
if desc > 0:
t = (-b - math.sqrt(desc)) / (2 * a)
if t < 0:
t = (-b + math.sqrt(desc)) / (2 * a)
else: # desv<0 ==> imag, f(x) > 0 for all x > 0
t = -1
return t
</code></pre>
<p>With scaled coefficients, we simply use a unitless threshold, e.g. \code{1e-6}, to check for degeneracies. If no positive real roots are found, the function simply returns <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>.</p>
<p>Now as we filter the initial step size in addition to non-interpenetration:</p>
<p><a name="imp:lec15:apply_filter"></a>
<strong>Implementation 15.3.3 (Apply filter, time_integrator.py).</strong></p>
<pre><code class="language-python"> alpha = min(BarrierEnergy.init_step_size(x, n, o, p), NeoHookeanEnergy.init_step_size(x, e, p)) # avoid interpenetration, tunneling, and inversion
</code></pre>
<p>and make sure all added data structures and modified functions are reflected in the time integrator, we can finally simulate the compressing square example from <a href="lec11-mov_DBC.html">Moving Boundary Condition</a> with guaranteed non-inversion (see <a href="#fig:lec15:compress_square_inv_free">Figure 15.3.1</a>).</p>
<figure>
<center>
<img src="img/lec15/inv_free_compress_square.jpg">
</center>
<figcaption><b><a name="fig:lec15:compress_square_inv_free"></a>
Figure 15.3.1.</b> A square is dropped onto the ground and compressed severely by a ceiling while maintaining inversion-free throughout the simulation. The ground has friction coefficient <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.11</span></span></span></span> so that the bottom of the square slides less than the top, where the ceiling has no friction. </figcaption>
</figure>
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