1. An Efficient Brush Model for Physically- Based 3D Painting Nelson S.-H. Chu Chiew-Lan Tai

2. Benefits Over Existing Models Brush flattening Brush flattening Bristle spreading Bristle spreading Plasticity of wetted brushes Plasticity of wetted brushes Resistance of paper surface Resistance of paper surface

3. Block Diagram Artists use a brush connected to several sensors to paint Artists use a brush connected to several sensors to paint The movements of the brush are interpreted by the brush, ink, and paper models The movements of the brush are interpreted by the brush, ink, and paper models The brush and strokes are the rendered on the screen The brush and strokes are the rendered on the screen

4. User Interface Sensors attached to real brush to provide input Sensors attached to real brush to provide input Ultrasound receivers and buzzer provide 6 degrees of freedom Ultrasound receivers and buzzer provide 6 degrees of freedom Gyroscopes provide brush orientation information Gyroscopes provide brush orientation information

5. Actual Brush Made of animal hairs Made of animal hairs Kernel is hard, outer layers are soft Kernel is hard, outer layers are soft Good brushes are elastic Good brushes are elastic Tip is less stiff than the root Tip is less stiff than the root

6. Brush Skeleton Brush is modeled by line segments Brush is modeled by line segments Spine nodes are made up of consecutively shorter line segments Spine nodes are made up of consecutively shorter line segments This makes the brush more flexible at the tip This makes the brush more flexible at the tip Lateral nodes are attached to each spine node Lateral nodes are attached to each spine node Connection points are modeled by bend springs Connection points are modeled by bend springs Lateral line segments are modeled by stretch springs Lateral line segments are modeled by stretch springs Columns of spine nodes and lateral nodes represent groups of bristles Columns of spine nodes and lateral nodes represent groups of bristles

7. Tuft Cross Section Spine nodes are represented at cylindrical groups of bristles Spine nodes are represented at cylindrical groups of bristles Lateral nodes are elliptical groups of bristles Lateral nodes are elliptical groups of bristles Entire tuft is modeled with ellipses at each spinal node Entire tuft is modeled with ellipses at each spinal node

8. Brush Surface Lateral nodes allow the simulation of bristle spreading Lateral nodes allow the simulation of bristle spreading An alpha map is used for fine bristle splitting effects An alpha map is used for fine bristle splitting effects Currently a static alpha map is used Currently a static alpha map is used

9. Brush Dynamics Exact Newtonian physics is not practical Exact Newtonian physics is not practical Instead, energy minimization is used Instead, energy minimization is used An energy function is set up for the system and its steady state is determined by finding a local energy minimum numerically. An energy function is set up for the system and its steady state is determined by finding a local energy minimum numerically. The state of the system at the previous time step is used as the new initial value The state of the system at the previous time step is used as the new initial value This is a ‘static contained minimization problem’ This is a ‘static contained minimization problem’ Sequential Quadratic Programming (SQP) is used to solve the energy minimization problem Sequential Quadratic Programming (SQP) is used to solve the energy minimization problem

10. Energy Minimization Initialize (θ f, φ f ) = (θ i, φ i ) Determine if any nodes penetrate the paper Set minimization constraints for those nodes Solve energy minimization problem for the state of the system and update accordingly

11. Energy Functions E = Edeform + Efrict Edeform has a spine and lateral component, each of which have bend and stretch components BendEnergy( θ)=kbend|θ|^3 Estretch(d, θ)=kstretch|d-(r+s(θ))|^3 d=distance from node, r=radius of node S is a function of the bend angle Efrict= u Sum(F*(kf|Xpar|+(1-kf)|Xperp)) F=normal force, kf=weighting value Xpar/perp=distance moved parallel and perpendicular

12. Plasticity Brushes have different ‘plasticities’ Brushes have different ‘plasticities’ Type of brush Type of brush Wetness of brush Wetness of brush Plasticity is the amount by which the brush returns to its original shape Plasticity is the amount by which the brush returns to its original shape BendEnergy( θ)=k|θ-p|^3 where: p=min(θ’, alpha) Alpha changes according to wetness of brush, larger alpha corresponds to more plasticity

13. Ink Depositing Ink and moisture information is stored at each node Ink and moisture information is stored at each node The brush footprint is the orthogonal projection of the penetrating portion of the brush onto the paper The brush footprint is the orthogonal projection of the penetrating portion of the brush onto the paper Ink is either subtracted from the tuft upon depositing, or maintained to allow continuous painting Ink is either subtracted from the tuft upon depositing, or maintained to allow continuous painting If the ink is subtracted the tuft alpha map is modified too reflect this If the ink is subtracted the tuft alpha map is modified too reflect this

14. Specs System written in Object Pascal using Borland Delphi 6 System written in Object Pascal using Borland Delphi 6 Runs real-time at 25 frames per second on a 1GHz Pentium-III with a GeForce2 Pro graphics card Runs real-time at 25 frames per second on a 1GHz Pentium-III with a GeForce2 Pro graphics card Most time frames require less then 10 SQP iterations Most time frames require less then 10 SQP iterations

15. Results