1. Physics Based Modeling
Lecture 1
Kwang Hee Ko
Gwangju Institute of Science and Technology

2. Introduction What is “Physics-based Modeling”???
The behavior and form of many objects are determined by the objects’ gross physical property.
This modeling technique uses “physics properties” to determine the shape and motions of objects.
Constraint-Based Modeling
Constraints are those which the parts of a model are supposed to satisfy.
Example 1: A model for human skeletons
Constraints on connectivity of bones, limits of angular motion on joints, etc.
Example 2: A sphere on a table

3. Introduction
Constraint-Based Modeling

4. Motivations Natural phenomena are characterized by physical laws.
Deformation is ubiquitous ranging from micro-scale to nano-scale.
Graphics/animation aims to model and simulate physical worlds.
Every component of graphics is relevant to physical laws.
Physics-based modeling gives rise to a large variety of applications in graphics, geometric design, visualization, simulation, etc.
Physics-based modeling has been a very powerful tool to tackle many real problems.

5. Applications of Physics-based Modeling
Geometric modeling using physics and energy
Interactive and dynamic editing
Geometric processing
Virtual surgery simulation
Haptic interface
Realistic rendering of natural phenomena
Fire, wave, etc.

6. Applications of Physics-based Modeling
Flow Simulation

7. Applications of Physics-based Modeling
Various Natural Phenomena

8. Applications of Physics-based Modeling
Fluid Simulation

9. Applications of Physics-based Modeling
Motion Animation/Synthesis

10. Applications of Physics-based Modeling
Virtual Surgery

11. Applications of Physics-based Modeling
Deformation Simulation

12. What we are going to study…
We will be studying concepts on
Physics on rigid and deformable bodies.
Physics on animation

13. A Bit of Mathematics….
Quaternion
Calculus …

14. Quaternion
Basics of Quaternion
Application to Rotation

15. Geometric Transformations
Rotation is defined by an axis and an angle of rotation.
Rotation in 3D is not as simple as translation.
It can be defined in many ways.

16. Quaternions
The second rotational modality is rotation defined by Euler’s theorem and implemented with quaternions.
Euler’s rotational theorem
An arbitrary rotation may be described by only three parameters.

17. Historical Backgrounds
Quaternions were invented by Sir William Rowan Hamilton in 1843.
His aim was to generalize complex numbers to three dimensions.
Numbers of the form a+ib+jc, where a,b,c are real numbers and i2=j2=-1.
He never succeeded in making this generalization.
It has later been proven that the set of three-dimensional numbers is not closed under multiplication.
Four numbers are needed to describe a rotation followed by a scaling.
One number describes the size of the scaling.
One number describes the number of degrees to be rotated.
Two numbers give the plane in which the vector should be rotated.

18. Basic Quaternion Mathematics
Quaternions, denoted q, consist of a scalar part s and a vector part v=(x,y,z). We will use the following form.
Let i2=j2=k2=ijk=-1, ij=k and ji=-k.
A quaternion q can be written:
q = [s,v] = [s,(x,y,z)] = s+ix+jy+kz.
The addition operator, +, is defined

19. Basic Quaternion Mathematics
Multiplication is defined:
Quaternion multiplication is not generally commutative.
Multiplication by a scalar is defined by
rq ≡ [r,0]q
Subtraction is defined
q – q’ ≡ q + (-1)q’
Let q be a quaternion. Then q* is called the conjugate of q and is defined by q* ≡ [s,v]* ≡ [s, -v].

20. Basic Quaternion Mathematics
Let p,q be quaternions. Then
(q*)* = q, (pq)* = q*p*, (p+q)* = p* + q*, qq* = q*q
The norm of a quaternion q.
||q|| = √qq*
The inner product is defined
q·q’ = ss’+v·v’ = ss’ + xx’ + yy’ + zz’
Let q,q’ be quaternions. Define them as the corresponding four-dimensional vectors and let α be the angle between them.
q·q’ = ||q|| ||q’|| cos α .

21. Basic Quaternion Mathematics
The unique neutral element under quaternion multiplication
I = [1,0]
Inverse under quaternion multiplication
qq-1=q-1q=I.
q-1=q*/||q||2

22. Basic Quaternion Mathematics
Unit quaternions
If ||q|| = 1, then q is called a unit quaternion.
Use H1 to denote the set of unit quaternions
Let q = [s,v], a unit quaternion. Then, there exists v’ and θ such that q = [cosθ ,v’sinθ ].
Let q, q’ be unit quaternions. Then
||qq’|| = 1
q-1 = q*
Etc…

23. Rotation with Quaternions
Let q=[cosθ,nsinθ] be a unit quaternion. Let r = (x,y,z) and p[0,r] be a quaternion. Then
p’= qpq-1 is p rotated 2θ about the axis n.
Any general three-dimensional rotation about n, |n|=1 can be obtained by a unit quaternion.
Choose q such that q=[cosθ/2,nsinθ/2]

24. Rotation with Quaternions
Let q1, q2 be unit quaternions. Rotation by q1 followed by rotation by q2 is equivalent to rotation by q2q1.
Geometric intuition

25. Comparison of Quaternions, Euler Angles and Matrices
Euler Angles/Matrices – Disadvantages
Lack of intuition
The order of rotation axes is important.
Gimbal lock
It is a concept originating from the air and space industry, where gyroscopes are used.
At a certain situation, two rotations act about the same axis.
Mathematically gimbal lock corresponds to loosing a degree of freedom in the general rotation matrix.

26. Comparison of Quaternions, Euler Angles and Matrices
Euler Angles/Matrices – Disadvantages
Gimbal lock
If we letβ=π/2, then a rotation with αwill have the same effect as applying the same rotation with -γ.
The rotation only depends on the difference and therefore it has only one degree of freedom. For β=π/2 changes of α and γ result in rotations about the same axis.

27. Comparison of Quaternions, Euler Angles and Matrices
Euler Angles/Matrices – Disadvantages
Implementing interpolation is difficult
Ambiguous correspondence to rotations
The result of composition is not apparent
The representation is redundant
Euler Angles/Matrices – Advantages
The mathematics is well-known and that matrix applications are relatively easy to implement.

28. Comparison of Quaternions, Euler Angles and Matrices
Quaternions – Disadvantages
Quaternions only represent rotation
Quaternion mathematics appears complicated
Quaternions – Advantages
Obvious geometrical interpretation
Coordinate system independency
Simple interpolation methods
Compact representation
No gimbal lock
Simple composition

29. Interpolation of Solid Orientations

30. Rigid Body (Single Particle) Newtonian Mechanics (Single Particle)
Newton’s Laws
A body remains at rest or in uniform motion unless acted upon by a force.
Force equilibrium
A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force.
P = mv. F = dP / dt = d(mv)/dt
If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

31. Rigid Body (Single Particle) Newtonian Mechanics (Single Particle)
Inertial Frame
The laws of motion to have meaning, the motion of bodies must be measured relative to some reference frame.
Equation of Motion for a Particle.
F = d(mv)/dt = m dv/dt = mr’’
Example
Projectile motion in two dimensions.

32. Rigid Body (Single Particle) Newtonian Mechanics (Single Particle)
Question
No air resistance.
Muzzle velocity of the projectile: v0.
Angle of elevation Θ.
Calculate the projectile’s displacement, velocity and range.

33. Rigid Body (Single Particle) Newtonian Mechanics (Single Particle)
Solution.

34. Rigid Body (Single Particle) Newtonian Mechanics (Single Particle)
Conservation Theorems
The total linear momentum P of a particle is conserved when the total force on it is zero.
P·s = constant
The angular momentum of a particle subject to no torque is conserved.
L = r X P.
The total energy E of a particle in a conservative force field is a constant in time.
E = T(kinetic) + U(potential)

35. Rigid Body (Single Particle) Newtonian Mechanics (Single Particle)
Example 2: A cylinder on a curved surface
A cylinder of mass m and radius R1 rolling without slippage on a curved surface of radius R.

36. Rigid Body (Single Particle) Newtonian Mechanics (Single Particle)
Solution
dE/dt = dT/dt + dU/dt = 0

37. Geometry of Deformable Models
The models are 3D solids in space.

38. Global Deformations The reference shape s is defined as
T: global deformation
e: geometric primitive defined parametrically in u and parametrized by the variables ai.
The vector of global deformation parameters

39. Local Deformations
The displacement d anywhere within a deformable model is represented as a linear combination of an infinite number of basis functions bj(u)
The diagonal matrix Si is formed form the basis functions and where qd,I are local degrees of freedom.

40. Kinematics and Dynamics
Kinematic and dynamic formulation of the deformable models
Kinematic formulation: the computation of a Jacobian matrix L
It allows the transformation of 3D vectors into q-dimensional vectors.
Dynamic formulation: based on Lagrangian dynamics and generalized coordinates.

41. Kinematics
The velocity of a point on the model

42. Kinematics Computation of R and B using Quaternions
The dual matrix of the position vector p(u) = (p1,p2,p3)T

43. Dynamics
Lagrange Equations of Motion

44. Dynamics
Kinetic Energy: Mass Matrix

45. Dynamics
Acceleration and Inertial Forces

46. Dynamics
Acceleration and Inertial Forces

47. Dynamics
Damping Matrix: Energy Dissipation

48. Dynamics
Stiffness Matrix: Strain Energy

49. Dynamics
Stiffness Matrix: Strain Energy

50. Dynamics
External Forces