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

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

Introduction Constraint-Based Modeling

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.

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.

Applications of Physics-based Modeling Flow Simulation

Applications of Physics-based Modeling Various Natural Phenomena

Applications of Physics-based Modeling Fluid Simulation

Applications of Physics-based Modeling Motion Animation/Synthesis

Applications of Physics-based Modeling Virtual Surgery

Applications of Physics-based Modeling Deformation Simulation

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

A Bit of Mathematics…. Quaternion Calculus …

Quaternion Basics of Quaternion Application to Rotation

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.

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.

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.

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

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].

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 .

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

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…

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]

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

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.

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.

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.

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

Interpolation of Solid Orientations

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.

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.

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.

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

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)

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.

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

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

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

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.

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.

Kinematics The velocity of a point on the model

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

Dynamics Lagrange Equations of Motion

Dynamics Kinetic Energy: Mass Matrix

Dynamics Acceleration and Inertial Forces

Dynamics Acceleration and Inertial Forces

Dynamics Damping Matrix: Energy Dissipation

Dynamics Stiffness Matrix: Strain Energy

Dynamics Stiffness Matrix: Strain Energy

Dynamics External Forces