Advanced Computer Graphics Spring 2014

K. H. Ko

School of MechatronicsGwangju Institute of Science and Technology

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Today’s Topics

Quaternions

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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.

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Two Rotational Modalities

Definition of Rotational ModalityThe combination of a definition and a corresponding

mathematical representation

Two popular modalitiesRotation defined by Euler angles represented by general

transformation matrices.Rotation defined by Euler’s theorem represented by

quaternions.

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Euler Angles

The space of orientations can be parameterized by Euler angles. A general orientation is written as a series of rotations about

three mutually orthogonal axes in space. Usually the x, y, and z axes in a Cartesian coordinate system are used.

To describe a general rotation, three Euler angles are required, each of which is the rotation angles about the x, y, and z axes, respectively.

The conversion from a general rotation to Euler angles is ambiguous. The same rotation can be obtained with different sets of Euler angles. The resulting rotation depends on the order in which the three rolls are

performed.

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Rotation Matrices

Rotation matrices are the typical choice for implementing Euler angles. For each type of roll, there is a corresponding rotation matrix. The matrices rotate by multiplying them to the position vector for a point in

space. The result is the position vector for the rotated point.

A rotation matrix is a 3x3 matrix but usually homogeneous 4x4 matrices are used in computer graphics.

Matrix multiplication is not generally commutative. It fits well with the fact that rotation in space do not commute.

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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.

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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.

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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

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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 is a quaternion. Then q* is called the conjugate of q and is defined by q* ≡ [s,v]* ≡ [s, -v].

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Basic Quaternion Mathematics

Let p,q 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’ quaternions. Define them as the corresponding four-dimensional vectors and let α be the angle between them. q·q’ = ||q|| ||q’|| cos α .

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Basic Quaternion Mathematics

The unique neutral element under quaternion multiplicationI = [1,0]

Inverse under quaternion multiplicationqq-1=q-1q=I.q-1=q*/||q||2

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Basic Quaternion Mathematics

Unit quaternionsIf ||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…

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Rotation with Quaternions

Let q=[cosθ,nsinθ] be a unit quaternion. Let r = (x,y,z) and p[0,r] be a quaternion. Thenp’= 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]

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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

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Comparison of Quaternions, Euler Angles and Matrices Euler Angles/Matrices – Disadvantages

Lack of intuitionThe 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.

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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.

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Comparison of Quaternions, Euler Angles and Matrices Euler Angles/Matrices – Disadvantages

Implementing interpolation is difficultAmbiguous correspondence to rotationsThe result of composition is not apparentThe representation is redundant

Euler Angles/Matrices – AdvantagesThe mathematics is well-known and that matrix

applications are relatively easy to implement.

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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

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Example Computation of Quaternion Rotation Question: Rotate a vector (1,0,0) by -90o

about (0,1,0). Solution??

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Rotational Matrix vs. Unit Quaternion The unit quaternion qR = [q0, q1, q2, q3]t represents

rotation around the axis (q1,q2,q3).

The rotation matrix R from the unit quaternion is given by

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Interpolation of Solid Orientations An application that benefits from a

quaternion representation of rotation is keyframe animation. The rotational keyframes must be interpolated

to produce reasonable in-between rotations. The quaternion representing the rotations can

themselves be interpolated in a natural manner.

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Interpolation of Solid Orientations Visualization of interpolation curves

Quaternion space is four-dimensional. Direct visualization is impossible.

We always interpolate between unit quaternions. The interpolated quaternions will always be unit quaternions.

We only need three dimensions to visualize the interpolated curves since they lie on the surface of the unit sphere.

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Interpolation of Solid Orientations Interpolation between two rotations

Given an arbitrary set M we interpolate between x0∈M and x1∈M parameterized by h∈[0,1]. The resulting interpolation curve γ : MⅹMⅹ [0,1] -> M with γ(x0,x1,0)=x0, γ(x0,x1,1)=x1.

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Interpolation of Solid Orientations Linear Euler Interpolations

Linear interpolation between two tuples of Euler angles between v0=(x0,y0,z0) and v1=(x1,y1,z1).

LinEuler(v0,v1,h) = v0(1-h)+v1h.

Not optimal and intuitively correct!!!

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Interpolation of Solid Orientations Linear Matrix Interpolations

Linear interpolation between rotation matrices – Linear interpolation of every single matrix element independently of the others.

LinMat(M0,M1,h) = M0(1-h) + M1h.

In general this interpolation method does not produce orthonormal matrices -> Involve translation, scaling,

projections etc. -> The interpolation can become wrong.

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Interpolation of Solid Orientations Linear Quaternion Interpolations

Linear interpolation between rotation quaternions.

Lerp(q0,q1,h) = q0(1-h)+q1h. The interpolation curve for linear interpolation gives a

straight line in quaternion space.

The angular velocity profile is not intuitively correct. Interpolated quaternions may not be unit quaternions.

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Interpolation of Solid Orientations

Spherical Linear Quaternion Interpolations The same interpolation curve as Lerp method

but the interpolation quaternions are unit quaternions.

Great arc interpolation on the quaternion unit sphere.

)sin(

)sin())1sin((),,(

)cos(

1010

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hqhq

hqqSlerp

qq

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Interpolation of Solid Orientations

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3D Registration

Problem Statement Input: Two corresponded point sets, pi, xi, where i =

1,…,Np, and Np = Nx.

Output: The optimal 3D rotation R and translation vector T so that

is minimized.

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3D Registration The centers of mass

The cross-covariance matrix of both points

ij

TpxpxijAAAA ,123123

4×4 matrix

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3D Registration The unit eigenvector (quaternion) qR = [q0, q1, q2, q3]t

corresponding to the maximum eigenvalue of the matrix Q(Σpx) is selected as the optimal rotation.

The optimal translation vector is given by

Rotation matrix R from a unit quaternion

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3D Registration

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3D Registration

Before registration After registration

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Virtual Trackball

It allows the user to define 3D rotation using mouse in 2D windows. It works similarly like the hardware trackball devices

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Virtual Trackball

Superimpose a hemi-sphere onto the viewport.

This hemi-sphere is projected to a

circle inscribed to the viewport.

The mouse position in 2D space is projected orthographically to this hemi-sphere.

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Virtual Trackball Keep track the previous mouse position and

the current position. Calculate their projection positions P1 and P2

to the virtual hemi-sphere. We then rotate the sphere from P1 to P2 by

finding the proper rotation axis and angle. This rotation is then applied to the object.

You should also remember to accumulate the current rotation to the previous modelview matrix.

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Virtual Trackball

The axis of rotation is given by the normal to the plane determined by the origin, p1 and p2.

The angle between p1

and p2 is given by

n = p1 p2

| sin | = ||||

||

21 pp

n

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Virtual Trackball

How to calculate P1 and P2? Assuming the mouse position is (x,z), then the sphere point P also has x and

z coordinates equal to x and z. Assume the radius of the hemi-sphere is 1. So the y coordinate of P is

If a point is outside the circle, project it to the nearest point on the circle.

(set y to 0 and renormalize (x,z))

22 zx1 y

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Virtual Trackball

Visualization of the algorithm