Rotation and Orientation: Fundamentals Questions

• How many DOFs f or 2D rot ati on? - Answer) One

• How many DOFs for 3D rotation? - Answer) Three

• How many DOFs for 4D rotation? - Answer) Six

중앙대학교 첨단영상대학원 박 경 주 What is Rotation?

• No t in tuiti ve – Formal definitions are also confusing

• Many different ways to describe – Rotation (direction cosine) matrix – Euler angles – Axis-angle – Rotation vector – Helical angles – Unit quaternions 중앙대학교 첨단영상대학원 박 경 주 Orientation vs . Rotation

• RttiRotation – Circular movement

• Orientation – The state of being oriented – Given a coordinate system, the orientation of an object can be represented as a rotation from a reference pose

중앙대학교 첨단영상대학원 박 경 주 3D Orientation

• Given t wo arbit rary ori ent ati ons of a ri gid o bjec t,

How many rotations do we need ttto transf orm one ori ent ttiation to the other?

중앙대학교 첨단영상대학원 박 경 주 Joints and Rotations

Rotational DOFs are widely used in character animation 3 translational DOFs 48 rot ati onal DOF s

Each joint can have up to 3 DOFs

1 DOF: knee 2 DOF: wrist 3 DOF: arm 중앙대학교 첨단영상대학원 박 경 주 Composite Transformations

A series of transformations on an object can be applied as a series of matrix multiplications

p: position in the global coordinate x: position in the local coordinate (h3, 0, 0)

중앙대학교 첨단영상대학원 박 경 주 Interpolation

• In ord er to “ move thi ngs” , we need b o th translation and rotation • Interpolating the translation is easy, but what about rotations?

중앙대학교 첨단영상대학원 박 경 주 Interpolation of Orientation

• HbtitltihtfthHow about interpolating each entry of the rotation matrix? Example: interpolate linearly from a positive 90 degree rotation about y axis to a negative 90 degree rotation about y

중앙대학교 첨단영상대학원 박 경 주 Leonhard Euler(1707-1783)

• 3D ro ta tion – Can be described by three parameters • Euler angles : a rotation about a single Cartesian axis that moves with the object

- roll, pitch and yaw 중앙대학교 첨단영상대학원 박 경 주 Euler angles

• Gim ble – Hardware implementation ofEf Eul er ang les – Aircraft, Camera

• Gimble Lock – When two rotational axes of an object pointing in the same direction, the rotation ends up losing one degree of freedom

- Poor interpolation 중앙대학교 첨단영상대학원 박 경 주 Quaternion

4lfl4 tuple of real num bers:

scalar vector

Same information as axis angles but in a different form

중앙대학교 첨단영상대학원 박 경 주 Quaternion math

UitUnit qua tern ion

Multiplication

중앙대학교 첨단영상대학원 박 경 주 Quaternion math

CjConjuga te

Inverse

the unit length quaternion

중앙대학교 첨단영상대학원 박 경 주 Quaternion Rotation

If q is a unit quaternion and

-1 Then qqpq results in p rotating about r by theta

중앙대학교 첨단영상대학원 박 경 주 Quaternion Rotation

중앙대학교 첨단영상대학원 박 경 주 Quaternion composition

If q1 and q2 are unit quat erni on the combined rotation of first rotating by q1 and then by q2 is equivalent to

중앙대학교 첨단영상대학원 박 경 주 Matrix form

중앙대학교 첨단영상대학원 박 경 주 Quaternion interpolation

1-angle rotation can be 2-angle rotation can be Represented by a unit circle represented by a unit sphere • Interpolation means moving on n-D sphere • NiNow imag ine a 4-DhD sphere f3for 3-anglttile rotation

중앙대학교 첨단영상대학원 박 경 주 Quaternion Interpolation

• MiMoving btbetween two po itints on th4Ditthe 4D unit sphere – A unit quaternion at each step-another point on the 4D unit sphere – Move with constant angular velocity along the great circle between the two points on the 4D unit sphere

중앙대학교 첨단영상대학원 박 경 주 Quaternion interpolation

Direct li near in terpo la tion does no t wor k Linearly interpolated intermediate points are not uniformly spaced when projected onto the circle

Spherical linear interpolation (SLERP)

Normalize to regain unit quaternion

중앙대학교 첨단영상대학원 박 경 주 Choose a representation

• Choose the bes t represen ta tion for the task – Input : Euler angles – Joint Limits : Euler angles , quaternion (harder) – Interpolation : quaternion – Compos iting : qua tern ion or or ien ta tion ma tr ix – Rendering : orientation matrix

중앙대학교 첨단영상대학원 박 경 주