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 tu itive – 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 bo 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 Eu ler 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 qua terni 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.