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Rotation and Orientation:Rotation and Orientation:Fundamentals
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QuestionsQuestions
H DOF f 2D t ti ?• How many DOFs for 2D rotation?- Answer) One
• How many DOFs for 3D rotation?• How many DOFs for 3D rotation?- Answer) Three
• How many DOFs for 4D rotation?- Answer) Six
중앙대학교 첨단영상대학원 박 경 주
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What is Rotation?What is Rotation?
N t i t iti• Not intuitive– 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
q
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Orientation vs RotationOrientation vs. Rotation
R t ti• Rotation– Circular movement
• Orientation• 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
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3D Orientation3D Orientation
Gi t bit i t ti f i id bj t• Given two arbitrary orientations of a rigid object,
How many rotations do we need t t f i t tito transform one orientation
to the other?
중앙대학교 첨단영상대학원 박 경 주
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Joints and RotationsJoints and RotationsRotational DOFs are widely used in character animationRotational DOFs are widely used in character animation
3 translational DOFs48 t ti l DOF48 rotational DOFs
Each joint can have up to 3 DOFs
중앙대학교 첨단영상대학원 박 경 주1 DOF: knee 2 DOF: wrist 3 DOF: arm
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Composite TransformationsComposite TransformationsA series of transformations on anA series of transformations on an
object can be applied as a series of matrix multiplicationsseries of matrix multiplications
p: position in the global coordinatep: position in the global coordinatex: position in the local coordinate
(h3 0 0)(h3, 0, 0)
중앙대학교 첨단영상대학원 박 경 주
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InterpolationInterpolation
I d t “ thi ” d b th• In order to “move things”, we need both translation and rotation
• Interpolating the translation is easy, but what about rotations?what about rotations?
중앙대학교 첨단영상대학원 박 경 주
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Interpolation of OrientationInterpolation of Orientation
H b t i t l ti h t f th• How about interpolating each entry of the rotation matrix?
fExample: interpolate linearly from a positive 90 degree rotation about y axis to a negative 90 degree rotation about yabout y
중앙대학교 첨단영상대학원 박 경 주
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Leonhard Euler(1707-1783)Leonhard Euler(1707 1783)
3D t ti• 3D rotation– Can be described by three parameters
• Euler angles: a rotation about a single Cartesian axis that moves g
with the object
- roll, pitch and yaw
중앙대학교 첨단영상대학원 박 경 주
y
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Euler anglesEuler angles
Gi bl• Gimble– Hardware implementation f E l lof Euler angles
– Aircraft, Camera
• Gimble Lock– When two rotational axes of an object pointing in the same direction, the rotationends up losing one degree of freedom
Poor interpolation
중앙대학교 첨단영상대학원 박 경 주
- Poor interpolation
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QuaternionQuaternion
4 l f l b4 tuple of real numbers:
scalarvector
Same information as axis angles but in a different formg
중앙대학교 첨단영상대학원 박 경 주
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Quaternion mathQuaternion math
U it t iUnit quaternion
Multiplication
중앙대학교 첨단영상대학원 박 경 주
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Quaternion mathQuaternion math
C j tConjugate
InverseInverse
the unit length quaternion
중앙대학교 첨단영상대학원 박 경 주
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Quaternion RotationQuaternion Rotation
If q is a unit quaternion and
Then qqpq-1 results in p rotating about r by theta
중앙대학교 첨단영상대학원 박 경 주
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Quaternion RotationQuaternion Rotation
중앙대학교 첨단영상대학원 박 경 주
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Quaternion compositionQuaternion composition
If 1 d 2 it t iIf q1 and q2 are unit quaternion
the combined rotation of first rotating by q1and then by q2 is equivalent toand then by q2 is equivalent to
중앙대학교 첨단영상대학원 박 경 주
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Matrix formMatrix form
중앙대학교 첨단영상대학원 박 경 주
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Quaternion interpolationQuaternion interpolation
1-angle rotation can beRepresented by a unit circle
2-angle rotation can berepresented by a unit sphereRepresented by a unit circle represented by a unit sphere
• Interpolation means moving on n-D sphereN i i 4 D h f 3 l t ti• Now imagine a 4-D sphere for 3-angle rotation
중앙대학교 첨단영상대학원 박 경 주
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Quaternion InterpolationQuaternion Interpolation
M i b t t i t th 4D it• Moving between two points on the 4D unit sphere– A unit quaternion at each step-another point
on the 4D unit spherep– Move with constant angular velocity along the
great circle between the two points on the 4Dgreat circle between the two points on the 4D unit sphere
중앙대학교 첨단영상대학원 박 경 주
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Quaternion interpolationQuaternion interpolation
Di t li i t l ti d t kDirect linear interpolation does not workLinearly 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
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Choose a representationChoose a representation
Ch th b t t ti f th• Choose the best representation for the task– Input : Euler angles– Joint Limits : Euler angles quaternion (harder)Joint Limits : Euler angles, quaternion (harder)– Interpolation : quaternion
C iti t i i t ti t i– Compositing : quaternion or orientation matrix– Rendering : orientation matrix
중앙대학교 첨단영상대학원 박 경 주