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2D Rotation and Orientation

Orientation vs. Rotation

  • Rotation
    • 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.
  • Analogy
    • (point : vector) is similar to (orientation : rotation)
    • Both represent a sort of (state : movement)

 

 

Representations for 2D rotations and orientations

  • Turning angles.
  • Complex numbers.
  • Rotation matrices.

 

 

Turning Angles

Rotate by a turning angle, θ˚

 

 

Turning Angles for 2D Rotations

Rotation between two states

 

Turning Angle from the Reference pose

 

  • Different rotational movements can result in the same final orientation
    • Turn 45 degree counter-clockwise
    • Turn -315 degree clockwise
    • Turn 405 degree counter-clockwise

 

Reference poses

 

 

Turning Angles for 2D Orientations?

Orientations by turning angles

 

 

One-to-one mapping between angles and orientations.

 

What is rotation between o1 and o2?

 

  • From o1 to o2 : o2o1=32π
  • From o2 to o1 : o1o2=32π
  • How can the shorter rotation be calculated?

 

Many-to-one mapping between angles and orientations.

 

 

Which one is better for 2D orientation?

 

  • Can it be one-to-one and continuous at the same time?

 

 

Complex Numbers

Extra Parameter

 

Complex number form

 

Circle group (or SO(2) group)

  • The circle group is a group of all complex numbers with absolute value 1.
    • x2+y2=1
  • One element in the circle group corresponds to one unique 2D orientation.

 

2×2 Matrix form

 

Exponential form of complex numbers

 

Rotation composition

 

2D rotations by complex numbers?

  • Complex numbers are good for representing 2D orientations(one-to-one mapping), but inadequate for 2D rotations.
  • A complex number cannot distinguish different rotational movements that result in the same final orientation.
    • Turn 120 degree counter-clockwise
    • Turn -240 degree clockwise
    • Turn 480 degree counter-clockwise

 

Conclusion

2D Rotation

  • The consequence of any 2D rotational movement can be uniquely represented by a turning angle.
  • A turning angle is independent of the choice of the reference orientation.

 

2D Orientation

  • The non-singular parameterization of 2D orientations requires extra parameters.
    • Eg) Complex numbers, 2×2 rotation matrices
  • The parameterization is dependednt on the choice of the reference orientation.

 

 

2D Slerp(Sherical Linear Interpolation)

Two orientations, oa and ob

  • oa=(xa,ya)
  • ob=(xb,yb)

 

Interpolation between oa and ob

  • oa=(xa,ya)
  • ob=(xb,yb)
  • ot=?

 

Linear Interpolation?

  • ot=oa+t(oboa)=(1t)oa+tob
    • otot
    • otSO(2)
  • Linear interpolation results a 2D vector that is not in the circular group.

 

2D Spherical Interpolation

 

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📖 Contents 📖
2D Rotation and OrientationOrientation vs. RotationRepresentations for 2D rotations and orientationsTurning AnglesRotate by a turning angle, θ˚Turning Angles for 2D RotationsRotation between two statesTurning Angle from the Reference poseReference posesTurning Angles for 2D Orientations?Orientations by turning anglesOne-to-one mapping between angles and orientations.What is rotation between o1 and o2?Many-to-one mapping between angles and orientations.Which one is better for 2D orientation?Complex NumbersExtra ParameterComplex number formCircle group (or SO(2) group)2×2 Matrix formExponential form of complex numbersRotation composition2D rotations by complex numbers?Conclusion2D Rotation2D Orientation2D Slerp(Sherical Linear Interpolation)Two orientations, oa and obInterpolation between oa and obLinear Interpolation?2D Spherical Interpolation