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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,
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 and ?
- From to :
- From to :
- 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.
- 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, and
Interpolation between and
Linear Interpolation?
-
- Linear interpolation results a 2D vector that is not in the circular group.
2D Spherical Interpolation
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