[Computer Animation] 2D Rotation and Orientation

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, $\theta˚$

 

 

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 $o_{1}$ and $o_{2}$?

 

• From $o_{1}$ to $o_{2}$ : $o_{2} - o_{1} = - \frac{3}{2} \pi$
• From $o_{2}$ to $o_{1}$ : $o_{1} - o_{2} = \frac{3}{2} \pi$
• 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.
  • $\sqrt {x^{2} + y^{2}} = 1$
• 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, $o_{a}$ and $o_{b}$

• $o_{a} = (x_{a}, y_{a})$
• $o_{b} = (x_{b}, y_{b})$

 

Interpolation between $o_{a}$ and $o_{b}$

• $o_{a} = (x_{a}, y_{a})$
• $o_{b} = (x_{b}, y_{b})$
• $o_{t} = ?$

 

Linear Interpolation?

• $o'_{t} = o_{a} + t(o_{b} - o_{a}) = (1 - t)o_{a} + to_{b}$
  • $o'_{t} ≠ o_{t}$
  • $o'_{t} \notin SO(2)$
• Linear interpolation results a 2D vector that is not in the circular group.

 

2D Spherical Interpolation