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Rotation and Orientation: Fundamentals. Jehee Lee Seoul National University. What is Rotation ?. Not intuitive Formal definitions are also confusing Many different ways to describe Rotation (direction cosine) matrix Euler angles Axis-angle Rotation vector Helical angles

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Rotation and orientation fundamentals

Rotation and Orientation:Fundamentals

Jehee Lee

Seoul National University

What is rotation

What is Rotation ?

  • 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

Orientation vs rotation

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



(point : vector) is similar to (orientation : rotation)

Both represent a sort of (state : movement)

Reference coordinate system



(point : vector) is similar to (orientation : rotation)

Both represent a sort of (state : movement)

point: the 3d location of the bunny

vector: translational movement

Reference coordinate system



(point : vector) is similar to (orientation : rotation)

Both represent a sort of (state : movement)

point: the 3d location of the bunny

vector: translational movement

orientation: the 3d orientation of the bunny

rotation : circular movement

Reference coordinate system

2d orientation

2D Orientation

Polar Coordinates

2d orientation1

2D Orientation

Although the motion is continuous,

its representation could be discontinuous

2d orientation2

2D Orientation

Many-to-one correspondences between 2D orientations and their representations

Extra parameter

Extra Parameter

Extra parameter1

Extra Parameter

2x2 Rotation matrix is yet another method of using extra parameters

Complex number

Complex Number



Complex exponentiation

Complex Exponentiation



2d rotation

2D Rotation

  • Complex numbers are good for representing 2D orientations, 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



2d rotation and orientation

2D Rotation and Orientation

  • 2D Rotation

    • The consequence of any 2D rotational movement can be uniquely represented by a turning angle

  • 2D Orientation

    • The non-singular parameterization of 2D orientations requires extra parameters

      • Eg) Complex numbers, 2x2 rotation matrices

Operations in 2d

Operations in 2D

  • (orientation) : complex number

  • (rotation) : scalar value

  • exp(rotation) : complex number

2d rotation and displacement

2D Rotation and Displacement



2d rotation and displacement1

2D Rotation and Displacement



2d orientation composition

2D Orientation Composition



2d rotation composition

2D Rotation Composition





3d rotation

3D Rotation

  • Given two arbitrary orientations of a rigid object,

3d rotation1

3D Rotation

  • We can always find a fixed axis of rotation and an angle about the axis

Euler s rotation theorem

Euler’s Rotation Theorem

In other words,

  • Arbitrary 3D rotation equals to one rotation around an axis

  • Any 3D rotation leaves one vector unchanged

The general displacement of a rigid body with

one point fixed is a rotation about some axis

Leonhard Euler (1707-1783)

Euler angles

Euler angles

  • Gimble

    • Hardware implementation of Euler angles

    • Aircraft, Camera

Euler angles1

Euler Angles

  • Rotation about three orthogonal axes

    • 12 combinations

      • XYZ, XYX, XZY, XZX

      • YZX, YZY, YXZ, YXY

      • ZXY, ZXZ, ZYX, ZYZ

  • Gimble lock

    • Coincidence of inner most and outmost gimbles’ rotation axes

    • Loss of degree of freedom

Euler angles2

Euler Angles

  • Euler angles are ambiguous

    • Two different Euler angles can represent the same orientation

    • This ambiguity brings unexpected results of animation where frames are generated by interpolation.



  • Euler angles in ZXZ

    • Z-axis by a  ( -p , p ]

    • X’-axis by b  ( -p/2 , p/2 ]

    • Z’’-axis by g  ( -p , p ]



  • Compare

    • Rotation about z-axis by 180 degree

    • Rotation about y-axis by 180 degree, followed by another rotation about x-axis by 180 degree

  • Rotations about x-, y-, and z-axes are dependent

Rotation vector

Rotation Vector

  • Rotation vector(3 parameters)

  • Axis-Angle(2+1 parameters)

3d orientation

3D Orientation

  • Unhappy with three parameters

    • Euler angles

      • Discontinuity (or many-to-one correspondences)

      • Gimble lock

    • Rotation vector (a.k.a Axis/Angle)

      • Discontinuity (or many-to-one correspondences)

Using an extra parameter

Using an Extra Parameter

  • Euler parameters



  • William Rowan Hamilton (1805-1865)

    • Algebraic couples (complex number) 1833




  • William Rowan Hamilton (1805-1865)

    • Algebraic couples (complex number) 1833

    • Quaternions 1843





William Thomson

“… though beautifully ingenious, have been an unmixed evil to those who have touched them in any way.”

Arthur Cayley

“… which contained everything but had to be unfolded into another form before it could be understood.”

Unit quaternions

Unit Quaternions

  • Unit quaternions represent 3D rotations

Rotation about an arbitrary axis

Rotation about an Arbitrary Axis

  • Rotation about axis by angle


Purely Imaginary Quaternion

Unit quaternion algebra

Unit Quaternion Algebra

  • Identity

  • Multiplication

  • Inverse

    • Opposite axis or negative angle

  • Unit quaternion space is

    • closed under multiplication and inverse,

    • but not closed under addition and subtraction

Unit quaternion algebra1

Unit Quaternion Algebra

  • Antipodal equivalence

    • q and –q represent the same rotation

    • ex) rotation by p-q about opposite direction

    • 2-to-1 mapping between S and SO(3)

    • Twice as fast as in SO(3)


3d orientations and rotations

3D Orientations and Rotations

Orientations and rotations are different in coordinate-invariant geometric programming

Use unit quaternions for orientation representation

  • 3x3 orthogonal matrix is theoretically identical

    Use 3-vectors for rotation representation

Tangent vector infinitesimal rotation

Tangent Vector(Infinitesimal Rotation)

Tangent vector infinitesimal rotation1

Tangent Vector(Infinitesimal Rotation)

Tangent vector infinitesimal rotation2

Tangent Vector(Infinitesimal Rotation)

Angular Velocity

Exp and log

Exp and Log



Exp and log1

Exp and Log





Rotation vector1

Rotation Vector

Rotation vector2

Rotation Vector

Rotation vector3

Rotation Vector

Rotation vector4

Rotation Vector

  • Finite rotation

    • Eg) Angular displacement

    • Be careful when you add two rotation vectors

  • Infinitesimal rotation

    • Eg) Instantaneous angular velocity

    • Addition of angular velocity vectors are meaningful

Spherical linear interpolation

Spherical Linear Interpolation

  • SLERP [Shoemake 1985]

    • Linear interpolation of two orientations

Spherical linear interpolation1

Spherical Linear Interpolation

Coordinate invariant operations

Coordinate-Invariant Operations

Adding rotation vectors

Adding Rotation Vectors

Affine combination of orientations

Affine Combination of Orientations



  • (point : vector) is similar to (orientation : rotation)

Rotation matrix vs unit quaternion

Rotation Matrix vs. Unit Quaternion

  • Equivalent in many aspects

    • Redundant, No singularity

    • Exp & Log, Special tangent space

  • Why quaternions ?

    • Fewer parameters, Simpler algebra

    • Easy to fix numerical error

      • Cf) matrix orthogonalization (Gram-shmidt process, QR, SVD decomposition)

  • Why rotation matrices ?

    • One-to-one correspondence

    • Handle rotation and translation in a uniform way

      • Eg) 4x4 homogeneous matrices

Rotation conversions

Rotation Conversions

  • In theory, conversion between any representations is always possible

  • In practice, conversion is not straightward because of difference in convention

  • Quaternion to Matrix

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