Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 1/26
Chapter 2
Representing a Moving Scene
Multiple View Geometry
Summer 2021
Prof. Daniel Cremers
Chair for Computer Vision and Artificial Intelligence
Departments of Informatics & Mathematics
Technical University of Munich
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 2/26
Overview
1 The Origins of 3D Reconstruction
2 3D Space & Rigid Body Motion
3 The Lie Group SO(3)
4 The Lie Group SE(3)
5 Representing the Camera Motion
6 Euler Angles
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 3/26
The Origins of 3D Reconstruction
The goal to reconstruct the three-dimensional structure of the
world from a set of two-dimensional views has a long history in
computer vision. It is a classical ill-posed problem, because the
reconstruction consistent with a given set of observations or
images is typically not unique. Therefore, one will need to
impose additional assumptions.
Mathematically, the study of geometric relations between a 3D
scene and the observed 2D projections is based on two types
of transformations, namely:
• Euclidean motion or rigid-body motion representing the
motion of the camera from one frame to the next.
• Perspective projection to account for the image formation
process (see pinhole camera, etc).
The notion of perspective projection has its roots among the
ancient Greeks (Euclid of Alexandria, ∼ 400 B.C.) and the
Renaissance period (Brunelleschi & Alberti, 1435). The study
of perspective projection lead to the field of projective geometry
(Girard Desargues 1648, Gaspard Monge 18th cent.).
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 4/26
The Origins of 3D Reconstruction
The first work on the problem of multiple view geometry was
that of Erwin Kruppa (1913) who showed that two views of five
points are sufficient to determine both the relative
transformation (motion) between the two views and the 3D
location (structure) of the points up to finitely many solutions.
A linear algorithm to recover structure and motion from two
views based on the epipolar constraint was proposed by
Longuet-Higgins in 1981. An entire series of works along these
lines was summarized in several text books (Faugeras 1993,
Kanatani 1993, Maybank 1993, Weng et al. 1993).
Extensions to three views were developed by Spetsakis and
Aloimonos ’87, ’90, and by Shashua ’94 and Hartley ’95.
Factorization techniques for multiple views and orthogonal
projection were developed by Tomasi and Kanade 1992.
The joint estimation of camera motion and 3D location is called
structure and motion or (more recently) visual SLAM.
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 5/26
Three-dimensional Euclidean Space
The three-dimensional Euclidean space E3 consists of all
points p ∈ E3 characterized by coordinates
X ≡ (X1,X2,X3)> ∈ R3,
such that E3 can be identified with R3. That means we talk
about points (E3) and coordinates (R3) as if they were the
same thing. Given two points X and Y , one can define a bound
vector as
v = X − Y ∈ R3.
Considering this vector independent of its base point Y makes
it a free vector. The set of free vectors v ∈ R3 forms a linear
vector space. By identifying E3 and R3, one can endow E3 with
a scalar product, a norm and a metric. This allows to compute
distances, curve length
l(γ) ≡
∫ 1
0
|γ̇(s)|ds for a curve γ : [0,1]→ R3,
areas or volumes.
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 6/26
Cross Product & Skew-symmetric Matrices
On R3 one can define a cross product
× : R3 × R3 → R3 : u × v =
u2v3 − u3v2u3v1 − u1v3
u1v2 − u2v1
∈ R3,
which is a vector orthogonal to u and v . Since u × v = −v × u,
the cross product introduces an orientation. Fixing u induces a
linear mapping v 7→ u × v which can be represented by the
skew-symmetric matrix
û =
0 −u3 u2u3 0 −u1
−u2 u1 0
∈ R3×3.
In turn, every skew symmetric matrix M = −M> ∈ R3×3 can be
identified with a vector u ∈ R3. The operator ̂ defines an
isomorphism between R3 and the space so(3) of all 3× 3
skew-symmetric matrices. Its inverse is denoted by
∨ : so(3)→ R3.
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 7/26
Rigid-body Motion
A rigid-body motion (or rigid-body transformation) is a family of
maps
gt : R3 → R3; X 7→ gt(X ), t ∈ [0,T ]
which preserve the norm and cross product of any two vectors:
• |gt(v)| = |v |, ∀ v ∈ R3,
• gt(u)× gt(v) = gt(u × v), ∀u, v ∈ R3.
Since norm and scalar product are related by the polarization
identity
〈u, v〉 =
1
4
(|u + v |2 − |u − v |2),
one can also state that a rigid-body motion is a map which
preserves inner product and cross product. As a consequence,
rigid-body motions also preserve the triple product
〈gt(u),gt(v)× gt(w)〉 = 〈u, v × w〉, ∀u, v ,w ∈ R3,
which means that they are volume-preserving.
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 8/26
Representation of Rigid-body Motion
Does the above definition lead to a mathematical
representation of rigid-body motion?
Since it preserves lengths and orientation, the motion gt of a
rigid body is sufficiently defined by specifying the motion of a
Cartesian coordinate frame attached to the object (given by an
origin and orthonormal oriented vectors e1,e2,e3 ∈ R3). The
motion of the origin can be represented by a translation
T ∈ R3, whereas the transformation of the vectors ei is given
by new vectors ri = gt(ei).
Scalar and cross product of these vectors are preserved:
r>i rj = gt(ei)
>gt(ej) = e
>
i ej = δij , r1 × r2 = r3.
The first constraint amounts to the statement that the matrix
R = (r1, r2, r3) is an orthogonal (rotation) matrix:
R>R = RR> = I, whereas the second property implies that
det(R) = +1, in other words: R is an element of the group
SO(3) =
{
R ∈ R3×3 | R>R = I, det(R) = +1
}
.
Thus the rigid body motion gt can be written as:
gt(x) = Rx + T .
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 9/26
Exponential Coordinates of Rotation
We will now derive a representation of an infinitesimal rotation.
To this end, consider a family of rotation matrices R(t) which
continuously transform a point from its original location
(R(0) = I) to a different one.
X trans(t) = R(t)X orig , with R(t) ∈ SO(3).
Since R(t)R(t)> = I, ∀t , we have
d
dt
(RR>) = ṘR> + RṘ> = 0 ⇒ ṘR> = −(ṘR>)>.
Thus, ṘR> is a skew-symmetric matrix. As shown in the
section about the ̂-operator, this implies that there exists a
vector w(t) ∈ R3 such that:
Ṙ(t)R>(t) = ŵ(t) ⇔ Ṙ(t) = ŵ(t)R(t).
Since R(0) = I, it follows that Ṙ(0) = ŵ(0). Therefore the
skew-symmetric matrix ŵ(0) ∈ so(3) gives the first order
approximation of a rotation:
R(dt) ≈ R(0) + dR = I + ŵ(0)dt .
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 10/26
Lie Group and Lie Algebra
The above calculations showed that the effect of any
infinitesimal rotation R ∈ SO(3) can be approximated by an
element from the space of skew-symmetric matrices
so(3) = {ŵ | w ∈ R3}.
The rotation group SO(3) is called a Lie group. The space
so(3) is called its Lie algebra.
Def.: A Lie group (or infinitesimal group) is a smooth manifold
that is also a group, such that the group operations
multiplication and inversion are smooth maps.
As shown above: The Lie algebra so(3) is the tangent space at
the identity of the rotation group SO(3).
An algebra over a field K is a vector space V over K with
multiplication on the space V . Elements ŵ and v̂ of the Lie
algebra generally do not commute.
One can define the Lie bracket
[ . , . ] : so(3)× so(3)→ so(3); [ŵ , v̂ ] ≡ ŵ v̂ − v̂ ŵ .
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 11/26
Sophus Lie (1841 – 1899)
Marius Sophus Lie was a Norwegian-born mathematician. He
created the theory of continuous symmetry, and applied it to
the study of geometry and differential equations. Among his
greatest achievements was the discovery that continuous
transformation groups are better understood in their linearized
versions (“Theorie der Transformationsgruppen” 1893). These
infinitesimal generators form a structure which is today known
as a Lie algebra. The linearized version of the group law
corresponds to an operation on the Lie algebra known as the
commutator bracket or Lie bracket. 1882 Professor in
Christiania (Oslo), 1886 Leipzig (succeeding Felix Klein), 1898
Christiania.
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 12/26
The Exponential Map
Given the infinitesimal formulation of rotation in terms of the
skew symmetric matrix ŵ , is it possible to determine a useful
representation of the rotation R(t)? Let us assume that ŵ is
constant in time.
The differential equation system{
Ṙ(t) = ŵR(t),
R(0) = I.
has the solution
R(t) = eŵ t =
∞∑
n=0
(ŵ t)n
n!
= I + ŵ t +
(ŵ t)2
2!
+ . . . ,
which is a rotation around the axis w ∈ R3 by an angle of t (if
‖w‖ = 1). Alternatively, one can absorb the scalar t ∈ R into
the skew symmetric matrix ŵ to obtain R(t) = ev̂ with v̂ = ŵ t .
This matrix exponential therefore defines a map from the Lie
algebra to the Lie group:
exp : so(3)→ SO(3); ŵ 7→ eŵ .
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 13/26
The Logarithm of SO(3)
As in the case of real analysis one can define an inverse
function to the exponential map by the logarithm. In the context
of Lie groups, this will lead to a mapping from the Lie group to
the Lie algebra. For any rotation matrix R ∈ SO(3), there exists
a w ∈ R3 such that R = exp(ŵ). Such an element is denoted
by ŵ = log(R).
If R = (rij) 6= I, then an appropriate w is given by:
|w | = cos−1
(
trace(R)− 1
2
)
,
w
|w |
=
1
2 sin(|w |)
r32 − r23r13 − r31
r21 − r12
.
For R = I, we have |w | = 0, i.e. a rotation by an angle 0. The
above statement says: Any orthogonal transformation
R ∈ SO(3) can be realized by rotating by an angle |w | around
an axis w|w| as defined above. We will not prove this statement.
Obviously the above representation is not unique since
increasing the angle by multiples of 2π will give the same
rotation R.
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 14/26
Schematic Visualization of Lie Group & Lie Algebra
Definition: A Lie group is a smooth manifold that is also a
group, such that the group operations multiplication and
inversion are smooth maps.
Definition: The tangent space to a Lie group at the identity
element is called the associated Lie algebra.
The mapping from the Lie algebra to the Lie group is called the
exponential map. Its inverse is called the logarithm.
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 15/26
Rodrigues’ Formula
We have seen that any rotation can be realized by computing
R = eŵ . In analogy to the well-known Euler equation
eiφ = cos(φ) + i sin(φ), ∀φ ∈ R,
we have an expression for skew symmetric matrices
ŵ ∈ so(3):
eŵ = I +
ŵ
|w |
sin(|w |) +
ŵ2
|w |2
(
1− cos(|w |)
)
.
This is known as Rodrigues’ formula.
Proof: Let t = |w | and v = w/|w |. Then
v̂2 = vv> − I, v̂3 = −v̂ , . . .
and
eŵ = ev̂ t = I+
(
t −
t3
3!
+
t5
5!
− . . .
)
︸ ︷︷ ︸
sin(t)
v̂+
(
t2
2!
−
t4
4!
+
t6
6!
− . . .
)
︸ ︷︷ ︸
1−cos(t)
v̂2.
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 16/26
Representation of Rigid-body Motions SE(3)
We have seen that the motion of a rigid-body is uniquely
determined by specifying the translation T of any given point
and a rotation matrix R defining the transformation of an
oriented Cartesian coordinate frame at the given point. Thus
the space of rigid-body motions given by the group of special
Euclidean transformations
SE(3) ≡
{
g = (R,T ) | R ∈ SO(3), T ∈ R3
}
.
In homogeneous coordinates, we have:
SE(3) ≡
{
g =
(
R T
0 1
) ∣∣∣∣∣ R ∈ SO(3), T ∈ R3
}
⊂ R4×4.
In the context of rigid motions, one can see the difference
between points in E3 (which can be rotated and translated) and
vectors in R3 (which can only be rotated).
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 17/26
The Lie Algebra of Twists
Given a continuous family of rigid-body transformations
g : R→ SE(3); g(t) =
(
R(t) T (t)
0 1
)
∈ R4×4,
we consider
ġ(t)g−1(t) =
(
ṘR> Ṫ − ṘR>T
0 0
)
∈ R4×4.
As in the case of SO(3), the matrix ṘR> corresponds to some
skew-symmetric matrix ŵ ∈ so(3).
Defining a vector v(t) = Ṫ (t)− ŵ(t)T (t), we have:
ġ(t)g−1(t) =
(
ŵ(t) v(t)
0 0
)
≡ ξ̂(t) ∈ R4×4.
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 18/26
The Lie Algebra of Twists
Multiplying with g(t) from the right, we obtain:
ġ = ġg−1g = ξ̂g.
The 4× 4-matrix ξ̂ can be viewed as a tangent vector along the
curve g(t). ξ̂ is called a twist.
As in the case of so(3), the set of all twists forms the tangent
space (at the identity) which is the Lie algebra
se(3) ≡
{
ξ̂ =
(
ŵ v
0 0
) ∣∣∣∣∣ ŵ ∈ so(3), v ∈ R3
}
⊂ R4×4
to the Lie group SE(3).
As before, we can define operators ∧ and ∨ to convert between
a twist ξ̂ ∈ se(3) and its twist coordinates ξ ∈ R6:
ξ̂ ≡
(
v
w
)∧
≡
(
ŵ v
0 0
)
∈ R4×4,
(
ŵ v
0 0
)∨
=
(
v
w
)
∈ R6.
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 19/26
Exponential Coordinates for SE(3)
The twist coordinates ξ =
(v
w
)
are formed by stacking the linear
velocity v ∈ R3 (related to translation) and the angular velocity
w ∈ R3 (related to rotation).
The differential equation system{
ġ(t) = ξ̂g(t), ξ̂ = const.
g(0) = I,
has the solution
g(t) = eξ̂t =
∞∑
n=0
(ξ̂t)n
n!
.
For w = 0, we have eξ̂ =
( I v
0 1
)
, while for w 6= 0 one can show:
eξ̂ =
(
eŵ (I−e
ŵ )ŵv+ww>v
|w|2
0 1
)
.
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 20/26
Exponential Coordinates for SE(3)
The above shows that the exponential map defines a
transformation from the Lie algebra se(3) to the Lie group
SE(3):
exp : se(3)→ SE(3); ξ̂ 7→ eξ̂.
The elements ξ̂ ∈ se(3) are called the exponential coordinates
for SE(3).
Conversely: For every g ∈ SE(3) there exist twist coordinates
ξ = (v ,w) ∈ R6 such that g = exp(ξ̂).
Proof: Given g = (R,T ), we know that there exists w ∈ R3 with
eŵ = R. If |w | 6= 0, the exponential form of g introduced above
shows that we merely need to solve the equation
(I − eŵ )ŵv + ww>v
|w |2
= T
for the velocity vector v ∈ R3. Just as in the case of SO(3), this
representation is generally not unique, i.e. there exist many
twists ξ̂ ∈ se(3) which represent the same rigid-body motion
g ∈ SE(3).
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 21/26
Representing the Motion of the Camera
When observing a scene from a moving camera, the
coordinates and velocity of points in camera coordinates will
change. We will use a rigid-body transformation
g(t) =
(
R(t) T (t)
0 1
)
∈ SE(3)
to represent the motion from a fixed world frame to the camera
frame at time t . In particular we assume that at time t = 0 the
camera frame coincides with the world frame, i.e. g(0) = I. For
any point X 0 in world coordinates, its coordinates in the
camera frame at time t are:
X (t) = R(t)X 0 + T (t),
or in the homogeneous representation
X (t) = g(t)X 0.
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 22/26
Concatenation of Motions over Frames
Given two different times t1 and t2, we denote the
transformation from the points in frame t1 to the points in frame
t2 by g(t2, t1):
X (t2) = g(t2, t1)X (t1).
Obviously we have:
X (t3) = g(t3, t2)X (t2) = g(t3, t2)g(t2, t1)X (t1) = g(t3, t1)X (t1),
and thus:
g(t3, t1) = g(t3, t2)g(t2, t1).
By transferring the coordinates of frame t1 to coordinates in
frame t2 and back, we see that:
X (t1) = g(t1, t2)X (t2) = g(t1, t2)g(t2, t1)X (t1),
which must hold for any point coordinates X (t1), thus:
g(t1, t2)g(t2, t1) = I ⇔ g−1(t2, t1) = g(t1, t2).
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 23/26
Rules of Velocity Transformation
The coordinates of point X 0 in frame t are given by
X (t) = g(t)X 0. Therefore the velocity is given by
Ẋ (t) = ġ(t)X 0 = ġ(t)g−1(t)X (t)
By introducing the twist coordinates
V̂ (t) ≡ ġ(t)g−1(t) =
ŵ(t) v(t)0 0
∈ se(3),
we get the expression:
Ẋ (t) = V̂ (t)X (t).
In simple 3D-coordinates this gives:
Ẋ (t) = ŵ(t)X (t) + v(t).
The symbol V̂ (t) therefore represents the relative velocity of
the world frame as viewed from the camera frame.
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 24/26
Transfer Between Frames: The Adjoint Map
Suppose that a viewer in another frame A is displaced relative
to the current frame by a transformation gxy : Y (t) = gxy X (t).
Then the velocity in this new frame is given by:
Ẏ (t) = gxy Ẋ (t) = gxy V̂ (t)X (t) = gxy V̂g−1xy Y (t).
This shows that the relative velocity of points observed from
camera frame A is represented by the twist
V̂y = gxy V̂ g
−1
xy ≡ adgxy (V̂ ).
where we have introduced the adjoint map on se(3):
adg : se(3)→ se(3); ξ̂ 7→ g ξ̂ g−1.
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 25/26
Summary
Rotation SO(3) Rigid-body SE(3)
Matrix repres.
R ∈ GL(3) :
R>R = I,
det(R) = 1
g =
(
R T
0 1
)
3-D coordinates X = RX 0 X = RX 0 + T
Inverse R−1 = R> g−1 =
(
R> −R>T
0 1
)
Exp. repres. R = exp(ŵ) g = exp(ξ̂)
Velocity Ẋ = ŵX Ẋ = ŵX + v
Adjoint map ŵ 7→ R ŵ R> ξ̂ 7→ g ξ̂ g−1
Representing a Moving
Scene
Prof. Daniel Cremers
The Origins of 3D
Reconstruction
3D Space & Rigid
Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the
Camera Motion
Euler Angles
updated April 12, 2021 26/26
Alternative Representations: Euler Angles
In addition to the exponential parameterization, there exist
alternative mathematical representations to parameterize
rotation matrices R ∈ SO(3), given by the Euler angles. These
are local coordinates, i.e. the parameterization is only correct
for a portion of SO(3).
Given a basis (ŵ1, ŵ2, ŵ3) of the Lie algebra so(3), we can
define a mapping from R3 to the Lie group SO(3) by:
α : (α1, α2, α3) 7→ exp(α1ŵ1 + α2ŵ2 + α3ŵ3).
The coordinates (α1, α2, α3) are called Lie-Cartan coordinates
of the first kind relative to the above basis.
The Lie-Cartan coordinates of the second kind are defined as:
β : (β1, β2, β3) 7→ exp(β1ŵ1) exp(β2ŵ2) exp(β3ŵ3).
For the basis representing rotation around the z-, y -, x-axis
w1 = (0,0,1)
>, w2 = (0,1,0)
>, w3 = (1,0,0)
>,
the coordinates β1, β2, β3 are called Euler angles.
The Origins of 3D Reconstruction
3D Space & Rigid Body Motion
The Lie Group SO(3)
The Lie Group SE(3)
Representing the Camera Motion
Euler Angles