Homogeneous Coordinates
Basilio Bona
DAUIN – Politecnico di Torino
Semester 1, 2015-16
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 1 / 32 Introduction
Homogeneous coordinates are an augmented representation of points and lines in Rn spaces, embedding them in Rn+1, hence using n + 1 parameters. This representation is useful in dealing with perspective and projective transformation (computer graphics, etc.) and for rigid displacement representation. We start introducing the concept on 2D spaces. 2 Given a point p in R , represented as P = (p1, p2), i.e., the vector T p = p1 p2 its homogeneous representation (using homogeneous coordinates) is T T p˜ = p˜1 p˜2 p˜3 ; with 0 0 0 not allowed
The vector representation is obtained dividing the first n homogeneous components by the (n + 1)-th, that is often called scale.
p1 =p ˜1/p˜3; p2 =p ˜2/p˜3
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 2 / 32 A geometric interpretation of the homogeneous coordinates is given in the following Figure.
Figure: Geometric interpretation of homogeneous coordinates.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 3 / 32 We embed the plane π : z = 1 in a 3D space and we consider the planar coordinates as the intersection of a projection line that goes through the origin of the RF {i, j, k}. As z goes to ∞ we can represent the origin, while as z goes to 0, we can represents points at infinity, along a well defined direction. The set of all points at infinity represents the line at infinity.
Figure: Points at ∞.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 4 / 32 The same approach is valid for 3D homogeneous coordinates: given a 3 point p in R , represented as P = (p1, p2, p3), or else by a vector
T p = p1 p2 p3 its homogeneous representation is
T p˜ = p1 p2 p3 p4
Hence p1 =p ˜1/p˜4; p2 =p ˜2/p˜4; p3 =p ˜3/p˜4
In robotics it is common to setp ˜4 = 1. We will adhere to this convention.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 5 / 32 Some author treats differently the homogeneous representation of a point (geometric vector vp) by that of an oriented segment (physical vector vab). In the first case v v˜ = p p 1 while in the second case v v v v˜ = b − a = ab ab 1 1 0 Not all textbooks adhere to this convention, since in this case the sum of two geometric vectors produces
v v v + v ˜v + v˜ = p + q = p q p q 1 1 2 This sum, once transformed back in 3D space, becomes not the usual parallelogram sum of segments, but the midpoint of the parallelogram (see also Affine Spaces).
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 6 / 32 Figure: Homogeneous sum of two geometric vectors.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 7 / 32 Affine spaces
Briefly, affine spaces are spaces where the origin is not considered to bea special point, and homogeneous coordinates are used to characterize the affine elements (points and planes).
Figure: Linear and affine subspaces.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 8 / 32 Rigid displacements and homogeneous coordinates
Using homogeneous representation, the generic displacement (R, t) is given by the following homogeneous transformation matrix
R t H ≡ H(R, t)= 0T 1
since R t v H˜v = = Rv + t 0T 1 1 Hence, the SE(3) group transformation g = (R, t) ∈ SE(3), is represented by the 4 × 4 homogeneous matrix H through the use of homogeneous coordinates: pa = gabpb ↔ p˜a = Hab(Rab, tab)p˜b
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 9 / 32 Transformation composition gabgbc is obtained by the homogeneous matrix product Hab(Rab, tab)Hbc (Rbc , tbc ) i.e., Rab tab Rbc tbc RabRbc Rabtbc + tab = 0T 1 0T 1 0T 1 Often it is convenient to write the time derivative of a homogeneous vector as p˙1 p˙ p˜˙ = 2 p˙3 0 This notation will prove useful when dealing with twists.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 10 / 32 Introduction to projective/perspective geometry
Homogeneous coordinates p˜ are used for describing point in projective spaces Pn of dimension n. Two points a˜ and b˜ are “equal” (a˜ ∼ b˜) if there exist a real λ = 0 such that a˜ = λb˜ → b˜ ∼ λb˜ collinear points are equal this also means that vectors are equal up to a scale factor. We say that a˜ and b˜ belong to the same line, i.e., to the same affine subspace of Rn+1. Consider the planar (2D) projection through a pinhole camera of a 3D point P, p = x y z T. The ray of light intercepts the image plane in the 2D image point Pi , whose homogeneous coordinates on the image plane are x T T x/f 1 x p˜ = y → p = = i i y/f f y f where f is the focal distance B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 11 / 32 Figure: Example of projection through a pinhole camera.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 12 / 32 Considering the similar triangles in Figure, we have: x X y Y = ; = f Z f Z If we set λ = Z/f , where f is known and the distance Z from π of the point p is unknown, we have
X = λx, Y = λy, Z = λf
Hence, if we compute the cartesian coordinates of pi on π from homogeneous coordinates, we have
fX x x x/f p = ↔ p˜ = fY ∼ λ y ∼ y i y/f i Z 1 f and every 3D point with homogeneous coordinates λ x y f T is projected onto the same point pi of the image plane π. Notice that the above transformation is a nonlinear one.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 13 / 32 The nonlinear mapping R3 → R2 can be expressed as a (linear) matrix transformation between the homogeneous spaces R4 → R3 as
X fX x f 0 0 0 Y p fY ∼ λ y = 0 f 0 0 ⇔ λp˜ = P Z i 1 Z 1 0 0 10 1 P Nonsingular (i.e., invertible) mappings between projective spaces are described by products of full rank square matrices T by homogeneous coordinates, p˜a = Tabp˜b An invertible mapping from Pn onto itself preserves collinearities, i.e., if two points belong to a line, also the transformed points will belong to a line. Affine transformations preserve not only collinearities, but also parallelism between lines and ratios of distances. Similarity transformations preserves also angles, but not scales, while Euclidean (rigid) transformations preserve also scales
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 14 / 32 Figure: Examples of different mappings.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 15 / 32 Hyperplanes
n A hyperplane in R is defined by the equation between a set of points pi and a set of (real) coefficients ℓi
ℓ1p1 + ℓ2p2 + + ℓnpn + ℓn+1 = 0
Using the homogeneous coordinates we obtain an homogeneous linear equation in Pn T ˜ℓ p˜ = 0 where ℓ1 p1 . . ˜ℓ = . ; p˜ = . ℓn pn ℓ 1 n+1 The two homogeneous vectors can exchange their roles; we say that they are dual.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 16 / 32 Remember that, given a vector p ∈V(F), the set of all linear transformations f (p) : V → R1 form another vector space V∗, and f ∈V∗ are called dual vectors or 1−forms. For example, the total time derivative of a scalar function f (p) on R3 is df (p(t)) = ∇ φ(p)p˙ (t) dt p is a dual vector wrt p˙ . Another dual vector is the divergence
T p1 ∂ ∂ . ∇ p = . ∂p1 ∂pn pn or the Laplacian (gradient of the divergence) ∂f T ∂p1 ∂ ∂ ∇2f = ∇ ∇f = . ∂p ∂p . 1 n ∂f ∂pn B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 17 / 32 Duality in projective spaces
The space of all hyperplanes in Rn defines another perspective space, called the dual of the original Pn Duality principle: for all projective results established using points and planes, a symmetrical result holds interchanging the role of planes and points. Example:
two points define a line ↔ two lines define a point
Notice that the projective plane P2 whose point are defined by homogeneous coordinates, has more point that the usual R2 plane, since it includes also points at ∞.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 18 / 32 A collineation given by a full rank matrix T transforms points as
p˜a = Tabp˜b
˜ℓT A hyperplane b p˜b = 0 is transformed as
˜ℓT ˜ℓT b p˜b = 0 ⇔ a p˜a = 0 hence ˜ℓT ˜ℓT ˜ℓT a Tabp˜b ⇒ b = a Tab yielding ˜ℓ −T ˜ℓT a = Tab b −T Matrix Tab is called the dual of Tab Comparing this last equation with the first one, we conclude that points and lines can be transformed, but using different (dual) collineation −T matrices Tab and Tab .
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 19 / 32 Points and lines
We restrict our analysis on the 2D projective space P2; for the sake of clarity, we drop the ˜ sign on top of the homogeneous vectors, i.e., a˜ is now indicated as a.
A line ℓ = ℓ1 ℓ2 ℓ3 can be characterized by three parameters:
1 the slope −ℓ1/ℓ2 2 the x-intercept −ℓ3/ℓ1 3 the y-intercept −ℓ3/ℓ2 Points p and lines ℓ are related by
ℓTp = 0
in the sense that a point is on a line or a line include a point iff the previous relation holds.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 20 / 32 Duality ℓ A line passing for two points p1 and p2 is defined as ℓ = p1 × p2 = S(p1)p2
A point p belonging to two lines (intercept) ℓ1 and ℓ2 is defined as
p = ℓ1 × ℓ2 = S(ℓ1)ℓ2
Points at infinity or ideal points have the following representation
T p∞ = p1 p2 0
All ideal points lie on the ideal line , represented by
T ℓ∞ = 0 0 1
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 21 / 32 Three points p1, p2, p3 lie on the same line (they are collinear) iff
det p1 p2 p3 = 0
Since the duality principle holds, three lines ℓ1, ℓ2, ℓ3 meet at the same point (they are concurrent) iff
det ℓ1 ℓ2 ℓ3 = 0
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 22 / 32 Parallelism
Two lines m1 n1 m = m ; n = n 2 2 m3 n3 are parallel when the intersection point is at infinity, i.e.,
p1 p = m × n = p2 0 This reduces to the conditions m n m n 1 = 1 or det 1 1 = 0 m n m2 n2 2 2 p The direction k 1 points toward the point at infinity. p2
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 23 / 32 Summary
T T point p = p1 p2 p3 line ℓ = ℓ1 ℓ2 ℓ3 incidence pTℓ = 0 incidence ℓTp = 0
ℓ ℓ ℓ line = p1 × p2 point p = 1 × 2
ℓ ℓ ℓ collinearity det p1 p2 p3 = 0 collinearity det 1 2 3 = 0
T T ∞ point p1 p2 0 ∞ line 0 0 ℓ3
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 24 / 32 Transformations in P2
The generic projective transformation (also called homographies) in P2 is represented by a non singular 3 × 3 matrix (acting on homogeneous vectors), whose elements are
t11 t12 t13 T = t21 t22 t23 t31 t32 t33 Since the transformations are equivalent up to a scale factor, the number of dof in T is 8. According to structural constraint, we have a hierarchy of transformations, from the most general to the least one; in particular Transformations in P2
(most general) projective → affine → similarity → Euclidean (rigid)
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 25 / 32 The affine transformation must preserve the ideal line and the ideal points (as well as parallelism); hence, for any arbitrary scaling factor λ
p1 p1 λ p2 = T p2 0 0 that implies t11 t12 t13 T = t21 t22 t23 → dof=6 0 0 t33
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 26 / 32 The similarity transformation must preserve angles and ratios of lengths; that implies cos θ − sin θ t13 T = sin θ cos θ t → dof=3 23 0 0 t33 with t33 = 1. The effects of the last column in T is to produce a shear i.e., a scale variation that is non uniform in all directions.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 27 / 32 The Euclidean (rigid) transformation must preserve also lengths; that implies cos θ − sin θ t13 T = sin θ cos θ t23 → dof=3 0 0 1
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 28 / 32 Historical notes Homogeneous coordinates were introduced in 1827 by M¨obius in the context of barycentric coordinates. Given a triangle, the barycentric coordinates of a point P are the weights required at the triangle vertices such that P become the center of gravity of the triangle. T The point is computed as p = wax wby wc z , with wa + wb + wc = 1
Figure: Example of barycentric coordinates.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 29 / 32 From homogeneous coordinates to quaternions
There exist a link between homogeneous coordinates and quaternions that goes through the M¨obius transformation.
Given two complex numbers x = x1 + jx2, y = y1 + jy2 ∈ C, the ordered pair of complex numbers
˜z = x y T ∈ C2, ˜z = 0 0 T
are the homogeneous coordinates of z = z1 + j z2
x x1 + jx2 z = = z1 + jz2 = y y1 + jy2 To every pair of complex numbers (x, y) a unique complex numbers z corresponds, while for every complex number z there is an infinite number of complex pairs (projective transformation).
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 30 / 32 Transformation between homogeneous representations
x u u a b x ax + by ˜z = → w˜ = where = = y v v c d y cx + dy where a, b, c, d are complex numbers. We can compute the transformed complex number as
x u ax + by a y + b az + b w = = = x = v cx + dy c y + d cz + d
This is nothing else that the M¨obius transformation. M¨obius transformation az + b w = ; a, b, c, d, w, z ∈ C cz + d
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 31 / 32 If we take a particular form of the M¨obius transformation, due to Gauss: az + b w = −b∗z + a∗ we know that every rotation of the sphere can be expressed by a complex function as the one above, and we notice that it can be represented by the matrix of its coefficients a b α + jβ γ + jδ ∗ ∗ = −b a −γ + jβ α − jβ 1 0 j 0 0 1 0 j = α + β + γ + δ 0 1 0 j −1 0 j 0 = α1 + βi + γj + δk = a quaternion
For further details see: T. Needham, Visual Complex Analysis, OUP, 1997.
B. Bona (DAUIN) Homogeneous Coordinates Semester 1, 2015-16 32 / 32