Wolfgang Förstner
It will be shown: the vector (L1, L», L3) is the direction of the line and the vector ( L4, L5, Le) is the normal of the plane
through the line and the origin. The Plücker condition expresses the orthogonality condition of these two vectors.
Incidence of two objects can use inner products, namely for points xand lines l in the plane, for points X and planes =
in 3D-space and for pairs (L, M) of 3D lines
<x,1>=x"1=x1=0 < X, A >= X'A = X-A =0 < L,M >= L'M = L-M =0
The first 2 relation result from the definition of the 2D line and the plane in their Hessian form. The last relation will be
proved below.
We can construct 2D lines / as join A of two points x and y and points x as intersection M of two lines 1, and 1,
]-x^Ay-—xxy x=1}nih=hx<hb
We also can construct 3D-lines L as join L — X A Y of two points or as intersection L = A N B of two planes, defined
via the dual line L = ANB= AAB
L-XAY-A(X)Y--A(YX L-AnB-AAB-A(A)--A(B)P
with the matrix at the same time being a Jacobian
T u0: dio
i Qon T 19 -V
LAY) dqo90. O0 oT ^W
AS 8y 7] 0 5 -W- Ve 09
6x4 W 0 -—-U 0
-V U 0 0
using the convention homogeneous matrices to be upright sans serif letters. Observe A(X)X = 0, VX and rkA(X) = 3.
The line coordinates obviously are bilinear in the homogeneous coordinates for the points and for the planes. Setting the
fourth coordinate of the two homogeneous vectors to 1, we find (L1, L2, L3)" — Y — X and (L4, Ls, Le): =X xY
with the Euklidean coordinates X and Y of the two points indicated with slanted bold face letters. The relations for the
planes exploit the duality of points and planes in 3D, specifically the duality of the join A and the intersection N.
We also obtain the plane coordinates as the join of a point and a line and the intersection of a line and a plane
A-XAL-A'(XL--B(LX X-AnL-A'(A)L- -B(L)A
with the Jacobian
0 La — La —L4
SOLAR), QUAL Lara 0. oly omLsul og onnt T
Ba X pan amer, SA
L4 Ls; Ls 0
where the last expression is valid for the line to be given by the intersection of two planes L — A (1 B. Observe
B(L)B(L) — 0 and rkB(L) — 2. The expressions for A 2 X AL and X = A 1L are consistent, as e. g. X € A due to
< X, A >= XTA"(X)L = 0 and L € A due to (X AL)NL = (-B)(D)(-B(L)X) = 0.
We now can prove the condition < L, M >= 0 for two lines to intersect. Let L be given as the join L = X ^ Y —
—A(Y)X. Then the intersection condition is equivalent to the condition the point X to lie in the plane A — M ^ Y —
A'(Y)M which leads to XTA = (XTAT(Y)) M 2 -L'M - 0.
We finally need conditions for two lines to intersect, in case they are given by two points or two planes.
(Xi,X4,X,X4|2-0 |A,A5A,A4|2-0 (XAY)n(AnB)-2 X'(AB'-BA')Y -0 (D
The first and second condition results from the coplanarity of the points or from the intersection condition for four planes.
The last condition uses X'R. — X'B(L)Y = 0 where the plane R. 2 L A Y is the join of L — A B and Y.
3 PROJECTION
31 POINTS
The projection of a 3D point P(X) onto the image plane yields the image point p'(x') via a direct linear transformation
(cf. Fig. 1).
X zPX or (u,v,w) z(L2,3) X 9(E-X,2X,9:X) "wi PD KR X;)
where (-]-) denotes concatenation. The 3 x 4 projection matrix P can be explicitely related to the 6 parameters of the
exterior orientation and 5 parameters of the interior orientation namely the Euclidean coordinates X , of the projection
centre O(X ;), the rotation matrix À, the principle distance c, the coordinates (x";, ÿ;;) of the principle point, the shear
5 and the scale difference of the z'- and the y'-coordinates. The parameters of the interior orientation are collected in the
3 x 3 calibration matrix
298 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
