1. C OMPUTER V ISION : P ROJECTIVE G EOMETRY 3D
IIT Kharagpur
Computer Science and Engineering,
Indian Institute of Technology
Kharagpur.
(IIT Kharagpur) Projective Geometry-3 Jan ’10 1 / 15

2. The projective geometry of 3D P3
A point X in 3-space is represented in homogeneous coordinates as:
 
 X1 
 
T
 
 X2 

X=

 X3  = X1 X2 X3 X4
 

 

 

 
X4
A projective transformation acting on P3 is a non-singular 4 × 4
matrix.
X = HX
The matrix H has 15 degrees of freedom.
The map is a collineation (lines are mapped to lines) which
preserves incidence relations such as intersection point of a line
with a plane, order of contact.
(IIT Kharagpur) Projective Geometry-3 Jan ’10 2 / 15

3. Planes
A plane in 3-space may be written as
π1 X + π2 Y + π3 Z + π4 = 0
This equation is unaffected by scalar multiplication.
The homogeneous representation of the plane is the 4-vector
π = (π1 , π2 , π3 , π4 )T
Homogenizing by replacements:
X → X1 /X4 , Y → X2 /Y4 , Z → X3 /X4
π1 X1 + π2 X2 + π3 X3 + π4 X4 = 0 πT X = 0
The normal to the plane is given by: n = (π1 , π2 , π3 )T
(IIT Kharagpur) Projective Geometry-3 Jan ’10 3 / 15

4. Join and incidence relations
A plane is defined uniquely by the join of 3 points, or the join of a
line and a point, (in general position).
Two distinct planes intersect in a unique line.
Three distinct planes intersect in a unique point.
(IIT Kharagpur) Projective Geometry-3 Jan ’10 4 / 15

5. Three points define a plane
A point Xi incident  T 
on a plane π would  X1
 

satisfy πT Xi = 0

 


 

π = 0
 T
 X 


 2 

 


 


 T 

X3
 
This is a 3 × 4 matrix with rank 3.
The intersection
 π1
 T 
point X of 3 planes  

πi is obtained using:

 


 

 π
 T 

 2 X = 0



 


 

 
π3
 T
 

(IIT Kharagpur) Projective Geometry-3 Jan ’10 5 / 15

6. Projective Transformation
Under the point transformation X = HX, a plane transforms as:
π = H−T π
(IIT Kharagpur) Projective Geometry-3 Jan ’10 6 / 15

7. Lines
A line is defined by the join of two points or the intersection of two
planes.
A line has 4 degrees of freedom in 3-space. A line can be defined
by its intersection with two orthogonal planes.
(IIT Kharagpur) Projective Geometry-3 Jan ’10 7 / 15

8. The hierarchy of transforms
A t
Projective: with 15 dof.
vT v
A t
Affine: with 12 dof.
0T 1
sR t
Similarity: with 7 dof.
0T 1
R t
Euclidean: with 6 dof.
0T 1
(IIT Kharagpur) Projective Geometry-3 Jan ’10 8 / 15

9. Invariants P3
Projective:
Intersection and tangency of surfaces in contact
Affine:
Parallelism of planes,
volume ratios,
centroids,
The plane at infinity π∞
Similarity:
The absolute conic
Euclidean:
Volume
(IIT Kharagpur) Projective Geometry-3 Jan ’10 9 / 15

10. Comparison
In planar P2 projective In 3-space P3 projective geometry
geometry
Identifying the line at Plane at infinity π∞
infinity l∞ allowed affine
properties of the plane
to be measured.
Identifying the circular
points on l∞ allows the Absolute conic Ω∞
measurement of metric
properties.
(IIT Kharagpur) Projective Geometry-3 Jan ’10 10 / 15

11. The plane at infinity
π∞ has the canonical position π∞ = (0, 0, 0, 1)T in affine 3-space.
Two planes are parallel, if and only if, their line of intersection is on
π∞ .
A line is parallel to another line, or to a plane, if the point of
intersection is on π∞ .
The plane π∞ is a geometric representation of the 3 degrees of
freedom required to specify affine properties in a projective
coordinate frame.
The plane at infinity is a fixed plane under the projective
transformation H if, and only if, H is an affinity.
(IIT Kharagpur) Projective Geometry-3 Jan ’10 11 / 15

12. Affine properties of a
reconstruction
Identify π∞ in the projective coordinate frame.
Move π∞ to its canonical position at π∞ = (0, 0, 0, 1)T .
The scene and the reconstruction are now related by an affine
transformation.
Thus affine properties can now be measured directly from the
coordinates of the entity.
(IIT Kharagpur) Projective Geometry-3 Jan ’10 12 / 15

13. The absolute conic Ω∞
The absolute conic Ω∞ is a (point) conic on π∞ .
In the metric frame π∞ = (0, 0, 0, 1)T and points on Ω∞ satisfy
X2 + X2 + X2 

1 2 3 
( X 1 , X 2 , X 3 ) I ( X 1 , X 2 , X 3 )T = 0

=0



X  4
Ω∞ corresponds to a conic C with matrix C = I.
It is a conic of purely imaginary points on π∞ .
The conic Ω∞ is a geometric representation of the 5 additional
degrees of freedom that are required to specify metric properties
in an affine coordinate frame.
The absolute conic Ω∞ is a fixed conic under the projective
transformation H if and only if H is a similarity transformation.
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14. The absolute conic Ω∞
The absolute conic Ω∞ is only fixed as a set by a general
similarity; it is not fixed pointwise. This means that under a
similarity transformation, a point on Ω∞ may travel to another point
on Ω∞ , but it is not mapped to a point off the conic.
All circles intersect Ω∞ in two points. These points are the circular
points of the support plane of the circle.
All spheres intersect π∞ in Ω∞ .
(IIT Kharagpur) Projective Geometry-3 Jan ’10 14 / 15

15. Metric Properties
Once Ω∞ and its support plane π∞ have been identified in
projective 3-space then metric properties, such as angles and
relative lengths, can be measured.
Consider two lines with directions (3-vectors) d1 and d2 . The
angle between these directions:
In Euclidean frame In a projective frame
dT d2
1 dT Ω∞ d2
cos θ = cos θ = 1
(dT d1 )(dT d2 )
1 2 (dT Ω∞ d1 )(dT Ω∞ d2 )
1 2
These expressions are equivalent since in the Euclidean world
frame Ω∞ = I
(IIT Kharagpur) Projective Geometry-3 Jan ’10 15 / 15