We will start by confining ourselves to the plane, let m and m’ be two distinct lines, and let P be a point not on either of them. It is possible to correspond the points on m to the ones on m’. Define a point A on m, and let correspond that point to a point A’ on m’, the point A’ is the point where the line PA meats m’ (see fig A.1).
In this way each point on either line is assigned a unique corresponding point on the other line. This kind of correspondence is called perspective, and the points on one line being transformed to the other are said to be transformed by a “perspective transformation with center P”. if this correspondence is done n finite times, meaning that points of the line m’ be transformed to ones on m’’ with center Q and so on, then we can say that every point on m corresponds to a unique point on _{}. This kind of correspondence being done n times is called projective, and the points of m are said to have been transformed to ones on _{}by a “projective transformation”.
Similarly, in 3D space, if lines are joining every point of a plane figure to a fixed point P not on the plane _{}, than the points in which this totality of lines meets plane _{}will form a new figure. Now every point or line on _{} has a corresponding point or line on _{}. We say that the figure on _{} has been transformed to the one on _{} by a perspective transformation. If this is being done n finite times with different centers than the final figure will still be such that its points and lines correspond to the ones in the original figure. Such a transformation is again called a projective transformation in projective transformation, two such figures are not regarded as different. In other words: projective geometry is concerned with those properties of figures that are left unchanged when the figures are subjected to a projective transformation. It is evident that no properties that involve essentially the notion of measurement can have any place in projective geometry.
Definition: If _{} are n+1 points not all on the same (n1) space, and _{} is an (n1) space on _{} the class _{} of all points on the lines joining _{} to the points of _{} is called the nspace, _{}, determined by _{} and _{}. Using this definition, a plane is called a 2space (_{}), a line 1space (_{}) and a point 0space (_{}).
A projective space has homogeneous coordinates _{}, if the coordinates can be complex then the space is denoted
as _{}, and if the coordinates must be real numbers, then we denote
the space _{}. Unlike affine coordinates, the only important issue is the
ratio of the coordinates, the point referred to by a set of coordinate is not
affected by multiplying the coordinates by a nonzero constant. Another way to
think of projective space _{}, is, as mentioned before, by the set of lines through the
origin in _{}. The projective space includes all the points at infinity.
For example, in fig A.2 one can see the projective space _{} of the Euclidean
space _{}
Figure A.2: The projective space
The basic object of the study of the projective geometry was the set of solutions to an algebraic equation or system of algebraic equations. Such a set is called a variety. The variety defined by the zeros of a singe homogeneous equation forms a space with one dimension less than the projective space we are working in. such a space if often referred to as a hypersurface.
The space of all nplanes thorough the origin
in _{}is
known as a Grassmann manifold or Grassmannian and is written as _{}, therefore for rays through the origin, _{}, that is the Grassmanian of lines through the origin is
simply a projective space. For _{} we simply take the nfold
antisymmetric product of m dimensional vectors. In this way _{} is embedded in projective
space of dimension_{}. Each coordinate is given by an nXn determinant that is
the minor of the of the following mXn matrix:
(A.1) _{}
For example: the
_{} is the Grassmanian of
2planes in _{}. That is each matrix is consist of two 4 elements
vector:
(A.2) _{}
This is the Plucker embedding, and the projective space is:
_{} Therefor the projective space is _{}.
Not every combination of the coordinates corresponds to planes. For _{}, the “Klein quadric”, only Plucker coordinates satisfying the quadratic relation (A.16) corresponds to 2planes. This relation is the Pfiaffian, that is the square root of the determinant of an antisymmetric matrix:
(A.3) _{}
notice that _{}
in general, the topology of a Grassmannian manifold is given by the fact that we can realize _{} as the quotient of a pair of groups:
_{}
Meaning that the Grassmanian manifolds are homogenous spaces.
Let au define a right hand coordinates system, where the z axis is
pointing out of the paper plane towards the reader (Fig A.3). We will also
define a line vector (say a force) _{} from _{}to _{}whose directionratios
are L and M so that:
(A.4) _{}
Let define the moment of _{} about the z axis as R, there for:
(A.5) _{}
in the positive sense about the zaxis.
Figure A.3: The coordinates of a line in a plane
Assigning
equation (A.4) in (A.5) will give:
(A.6) _{}
Equation (A.4) and (A.6) can be obtained by three determinants of a
single matrix defined as:
(A.7) _{}
The direction or gradient of the line ig given by:
(A.8) _{}
and the shortest distance from the line to the origin is given by:
(A.9) _{}
Sometimes it is easier to consider the unite
vector along the line, and then write the coordinates of the line as l, m and r
with _{}.
Whether or not a line has unit values as
coordinates, L, M and R are not homogeneous coordinates. Establishing a
homogeneous coordinates system to a line can be done easily by rewriting matrix
(A.7) as:
(A.10) _{}
Then, starting from the right this time we can get:
(A.11) _{}
Having driven line coordinates in an
orthogonal xyz system can be done easily by a
third direction ratio N to (A.4):
(A.12) _{}
The moments about the x,y and z axis are P,
Q, and R respectively and are calculated as followed:
(A.13) _{}
In homogeneous coordinates the line can be represented as (see_{},eq A.2):
(A.14) _{}
And the plucker’s coordinates of the line will be:
(A.15) _{}
The six coordinates of the line can be
written as (L,M,N,P,Q,R) or, to emphasize homogeneity, as _{}.
Not all points of _{}, however represent
lines. For a line _{} being the line joining two points and V being
its moment about the origin, therefore _{}. In terms of
Plucker’s coordinates of the line, this gives:
(A.16) _{}
This is a homogeneous equation of degree 2,
and hence its solutions lie on a four dimensional quadric hypersurface denoted
as _{} in _{} (see table 2). This quadric is called “Klein quadric”
[Klein, 1871].
Points of _{} not on the Klein quadric cannot be line
presentation. On the other hand, no line in _{}has _{}=0 (we can have V=0
as line through the origin). There is a twodimensional plane in the quadric,
define by _{}=0 whose points are
not real lines, they are usually referred as line at infinity.
To summarize, lines in _{}are in oneto one
correspondent to points on a fourdimensional quadric, excluding the points on
a 2plane.
Finally, suppose the plucker’s coordinates of
a line are given, how do points in 3D that lie on the line can be found? Let _{} and V be the direction and moment of the
line, then if q is a point on the line it must satisfy:
(A.17) _{}
Let _{} and _{}, and for the point _{} . Parametrically q can be described as:
(A.18) _{}
Line coordinates can be represented as two vectors _{}, therefore (A.16) can be written as:
(A.19) _{}
Which is a quadric equation in _{}.
Tables 1 and 2 state some properties and accepted terminology that
apply to varieties in _{} of degree 1 and
2.


Dimension of the linear subspace 
Number of linear equations (conditions) 
1(a) 
Hyperplane 
4 
One 
1(b) 
Linear subspace (identifiable with _{} 
3 
Two 
1(c) 
Plane (identifiable with _{}, i.e a plane in _{}) 
2 
Three 
1(d) 
Line(identifiable with _{}, i.e a line in _{}) 
1 
Four 
1(e) 
Point 
0 
Five 
Note 1:
Two, three, four pyperplanes [1(a)] in general intersect respectively in (i)
a linear subspace of dimention 3 [1(b)], (ii) a plane [1(c)], (iii) a line
[1(d)]. Note 2: A
Hyperplane and a plane in general intersect in one line [1(d)]. Note 3: Two Planes [1(c)] do not in general intersect. 
Table A.1: Varieties of degree 1 in projective space _{}[Hunt, and Gibson, 1990].


Dimension of the quadric 
Number and degree of equations (conditions) 
2(a) 
Hyperquadric 
4 
One (degree 2) 
2(b) 
Quadric 
3 
One (degree 2), one linear 
2(c) 
Quadric (identified with a quadric surface in _{}) 
2 
One (degree 2), two linear 
2(d) 
Conic (identified with a conic in _{}) 
1 
One (degree 2), three linear 
2(e) 
Two points 
0 
One (degree 2), four linear 
Note 1: A hyperquadric [2(a)] in general
intersects a hyperplane [1(a)], a linear subspace of dimention 3 [1(b)], a
plane [1(c)], a line [1(d)], respectively in (i) a Quadric of dimention 3
[2(b)], (ii) a quadric of dimention 2 [2(c)], (iii) a conic [2(d)], (iv) two
points. Note 2: A
nonsingular hyperguadric [2(a)] wholly contains two three parameter
families of planes that are its generators. 
Table A.2: Varieties of degree 2 in projective space _{}[Hunt, and Gibson, 1990].
Each rigid body’s infinitesimal transformation can be expressed as a combination of a rotation relative to a fixed axis and a translation along the same axis. This motion combined by a rotation and a translation is called a screw displacement or twist. The axis relative to the rotation and translation is carried out is called the screw axis of the displacement. Other parameter defined by this motion is called the pitch that is defined as the ratio translation to rotation. To be more specific, if we denote the pitch as p then:
(A.20) _{}
For finite displacements, and:
(A.21) _{}
For infinitesimal displacement. In order to represent a screw _{} it is more
convenient to use a screw coordinate system composed of two vectors [Ball,
1900; Beyer, 1958; Dimentberg, 1965; Roth, 1984]:
(A.22) _{}
Where, s, is a unit vector along the direction of the screw axis and _{}, is a position vector of any point located on the screw
axis. The vector _{} is the moment of
the screw axis relative to the origin of a reference frame. Since both the
screw axis and the moment are directed at right angles to one another, their
dot product is identically zero:
(A.23) _{}
Hence only five of the six coordinates are independent (see Plucker coordinates and Klein Quadric (A.16), and (A.19)). Another explanation to that is: five independent quantities, four for the axis and one for the pitch can uniquely specify A screw.
For a revolute joint, _{}meaning that the screw is reduced to:
(A.24) _{}
For a prismatic joint, _{}, meaning that the screw is reduced to:
(A.25) _{}
As can be seen from equation (A.22)(A.25) the Plucker coordinates of a
line can be related to the screw system as follows:
(A.26) _{}
Applying Plucker’s condition to (A.26) gives:
_{}
Meaning that a revolute joint (_{}) is represented as a line in Plucker coordinates or as a
point on the Klein Quadric, and a prismatic Joint (_{}) is represented as a line at infinity (A.25) contained in a
plane perpendicular to the line represented at (A.24). Of course the magnitude
of the vector along such a line in the plane at infinity must be taken to be
zero, r at the best infinitesimal By virtue of the first three of its line
coordinates being zero but its moment about the origin can be regarded as
meaningful and finite [Hunt, 1978].
The displacement of a rigid body can not be completely determined
without the specification of the amplitude or intensity of the screw axis is
specified. Let _{} be the intensity
of a twist, then the twist can be expressed as:
(A.27) _{}
Where _{}for a revolute joint, and _{} for a prismatic
one.
The infinitesimal motion of a rigid body can generally be considered as
the resulting motion of several instantaneous screws of arbitrary pitches. To
be more specific, the motion of a rigid body with n degrees of freedom _{} , can be
generally described by an nsystem. For instance: the screw system that permits
a rigid body to move instantaneously with 1 DOF consists of a single screw
(1system) [Hunt, 1978].
(A.28) _{}
A unit wrench is defined as:
(A.29) _{}
Where _{} is a unit vector
in the direction of the screw axis and _{}is a position vector of any point on the axis. The vector _{} defines the
moment of the screw axis about the origin. For a pure force _{}: (see the definition of line)
(A.30) _{}
For a pure couple _{}: (see the definition of line at infinity or a prismatic
joint at the previous chapter)
(A.31) _{}
A wrench of intensity _{} can be expressed
as:
(A.32) _{}
Note that the first three components of the wrench represent the force
f, and the last three represent the resulting moment due to the combined
effect of the force f and the couple c about the origin of the
reference frame.
The concept of reciprocal screw was first
introduced by Ball (1900), Waldron (1969), Hunt (1970,1978), Roth (1984), and
others. The main idea of this concept is that if a wrench acts on a rigid body
in such a way that that it produced no work while the body undergoes an
infinitesimal twist, then both screws representing the twist and the wrench are
said to be reciprocal to each other.
As can be seen in fig 4, two screws are acting on a rigid body, the
first screw representing a wrench: _{} and a twist _{}. The Virtual work performed between the twist and the wrench
is given by:
(A.33) _{}
(the first three components of the twist are angular velocity and the last three are linear velocity of a point that is instantaneously coincident with the origin of the reference frame).
Applying basic algebra to (A.33) gives:
(A.34) _{}
From fig 4, we know that:
(A.35) _{}
(the dot product of two unit vector)
and also from fig A.4:
(A.36) _{}
Where a is a vector along the common perpendicular from _{} to _{}, and _{} is the angle
between the axes of _{}and _{}(from _{}to_{}) about a, according to the right hand rule.
Substituting (A.35) and (A.36) to (A.34) gives:
(A.37) _{}
By definition the virtual work should be equal to zero, since usually _{}and _{}are not zero, the reciprocal condition is reduced to:
(A.38) _{}
_{}is the virtual coefficient between the twist and the wrench
and is independent of the coefficient of the two. Now if _{} then the wrench
will not produce any work while the body undergoes an infinitesimal twist, more
over, since _{}is symmetrical in _{}and _{}, then the two screws can be interchanged without effecting
their mutual reciprocal product.
Let define the transpose of a screw as:
_{}
Therefor the orthogonal product of two screws is given by:
_{}
Then the reciprocal condition can be stated as:
(A.39) _{}
Equations (A.38) and (A.39) impose one geometric constraint on the two
reciprocal screws. If we take into consideration that a unit screw is defined
by five independent parameters (for location and pitch), then there is a
quadruple infinitude _{} of screws reciprocal
to a given screw. Thus all screws that are reciprocal to a single
screw form a 5system in 3D space.
Example:
In fig 5, a body is constraint about an instantaneous screw axis, L,
its instantaneous angular velocity is _{} and its
transnational velocity is _{} the pitch, h, is
defined by (A.21) as _{}.
The screw L’ contains a wrench having a pitch h’; intensity F, then its
couple is given by (A.28) as _{}. We will also define the shortest distance between L and L’
as r and the angle between the two as _{} (according to
the right hand rule).
By taking all the parameters into consideration and implying it to
(A.37) gives:
_{}
Replacing _{}and c by the two pitches h and h’ gives:
_{}
now according to (A.38), we get:
(A.40) _{}
and the two screws are reciprocal to each other. (A.38) and (A.40) are
the reciprocity condition and each is the fundamental equation for the entire
theory of screw systems (Ball, 1900). Now Timerding (1908) gives a concise
account of reciprocity in screw systems, in his book he is doing it by starting
with the fundamental work of MÖbius.
Given a screw L with a pitch h and any line s’, r and _{} can be determine
according to (A.40) so that both s’ and L would be reciprocal to each other.
Therefore the pitch of a single screw L’ (reciprocal to L), lying along s’ is _{}. Table A.3 summarizes all the instances of reciprocal screws
according to (A.40).
For example (from table A.3): if r=0 and _{}, _{}is indeterminate and h’ can take any value. When _{} the screw L is
no longer confined to a line but is characterized by a direction, and
reciprocal screws of all pitches lie along all lines perpendicular to this
direction.
We have already shown how to represent a
screw of infinite pitch using Plucker line coordinates. Each line (screw) is
represented as a point in _{}contained in a Quadric (Klein Quadric). Evidently, screws
with nonzero pitches, are represented by other points in _{}, those points are laying on other Quadrics. Hence each
Quadric in _{} represents a
group of screws with a characterized pitch.
Now if we write eq (A.26) as [Hunt and Gibson,1990]:
(A.41) _{}
Then _{} define a pitch
hyperquadric in _{}. To a general point $ on _{}, corresponds in _{} a screw $ of
pitch p. when p=0, _{} becomes the
Klein quadric _{}, and so a general point on _{}, corresponds to a line in _{}. When _{} (A.26) becomes:
(A.42) _{}
A screw of infinite pitch in _{}has its corresponding point in _{} on _{}.
The totality of pitch hyperquatrics_{}, is now given as a linear combination of _{} and _{}, namely, by a pencil of hyperquatrics:
(A.43) _{}
for a particular _{} _{} for _{}, _{} and for _{}, _{}. The hyperquadric _{} fill _{}, so as long as p is finite every point in _{} lies on one and
only one _{}, and corresponds to a screw of pitch p in _{}. A point corresponding to a screw with infinite pitch lies
on every _{} since (54) is a
real plane of dimension 2 in _{}(defined by three linear equations _{}, see case 1(c) table A.3).
In the next chapter screw systems will be presented [Hunt, 1978], not all cases will be discussed but only the relevant ones.
One system (or first order system) consists
of one screw, which can be placed along the x axis and called the one system
principal screw _{}with pitch _{}. The pitch, _{}, can be provided with any finite value, this line have as
its Plucker coordinates, all parameters as zero but _{} which is equal
to:
(A.44) _{}
(From the Klein Quadric equation it follows that _{}.)
As we have five linear equations in five independent variables it will
have a unique solution, and we conclude that there is only one screw reciprocal
to five given screws.
This system comprise _{}screws, that is all possible resultants of twists about any
two screws linearly independent (not both belong to the same onesystem). For
two lines there are only two possibilities to be linearly dependent [Dandurand,
1984]: either they lie on the same axis and have the same finite pitch, or they
both have infinite pitch and are parallel to one another. The general ruled
surface on which the screws of the twosystem lie is the cylindroid which has
two principal screws a and b along the x and y axes, respectively. And the two
screws have different finite pitches _{}. This system is defined by Plucker coordinates as:
(A.45) _{}
For every h between _{} two real
conjugate screws exist placed at equal distant on either side of the central
plane (xy plane) of the cylindrid. For _{} there exist only
one screw, the principal screw. As (A.45) have four equations in five
independent unknowns of a screw there will is _{}screws reciprocal to four given screws. Taking the moment
into account there yields a twodegree of freedom system.
There
are three equations with five unknowns describing this system, therefore there
are two free variables for this system, and there are _{}screws reciprocal to three given screws. Taking into account
the magnitude of the twists or wrenches we have a threedegree of freedom
system. The system’s screws are ones whose axes make up a (3,2) congruence of
lines. As mentioned, the system has three principal screws _{}of pitches _{}, along the x, y, z axes respectively. Moreover, the system
does not contain screws with pitch grater then the largest or less than the
least of _{}. Each pitch h has a characteristic regulus on a hyperbeloid
described by:
(A.46) _{}
In Plucker coordinates:
(A.47) _{}
it is convenient to take _{} for the
principal screws.
This
system comprised two equations in five independent variables since we are using
screw coordinates. Therefore we have _{} screws
reciprocal to two given screws. If we take into account the magnitudes of the
twists or wrenches we have four degree of freedom, hence a socalled
foursystem. The screws of a four
system are reciprocal to those of a two system. A screw _{}of pitch _{}lying along the zaxis is reciprocal to all three screws _{}see equation (A.40).
[Ball,1900], based hi system of screws on six coreciprocal screws _{} whose pitches
are _{}, respectively. In effect of that four of these screws will
be accepted for a four system and later five of them to fivesystem. For
segmentation, screws from afoursystem shall retain the primes on the symbols h
viz. h’ etc.
In Plucker coordinates the screws of a four system are given by:
(A.48) _{}
the difference between this system to twosystem (A.45) is that both _{} are not
restricted, hence not necessarily zero.
The
fifthorder screw system ordinarily comprises _{}screws, since there is only one condition to be satisfied by
a reciprocal screw and five parameters (four for the axis and one for the
pitch). Hens there are _{} screws
reciprocal to a given screw. In Plucker coordinates a five system is
represented by a single equation:
(A.50) _{}
Where _{}is its single principal screw along the xaxis. The rest of
the Plucker coordinates are not restricted. For a given value of h’ equation
(A.50) represents a linear complex of pitch _{}with central axis along Ox. In its general form the pitch of
the principal screw _{} is finite.
Accordingly the screws of pitch _{} lie along all
lines of a special linear complex namely the _{}lines intersecting the xaxis. Also when _{} the screws of
infinite pitch lie along all lines in all planes perpendicular to the xaxis,
and can be represented by all screws linearly dependent on two screws of
infinite pitch, (0,0,0;0,1,0) and (0,0,0;0,0,1).
For more details on screw systems and special cases of screw suystem,
information can be found in Hunt [Hunt (chapter 12),1987].
The screws and reciprocal screws associated with some frequently used joints are listed below:
Revolute joint:
The
unit screw associated with this kind of joint is one with zero pitch pointing
along the joint axis. The reciprocal screws form a five system (see
fivesystem). In particular, those zeropitch reciprocal screws lie on all
planes containing the axis of the revolute joint.
Prismatic joint: the unit screw associated to
this joint is one with infinite pitch pointing along the sliding direction. The
screws reciprocal to this screw, also forma fivesystem, and the zeropitch
reciprocal screws lie on all planes perpendicular to the axis of prismatic
joint.
Spherical joint: the unit screws associated
to this joint form a threesystem of zeropitch screws passing through the
center of the joint. The reciprocal screws also form a threesystem of zero
pitch passing through the center of the sphere.
Universal joint: the unit screws associated
with this joint form a twosystem of zero pitch. It is a planar pencil
radiating from the center of the universal joint and lying on a plane that
contains the two axes of revolution. Th reciprocal screws form a foursystem.
All zeropitch reciprocal screws either pass through the center of the
universal joint or lie on a plane defined by the axes of the universal joint.
Furthermore, there exists an infinitepitch reciprocal screw that passes through
the center of the universal joint and is perpendicular to both joint axes.