Appendix A

Line Geometry and Projective geometry

 

A1.    Projective and metric geometry - an introduction

 

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).


 


Figure A.1: perspective

 

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 3-D 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 (n-1) space, and  is an (n-1) space on  the class  of all points on the lines joining  to the points of  is called the n-space, , determined by  and . Using this definition, a plane is called a 2-space (), a line 1-space () and a point 0-space ().


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 non-zero 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. 

A.2.   Grassmannians

The space of all n-planes 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 n-fold anti-symmetric 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 2-planes 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 2-planes. This relation is the Pfiaffian, that is the square root of the determinant of an anti-symmetric 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.

 

 

 

A.3.   Plucker’s line coordinates

 

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 direction-ratios 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 z-axis.


 


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 two-dimensional 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 one-to one correspondent to points on a four-dimensional quadric, excluding the points on a 2-plane.

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 non-singular 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].

 

A.4    Screw coordinates

 

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 n-system. For instance: the screw system that permits a rigid body to move instantaneously with 1 DOF consists of a single screw (1-system) [Hunt, 1978].

A.4.1   Wrenches and reciprocal screws

 

Just like in the previous chapter where we represented a finite or infinite motion (displacement) of a rigid body in 3D space, by using the concept of twist, for static a similar concept can be defined. As we all know from basic static courses, it is possible to reduce a system of forces and couples to a resultant force and couple about any point of interest.  In general to resultant force and couple are not collinear, however, one can show that it is possible to fined a unique axis of which both the resulting force f and the resulting couple c could be acted along. This force-couple combination is called a wrench (Roth, 1984; Yuan et al, 1971). The unique axis is called the wrench axis or screw axis of the system of forces and couples. The pitch of the wrench is defined as the ratio of the couple to the force:

(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.

 

Figure A.4. Twist and wrench in 3D-Space

 


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 5-system 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 . 


 


Figure A.5: System of two screws

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.

 


Table A.3: Some reciprocal screws

 


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 non-zero 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).

 

 

 

 

A.5    Screw systems

 

In the next chapter screw systems will be presented [Hunt, 1978], not all cases will be discussed but only the relevant ones.

A.5.1   The one-system (one DOF): Screws Reciprocal to Five screws

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.

A.5.2   The Two-system (Two DOF):  Screws Reciprocal to Four screws 

This system comprise screws, that is all possible resultants of twists about any two screws linearly independent (not both belong to the same one-system). 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 two-system 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 (x-y 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 two-degree of freedom system.

A.5.3   The three-system (Three DOF):  Screws Reciprocal to three screws 

 

            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 three-degree 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.

 

A.5.4   The Four-system (Four DOF):  Screws Reciprocal to two 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 so-called four-system.  The screws of a four system are reciprocal to those of a two system. A screw of pitch lying along the z-axis is reciprocal to all three screws see equation (A.40).  [Ball,1900], based hi system of screws on six co-reciprocal 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 five-system. For segmentation, screws from afour-system 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 two-system (A.45) is that both  are not restricted, hence not necessarily zero.

 

A.4.5   The Five-system (Five DOF):  Screws Reciprocal to one screws 

 

            The fifth-order 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 x-axis. 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 x-axis. Also when  the screws of infinite pitch lie along all lines in all planes perpendicular to the x-axis, 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].

 

A.6    Reciprocal Screws of Some Kinematic Pairs

 

            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 five-system). In particular, those zero-pitch 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 five-system, and the zero-pitch reciprocal screws lie on all planes perpendicular to the axis of prismatic joint.

 

Spherical joint:   the unit screws associated to this joint form a three-system of zero-pitch screws passing through the center of the joint. The reciprocal screws also form a three-system of zero pitch passing through the center of the sphere.

 

Universal joint:   the unit screws associated with this joint form a two-system 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 four-system. All zero-pitch 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 infinite-pitch reciprocal screw that passes through the center of the universal joint and is perpendicular to both joint axes.