of points of R

or

is a system of four points of the plane {A, B, C, D} represented by corresponding vectors {[a], [b], [c], [d]}

of R

(i) {a,b,c} build a basis of R

(ii) d = a + b + c.

The projective base defines a projective coordinate system in

which we write X = u[a] + v[b] + w[c], {u,v,w} being defined modulo a non-zero

multiplicative constant, and resulting from the corresponding representetion of [x] = X, through

the basis vectors: x = ua + vb + wc.

We write simply X = uA + vB + wC (meaning a weighted sum of points of the projective plane).

{u,v,w} are the

In particular, points {A,B,C,D} have correspondingly the coordinates {(1,0,0), (0,1,0), (0,0,1), (1,1,1)}.

{A,B,C} are often called the

D is often called the

of the triangle defines a homogeneous system of coordinates. The mechanism to do this is

simply described by representing the triangle through coordinates in space R

the plane z=1. Thus, the vertices of the triangle and D being represented through the lines

(= points of P

The first three column vectors are those used in the definition. The fourth is determined by solving

the system:

This system admits a unique solution (by the non-collinearity assumption) and determines

the vectors necessary to find the homogeneous coordinates according to the definition.

In fact, the vectors to be used are:

To find the homogeneous coordinates in this system for a point X represented by a non-zero vector

X=[

(0)

The triple (u,v,w) gives the homogeneous coordinates of X=[

defined by the triangle ABC and the coordinator (or unit) point D. Point D has the coordinates (1,1,1)

as required.

There are two particular systems of projective coordinates worth mentioning for their use in the

geometry of the triangle: (i) the barycentric homogeneous coordinates (see BarycentricCoordinates.html ),

and (ii) the trilinear homogeneous coordinates (see Trilinears.html ). The barycentric system for the

triangle of reference ABC results by adopting for unit the centroid G = (1/3)(A+B+C) of the triangle.

The trilinear or

the incenter I of the triangle of reference.

The barycentric system is in some sense universal and simpler than the others since in this case the

coefficients in (*) are equal and can be taken 1. In addition to the resulting simplification, the barycenter

is an affine invariant.

Note also that in (*) used to define the system through the unit D = (d

coefficients:

are the

system of homogeneous coordinates this triple is (a,b,c), where the numbers are the lengths of the corresponding

sides of the triangle of reference.

cartesian (some times referred to as

homogeneous coordinates using the basic matrix defined by the vertices of the triangle and equation (0).

It is interesting here to notice that

with x

Points

satisfying the equation:

The inverse relation is also interesting to note. The formula below is calculated in BarycentricsFormulas.html .

S denotes here the area of the triangle of reference ABC.

results from the previous by multiplying the appropriate matrices:

In particular keeping fixed the triangle ABC and changing only D to D' gives for the corresponding

transition functions relating the two systems of homogeneous coordinates the simple rule:

coordinates of the point the tripple (x,y,z). It is a projective system in which the line at infinity is z=0.

This system can be considered as one example of those described in section-2 for which the triangle of reference has

one side identical with the line at infinity (z=0). Some of the assertions for general systems are not valid in this case.

For example the coordinates of the unit are (1,1,1) but these are not the barycentric coordinates with respect to the triangle

of reference, since this is infinite.

cartesian systems and transforming by the transformation rule of section-3. Thus, the line represented in cartesian

coordinates by an equation of the form ax+by+c=0 goes over to homoneous ax+by+cz=0, which transforming the

coordinates by an invertible matrix (x,y,z)

a'x'+b'y'+c'z'=0. Analogously is shown that a quadratic equation transforms to a quadratic equation etc..

whose equations are x

we obtain z((a-a')x+(b-b')y+(c-c')z)=0. This represents the product of two lines: the line at infinity (z=0) and the

radical axis in homogeneous coordinates ( (a-a')x+(b-b')y+(c-c')z=0). By transforming to a general homogeneous system

the expression x

terms transform to a product of lines L(x',y',z')L

Thus, in the general coordinate system, in which the equation of the circle is written as a sum Q

in which Q

the difference L-L'=0 of the linear parts of the equations of two circles represents the radical axis of these circles.

See Remark-1 in section-2 of BarycentricsFormulas.html [LoneyII, p. 69].

projective plane. The basic rule is that point X and point rX are the same. If (x,y,z) are the (homogeneous)

coordinates of X with respect to some fixed system, then rX can be considered to have coordinates

(rx,ry,rz) which define the same point. Every point of the plane can be represented as:

(1) X = uA + vB + wC.

Every point of the line defined by points {X,Y} can be written in the form:

(2) P

An equation of the form pX + qY = p'X + q'Y denotes a point Z which is common to the two lines

defined by {X,Y} and {X',Y'}: Z = pX + qY = p'X + q'Y'.

In a combination Y = t

the resulting point (tt

If points {X

(3) t

then for every other point X there are three numbers, defined up to multiplicative constant, such that:

(4) X = t

Often using such representations of points and these rules of computation one can give simple proofs

of seemingly complicated theorems. As an example let us prove Desargue's theorem (see Desargues.html ):

they are also

same point D.

at three collinear points on a line d.

Assume now that ABC, A'B'C' are point-perspective and there is a point D such that:

(i) D = aA+a'A' (is on line AA'), (ii) D = bB+b'B' (is also on BB'), (iii) D = cC+c'C' (is also on CC').

Thus aA + a'A' = bB+b'B' => aA-bB = -a'A'+b'B = X is a point on both lines AB and A'B' hence it is

their intersection. Analogously bB-cC = -b'B'+c'C' = Y is the intersection of BC and B'C' and

cC-aA = -c'C+a'A' = Z is the intersection of CA and C'A'. But X+Y+Z = (aA-bB)+(bB-cC)+(cC-aA)=0

implies that the three points are collinear.

their cross ratio is the number (see ProjectiveBase.html ):

(5) (A,B,C,D) = [(a-c)/(b-c)]:[(a-d)/(b-d)].

Four points are said to form a

Given three points {A,B,C} the

to {A,B} is a point D, denoted some times with D=(A,B)/C and satisfying (A,B,C,D)=-1. This relation of

harmonicity is handled in section-3 of ProjectiveBase.html , where it is seen that the harmonic conjugate

of Z = X+zY with respect to {X,Y} is Z'=(X,Y)/Z = X-zY.

An interesting construct related to cross-ratio and harmonicity is the

respect to a triangle, handled also in the aforementioned file.

by (0). First the intersection points of the axes {u=0, v=0, w=0} with this line are correspondingly points:

According to the previous section the harmonic conjugates {A''=(B,C)/A', B''=(C,A)/B', C''=(A,B)/C'} are:

Lines {AA'', BB'', CC''} intersect at a point E, which can be located by solving the following equations

with respect to {x,y,z}:

For example the first equation is equivalent to:

xA + A'' = k(yB+B''),

since multiplication by arbitrary constants is allowed. Replacing in this the values for A'', B'' gives:

By the independence of {A,B,C} this implies the vanishing of the coefficients, hence:

Introducing this into the expression for the common point E = xA+A'' we obtain:

This, by the invariance for multiples with scalars, is equivalent to (see Trilinears.html , section [18]):

(5) ax + by + cz = 0.

They can be also described in parametric form using the projective basis of the system (A,B,C):

(6) X

Higher degree curves such as conics, cubics, quartics, quintics etc.. are of interest in projective geometry.

Analogously to (5) they are defined by homogeneous functions of the three homogeneous coordinates

(x,y,z) through equations such as:

(7) ax

(8) ax

Some properties of the conics expressed through homogeneous coordinates are discussed in

Conics_In_Homogeneous.html .

BarycentricCoordinates.html

Trilinears.html

BarycentricsFormulas.html

Desargues.html

Conics_In_Homogeneous.html

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