..Purpose..Sets of
multiples..Homogeneous
coordinates of a point..Points at
infinity ; Ideal points..Formulas
for ideal points..Affine
and projective plane -- projective points..Dividing
ratio -- another parameter on a line..Dividing
ratio and cartesian coordinates..Dividing
ratio of the ideal point of a line a..Equation
of a line a..The ideal
line and regular lines...Line
coordinates..Duality
between points and lines..Formulas
and properties in the projective plane
..Homogeneous
equation of a curve.
Purpose
To make all
formulas about lines, points, intersection, and equations of curves far more
homogeneous, we take a more abstract point of view.
Sets of multiples
S is the set of all ordered triples (a,b,c) ; with a,b,c in R.Now, we remove the element (0,0,0). So = S \ {(0,0,0)}.
In So, we say that (a',b',c') is a multiple of (a,b,c) if and only if there is a real number r such that (a',b',c') = r (a,b,c).
In So, we denote the set of all these multiples of (a,b,c) as ||a,b,c||.
Homogeneous coordinates of a point
We choose an x-axis and a y-axis in a plane.With each point corresponds just one couple of cartesian coordinates (x,y) and with that couple just one set of multiples ||x,y,1||.
Conversely, with the set ||x,y,1|| corresponds just one couple of coordinates (x,y) of a point.
We say that each element of the set ||x,y,1|| is a triple homogeneous coordinates of that point.
Example:
Take the point P(2,3). With P corresponds the set ||2,3,1||. (2,3,1); (20,30,10); (4,6,2) are different homogeneous coordinates of the same point P. So, P has an infinite number of homogeneous coordinates.
Knowing a triple homogeneous coordinates, it is easy to find the cartesian coordinates.
Points at infinity ; Ideal points
Point Po(xo,yo) is a fixed point of the line a.Suppose, the vector v(a,b) is a direction vector of that line.
Then, for point P different from Po, we have:
P(x,y) is on line a
<=>
There is a real number r such that
x = xo + r.a and y = yo + r.b
<=>
P has homogeneous coordinates (xo + r.a, yo + r.b, 1) with r not 0.
<=>
P has homogeneous coordinates
xo yo 1
(--- + a,--- + b,---)
r r r
If r grows infinitely, P recedes indefinitely along that line and the
homogeneous coordinates of P approach to (a,b,0).But (a,b,0) are not the homogeneous coordinates of a point!
Now, we add to the line a one extra point. We call that point 'the point at infinity of the line a' or 'the ideal point of a'.
That special point receives, by definition, (a,b,0) and each real multiple of (a,b,0) as homogeneous coordinates.
Thus with the ideal point of the line a corresponds the set ||a,b,0|| of homogeneous coordinates.
Conversely, with each set ||a,b,0|| corresponds an ideal point of a line with direction vector (a,b).
From this, it follows that parallel lines have the same ideal point and lines with the same ideal point are parallel. We say that parallel lines intersect in the common ideal point.
A point that is not ideal is called a regular point.
From this theory we deduce
Formulas for ideal points
A line with direction vector (a,b) has (a,b,0) as ideal point.The x-axis has (1,0,0) as ideal point.
The y-axis has (0,1,0) as ideal point.
A line with slope m has (1,m,0) as ideal point.
A line through P(x1,y1) and Q(x2,y2) has (x2-x1,y2-y1,0) as ideal point.
A line with equation ux+vy+w =0 has (v,-u,0) as ideal point.
Note that each non-zero multiple of these homogeneous coordinates is also a triple of homogeneous coordinates of the same ideal point.
Affine and projective plane -- projective points
The set of all regular points is called the affine plane.The set of all regular and all ideal points is called the completed affine plane.
If we make no distinction at all, between ideal points and regular points, then we call that set of points the projective plane.
If we say 'projective point', we intensify the notion that we make no difference between ideal points and regular points. A projective point can be ideal or regular!
Dividing ratio -- another parameter on a line
Take P1 and P2 as two different points on a line a, and take point P different from P2. Now, we have (vectors in bold) :
P is on line a
<=>
There is a real number k such that
PP1 = k. PP2
The real number k is called the dividing ratio of point P with respect to
(P1,P2).We denote this ratio k as (P1,P2,P).
Note that
PP1
----- = (P1,P2,P)
PP2
We say that the point P2 of the line P1P2
has infinity as dividing ratio.
Dividing ratio and cartesian coordinates
Take P1 and P2 as two different points on a line a, and take point P different from P2. Now, we have :
P is on line a with (P1,P2,P) = k
<=>
PP1 = k. PP2
<=>
P1 - P = k (P2 - P)
<=>
P1 - k P2
P = -----------
1 - k
then we have for the cartesian coordinates
<=>
x1 - k x2 y1 - k y2
x = ------------ and y = ---------
1 - k 1 - k
From this we see that a variable point P of line a has homogeneous
coordinates
x1 - k x2 y1 - k y2
( ---------- , --------- , 1 )
1 - k 1 - k
or
( x1 - k x2 , y1 - k y2 , 1 - k )
When k varies, the point P describes the line a.
Dividing ratio of the ideal point of a line a
A variable point on a line a, has homogeneous coordinates
( x1 - k x2 , y1 - k y2 , 1 - k )
with k = (P1,P2,P).
For k = 1 we have
( x1 - x2 , y1 - y2 , 0 )
This is the ideal point of the line a
So, it is obvious to say that 1 is the dividing ratio of the ideal point
of the line a.
Equation of a line a
Take a line A with equation u x + v y + w = 0.- P is a regular point of the line a
P(x,y,z) is on line a <=> P(x/z , y/z) is on line a <=> u . (x/z) + v . (y/z) + w = 0 <=> u x + v y + w z = 0 - P is the ideal point of the line a
P(x,y,z), with z = 0, is on line a <=> V(x,y) is a direction vector of line a <=> u x + v y = 0 <=> u x + v y + w z = 0 (since z = 0)
P(x,y,z) is on line a
<=>
u x + v y + w z = 0
and this is true for all points (ideal or regular).Therefore we say that u x + v y + w z = 0 is the homogeneous equation of the line a. Each non-zero multiple of this equation is a homogeneous equation of the line a.
The ideal line and regular lines.
P(x,y,z) is a ideal point
<=>
z = 0
<=>
0 x + 0 y + 1 z = 0
Any ideal point P belongs to the curve with equation z = 0.Because this equation has the form u x + v y + w z = 0, we say that z = 0 is the equation of the 'ideal line'. All other lines are regular lines.
If we make no distinction at all, between ideal lines and regular lines, then
we call that set of lines the projective lines.
If we say 'projective line',
we intensify the notion that we make no difference between ideal lines and
regular lines. A projective line can be ideal or regular!
Line coordinates
With each line a with equation u x + v y + w z = 0 corresponds exactly one set ||u,v,w||.Each element of that set is called 'line coordinates' of line a.
So, we can write : line a(u, v, w)
(u,v,w) are homogeneous coordinates of the line.
Example :
Line l has equation 5x + 3y -4 = 0.
Line coordinates of l are
( 5, 3, -4) and (-50,-30,40) and ...
Duality between points and lines
The equation of a line, u x + v y + w z = 0, is the necessary and sufficient condition for the coordinates (x,y,z) of a point, to be on the line (u,v,w).The equation of a point, u x + v y + w z = 0, is the necessary and sufficient condition for the coordinates (u,v,w) of a line, to contain the point (x,y,z).
This resemblance is called the 'duality' between points and lines.
Formulas and properties in the projective plane
Using all previous properties, we can make formulas about lines, points of intersection, and equations of lines far more homogeneous. Since we work in the projective plane, all points can be ideal or regular. We make no distinction about that. So, all points are projective points.Three points on a line in the projective plane
P1(x1,y1,z1), P2(x2,y2,z2), P3(x3,y3,z3) are projective points.
P1(x1,y1,z1), P2(x2,y2,z2), P3(x3,y3,z3) are on a line
<=>
There are numbers u,v,w , not all zero, such that
P1(x1,y1,z1), P2(x2,y2,z2), P3(x3,y3,z3) are on
the line ux +vy + w z = 0
<=>
There are numbers u,v,w , not all zero, such that
u x1 + v y1 + w z1 = 0
u x2 + v y2 + w z2 = 0
u x3 + v y3 + w z3 = 0
<=>
The homogeneous system
u x1 + v y1 + w z1 = 0
u x2 + v y2 + w z2 = 0
u x3 + v y3 + w z3 = 0
has a solution different from (0,0,0)
<=>
| x1 y1 z1 |
| x2 y2 z2 | = 0
| x3 y3 z3 |
This is the necessary and sufficient condition for
collinearity.
Three lines with common point in the projective plane
a1(u1,v1,w1), a2(u2,v2,w2), a3(u3,v3,w3) are projective lines.This means that it does not matter if these lines are ideal or regular.
a1(u1,v1,w1), a2(u2,v2,w2), a3(u3,v3,w3) have a common point
<=>
There is a point P(x,y,z) such that P is on the three lines
<=>
There are numbers (x,y,z) , not all zero, such that
u1 x + v1 y + w1 z = 0
u2 x + v2 y + w2 z = 0
u3 x + v3 y + w3 z = 0
<=>
The homogeneous system
u1 x + v1 y + w1 z = 0
u2 x + v2 y + w2 z = 0
u3 x + v3 y + w3 z = 0
has a solution different from (0,0,0)
<=>
| u1 v1 w1 |
| u2 v2 w2 | = 0
| u3 v3 w3 |
This is the necessary and sufficient condition for concurrency of the
three projective lines.
Equation of a line in the projective plane
Take a line P1P2 with P1(x1,y1,z1), P2(x2,y2,z2).
Point P(x,y,z) is on the line P1P2
<=>
P, P1,P2 are on one line
<=>
| x y z |
| x1 y1 z1 | = 0
| x2 y2 z2 |
This is the formula for the line P1P2.
Variable point of a line in the projective plane
Take a line P1P2 with P1(x1,y1,z1), P2(x2,y2,z2).
Point P(x,y,z) is on the line P1P2
<=>
| x y z |
| x1 y1 z1 | = 0
| x2 y2 z2 |
<=>
| x1 y1 z1 |
| x2 y2 z2 | = 0
| x y z |
Since P1 and P2 are different (x2,
y2, z2) is not a real multiple of (x1,
y1, z1) and from
| x1 y1 z1 |
| x2 y2 z2 | = 0
| x y z |
it follows that the third row is a linear combination of the other
rows.Therefore there are real numbers k and l (not all 0) such that
x = k x1 + l x2
y = k y1 + l y2
z = k z1 + l z2
Thus, a variable point P of P1P2 has coordinates
(k x1 + l x2, k y1 + l y2, k z1 + l z2)
The numbers k and l are homogeneous parameters.For k = 0 point P = P2. If P is different from P2, then k is not 0 and dividing by k we find for the homogeneous coordinates of P
(x1 + (l/k) x2, y1 + (l/k) y2, z1 + (l/k) z2)
Say (l/k)= h , we find for the homogeneous coordinates of P
(x1 + h x2, y1 + h y2, z1 + h z2)
The number h is a non-homogeneous parameter.
Variable line defined by two different lines in the projective plane.
a1(u1,v1,w1), a2(u2,v2,w2), a(u,v,w) are projective lines.The necessary and sufficient condition for concurrency is
| u1 v1 w1 |
| u2 v2 w2 | = 0
| u v w |
Since a1 and a2 are different,(u2,v2,w2)
is not a real multiple of (u1,v1,w1) and
therefore we have that the third row is a linear combination of the other
rows.There are real numbers k and l such that
u = k u1 + l u2
v = k v1 + l v2
w = k w1 + l w2
So, a variable line a through the intersection point of a1 and a2 has
homogeneous coordinates
(k u1 + l u2, k v1 + l v2, k w1 + l w2)
The numbers k and l are homogeneous parameters.
That variable line a has homogeneous equation
(k u1 + l u2)x + (k v1 + l v2)y + (k w1 + l w2)z = 0
<=>
k(u1 x + v1 y + w1 z) + l(u2 x + v2 y + w2 z) = 0
The numbers k and l are homogeneous parameters.Denote the homogeneous equation of line a1 as A = 0.
Denote the homogeneous equation of line a2 as B = 0.
Then the equation of a is kA +lB = 0.
The numbers k and l are homogeneous parameters.
For k = 0, the line a = a2. If line a is different from a2, then k is not 0 and dividing by k we find for the equation of line a
A+(l/k)B=0.
Say (l/k)= h , then we find for line a
A + h B = 0 .
The number h is a non-homogeneous parameter.
Example :
Line a has equation x - y + 2 z = 0 .
Line b has equation 2x
- y + 3 z = 0 .
A variable line, different from b, through the intersection
point of a and b has equation
(x - y + 2 z) + h (2x - y + 3 z) = 0
Intersection point defined by two different lines
a: u1 x + v1 y + w1 z = 0
b: u2 x + v2 y + w2 z = 0
The coordinates of the intersection point of these lines is a solution of
the linear homogeneous system
/ u1 x + v1 y + w1 z = 0
\ u2 x + v2 y + w2 z = 0
Since the lines a and b are different,
(u1,v1,w1) and
(u2,v2,w2) are not proportional and from this
we know from the theory about homogeneous systems that it is a homogeneous
system of the second kind. We can choose all the side unknowns arbitrarily. With
each choice of these side unknowns corresponds exactly one solution of the
system. Say z is the side unknown, then we can write the system as
/ u1 x + v1 y = - w1 z
\ u2 x + v2 y = - w2 z
Solving with Cramer we find:
| -w1 z v1|
| -w2 z v2|
x = ---------------- and
| u1 v1|
| u2 v2|
| u1 -w1z|
| u2 -w2 z|
y = -----------------
| u1 v1|
| u2 v2|
<=>
| -w1 v1|
| -w2 v2|
x = --------------.z and
| u1 v1|
| u2 v2|
| u1 -w1 |
| u2 -w2 |
y = ----------------.z
| u1 v1|
| u2 v2|
For each choice of z we have a solution. We choose
| u1 v1|
z = | u2 v2|
then
| -w1 v1|
x = | -w2 v2| and
| u1 -w1 |
y = | u2 -w2 |
<=>
| v1 w1|
x = | v2 w2| and
| u1 w1 |
y = - | u2 w2 | and
| u1 v1|
z = | u2 v2|
The coordinates of the intersection point defined by two different lines
are
| v1 w1| | u1 w1 | | u1 v1|
( | v2 w2| , - | u2 w2 | ,| u2 v2| )
These formulas give a efficient method to calculate the intersection point
of two lines. Remark: If in the previous system x or y is the side unknown, the
resulting coordinates are the same.
Example:
We calculate the intersection point of the lines
x - y + 2 z = 0
2x - y + 3 z = 0
The coordinates of the intersection point are
| -1 2 | | 1 2 | | 1 -1|
( | -1 3 | , - | 2 3 | ,| 2 -1| )
<=>
(-1,1,1)
Homogeneous equation of a curve.
An cartesian equation of a curve is the necessary and sufficient condition for the cartesian coordinates of a point in order that the point is on the curve.A homogeneous equation is such condition for the homogeneous coordinates of the point.
Say F(x,y)=0 is a cartesian polynomial equation of a curve c.
- P is a regular point.
P(x,y,z) is on c <=> P(x/z , y/z) is on c <=> F(x/z , y/z) = 0 Since the equation is a polynomial equation, it can be written as a homogeneous polynomial equation in x,y,z. <=> G(x,y,z) = 0 - P is the ideal point.
By DEFINITION we say :P(x,y,z) is on c <=> G(x,y,z) = 0
G(x,y,z) = 0 is a homogeneous equation of c.
Each real multiple is also a homogeneous equation of c.
Example:
y2 - 2p x = 0
is the cartesian equation of a parabola .We'll transform this equation to a homogeneous equation.
P(x,y,z) is on the parabola
<=>
P(x/z , y/z) is on the parabola
<=>
(y/z)2 - 2 p (x/z) = 0
<=>
y2 - 2 p x z = 0
The last equation is the homogeneous equation of the parabola.The point (1,0,0) is an ideal point of that parabola.
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