Editing Real projective plane (section)

== Homogeneous coordinates ==
{{main|Homogeneous coordinates}}
The points in the plane can be represented by [[homogeneous coordinates]]. A point has homogeneous coordinates [''x''&nbsp;:&nbsp;''y''&nbsp;:&nbsp;''z''], where the coordinates [''x''&nbsp;:&nbsp;''y''&nbsp;:&nbsp;''z''] and [''tx''&nbsp;:&nbsp;''ty''&nbsp;:&nbsp;''tz''] are considered to represent the same point, for all nonzero values of ''t''.  The points with coordinates [''x''&nbsp;:&nbsp;''y''&nbsp;:&nbsp;1] are the usual [[real plane]], called the '''finite part''' of the projective plane, and points with coordinates [''x''&nbsp;:&nbsp;''y''&nbsp;:&nbsp;0], called '''points at infinity''' or '''ideal points''', constitute a line called the '''[[line at infinity]]'''.  (The homogeneous coordinates [0&nbsp;:&nbsp;0&nbsp;:&nbsp;0] do not represent any point.)

The lines in the plane can also be represented by homogeneous coordinates. A projective line corresponding to the plane {{nowrap|''ax'' + ''by'' + ''cz'' {{=}} 0}} in '''R'''<sup>3</sup> has the homogeneous coordinates (''a''&nbsp;:&nbsp;''b''&nbsp;:&nbsp;''c''). Thus, these coordinates have the equivalence relation (''a''&nbsp;:&nbsp;''b''&nbsp;:&nbsp;''c'') = (''da''&nbsp;:&nbsp;''db''&nbsp;:&nbsp;''dc'') for all nonzero values of ''d''. Hence a different equation of the same line ''dax''&nbsp;+&nbsp;''dby''&nbsp;+&nbsp;''dcz''&nbsp;=&nbsp;0 gives the same homogeneous coordinates. 
A point [''x''&nbsp;:&nbsp;''y''&nbsp;:&nbsp;''z''] lies on a line (''a''&nbsp;:&nbsp;''b''&nbsp;:&nbsp;''c'') if ''ax''&nbsp;+&nbsp;''by''&nbsp;+&nbsp;''cz''&nbsp;=&nbsp;0.  
Therefore, lines with coordinates (''a''&nbsp;:&nbsp;''b''&nbsp;:&nbsp;''c'') where ''a'',&nbsp;''b'' are not both 0 correspond to the lines in the usual [[real plane]], because they contain points that are not at infinity.  The line with coordinates (0&nbsp;:&nbsp;0&nbsp;:&nbsp;1) is the line at infinity, since the only points on it are those with&nbsp;''z''&nbsp;=&nbsp;0.

=== Points, lines, and planes ===
[[Image:Proj geom1.PNG|right]]
A line in '''P'''<sup>2</sup> can be represented by the equation ''ax'' + ''by'' + ''cz'' = 0. If we treat ''a'', ''b'', and ''c'' as the column vector '''ℓ''' and ''x'', ''y'', ''z'' as the column vector '''x''' then the equation above can be written in matrix form as:
: '''x'''<sup>T</sup>'''ℓ''' = 0 or '''ℓ'''<sup>T</sup>'''x''' = 0.

Using vector notation we may instead write '''x''' &sdot; '''ℓ''' = 0 or '''ℓ''' &sdot; '''x''' = 0.

The equation ''k''('''x'''<sup>T</sup>'''ℓ''') = 0 (which k is a non-zero scalar) sweeps out a plane that goes through zero in '''R'''<sup>3</sup> and ''k''(''x'') sweeps out a line, again going through zero. The plane and line are [[linear subspace]]s in [[real coordinate space|'''R'''<sup>3</sup>]], which always go through zero.
{{Clear}}

=== Ideal points ===
[[Image:prj geom.svg|right]]

In '''P'''<sup>2</sup> the equation of a line is {{nowrap|''ax'' + ''by'' + ''cz'' {{=}} 0}} and this equation can represent a line on any plane parallel to the ''x'', ''y'' plane by multiplying the equation by ''k''.

If {{nowrap|''z'' {{=}} 1}} we have a normalized homogeneous coordinate. All points that have ''z'' = 1 create a plane. Let's pretend we are looking at that plane (from a position further out along the ''z'' axis and looking back towards the origin) and there are two parallel lines drawn on the plane. From where we are standing (given our visual capabilities) we can see only so much of the plane, which we represent as the area outlined in red in the diagram. If we walk away from the plane along the ''z'' axis, (still looking backwards towards the origin), we can see more of the plane. In our field of view original points have moved. We can reflect this movement by dividing the homogeneous coordinate by a constant. In the adjacent image we have divided by 2 so the ''z'' value now becomes 0.5. If we walk far enough away what we are looking at becomes a point in the distance. As we walk away we see more and more of the parallel lines. The lines will meet at a line at infinity (a line that goes through zero on the plane at {{nowrap|''z'' {{=}} 0}}). Lines on the plane when {{nowrap|''z'' {{=}} 0}} are ideal points. The plane at {{nowrap|''z'' {{=}} 0}} is the line at infinity.

The homogeneous point {{nowrap|(0, 0, 0)}} is where all the real points go when you're looking at the plane from an infinite distance, a line on the {{nowrap|''z'' {{=}} 0}} plane is where parallel lines intersect.
{{Clear}}

=== Duality ===
[[Image:Projective geometry diagram 2.svg|200px|right]]
In the equation {{nowrap|'''x'''<sup>T</sup>'''ℓ''' {{=}} 0}} there are two [[column vector]]s. You can keep either constant and vary the other. If we keep the point '''x''' constant and vary the coefficients '''ℓ''' we create new lines that go through the point. If we keep the coefficients constant and vary the points that satisfy the equation we create a line. We look upon '''x''' as a point, because the axes we are using are ''x'', ''y'', and ''z''. If we instead plotted the coefficients using axis marked ''a'', ''b'', ''c'' points would become lines and lines would become points. If you prove something with the [[data plot]]ted on axis marked ''x'', ''y'', and ''z'' the same argument can be used for the data plotted on axis marked ''a'', ''b'', and ''c''. That is duality.
{{Clear}}

==== Lines joining points and intersection of lines (using duality) ====
The equation {{nowrap|'''x'''<sup>T</sup>'''ℓ''' {{=}} 0}} calculates the [[dot product|inner product]] of two column vectors. The inner product of two vectors is zero if the vectors are [[orthogonal]]. In  '''P'''<sup>2</sup>, the line between the points '''x'''<sub>1</sub> and '''x'''<sub>2</sub> may be represented as a column vector '''ℓ''' that satisfies the equations {{nowrap|'''x'''<sub>1</sub><sup>T</sup>'''ℓ''' {{=}} 0}} and {{nowrap|'''x'''<sub>2</sub><sup>T</sup>'''ℓ''' {{=}} 0}}, or in other words a column vector '''ℓ''' that is orthogonal to '''x'''<sub>1</sub> and '''x'''<sub>2</sub>. The [[cross product]] will find such a vector: the line joining two points has homogeneous coordinates given by the equation {{nowrap|'''x'''<sub>1</sub> × '''x'''<sub>2</sub>}}. The intersection of two lines may be found in the same way, using duality, as the cross product of the vectors representing the lines, {{nowrap|'''ℓ'''<sub>1</sub> × '''ℓ'''<sub>2</sub>}}.