Projective geometry bases
Just a few "recalls" of definitions, not too rigourous.
The projective geometry is axiomatically defined in terms of vector spaces.
Here we have a more intuitive point of view.
- Projective line
- Intuitively, a usual line completed by one "point at infinity"
- Projective plane
- Intuitively, a usual plane completed by one "point at infinity" in each direction
- Homogenous coordinates
- A point on a projective line has coordinates on that line (X,Y), Y=0 for a "point at infinity"
(X,Y) and (λX,λY), for any real λ ≠ 0 is the same point.
A point on a projective plane has coordinates (X,Y,Z) = (λX,λY,λZ).
Z=0 for a "point at infinity".
Z=0 is the the equation of the "line at infinity", set of all "points at infinity".
- Correlation between the projective line aX + bY + cZ = 0 and the projective point (a,b,c).
- Anharmonic ratio [cross ratio]
- Given 4 points A,B,C,D on a projective line, the cross ratio, noted
(A,B,C,D) is the ratio of ratios : (AC/AD) / (BC/BD)
A specific case is (A,B,C,D) = -1, harmonic ratio.
- A transform that keeps in line points and anharmonic ratio.
May be restricted to homography between two projective lines, or of a line onto itself,
or by duality between pencils of lines.
An homography on the plane is defined by the images of 4 non colinear points.
An homography between projective lines is defined by the images of 3 distinct points.
- Axis of homography
- Given an homography of a line (d) to a line (d') transforming M → M' and P → P',
the intersect points M∩P' and M'∩P are on a fixed line named the homography axis.
- Center of homography
- By duality, an homography between two pencils of lines A* → B* transforming a line (p)
of A* into (p') of B* and (q) → (q').
The line through the intersection points (p)∩(q') and (p')∩(q) goes through a fixed point
named the homography center.
Given two lines (d) and (d') and a point S not on these lines.
A projection (d) → (d') is the application that transforms any point M of (d)
into point M' of (d') with SMM' in line.
An homography between two lines (d) et (d') is in infinitely many ways the product of two projections.
An homograph betwen (d) et (d') is a projection iff the point d∩d' is invariant.
We consider in a similar way a projection of a projective plane on another, from a point S outside the planes.
And by par duality, the projection of a pencil A* on B* (the center becoming then the axis of projection).
- Projective homology
- An homography with a fixed point P (homology center) and a line of fixed points (d) : homology axis.
Any homography is in infinitely many ways the product of two homologies.
An homography h is an homology iff, for any point M the line Mh(M) goes through a fixed point P.
- A transform which is its own inverse h = h-1, or also h² = I identity.
The involutions on the projective plane are homologies with cross ratio -1.
The cross ratio of an homology is the cross ratio (O,I,M,M') where O is the center of homology,
I the intersection of OM with axis and M' ≠ M the image of M.
This cross ratio doesn't depend on M.
- The real projective plane may be "extended" by points with coordinates in
ℂ, that is (X,Y,Z) in ℂ³
instead of ℝ³.
Any quadratic equation has then always two roots, or a double root,
hence any line always intersects any conic section in two points.