Cross ratio: given four points, a, b, c and d, then, (a, b; C, d) = (ab CD)/(AC BD) is the intersection ratio of these four points, in other words, it is "intersection ratio".

Cross ratio theorem: projective transformation keeps the cross ratio unchanged. Let E be the projection center, and the straight line connecting points A, B, C, D and E on the straight line M intersects with the straight line N at A', B', C' and D'. Then, (a, b; c，D)=(A '，B '； c '，D ').

Next, the area method is used to prove this problem. Look at the picture.

Homogeneous coordinates and duality principle

If the Cartesian coordinates of a point are (a/c, b/c), the homogeneous coordinates are (a/c, b/c, 1) or (a, b, c); As long as this ratio is the same, it always represents the same point. Homogeneous coordinates include point coordinates and line coordinates, and the shapes of homogeneous point coordinates and homogeneous line coordinates are the same, so from the algebraic point of view, the dual situation of point * * * line is the point of line * * *. This is the main content of duality principle.

In my opinion, the function of homogeneous coordinates is to unify the operation of points and lines.

Projective transformation on conic

There is a very profound conclusion here: the locus of the intersection of two projective beams whose centers are not coincident is a quadratic curve passing through the two centers. This is the most important conclusion in projective geometry, and its proof process, if it is very concise by algebraic method, will not be repeated here; But as compensation, it is demonstrated by a legend: on the plane, there are three fixed points A, B and C, and four fixed lines (black); J and n are moving points on straight lines M and N, respectively, and the projections of wire harnesses A(J) and B(N) correspond; The straight line AJ intersects BN and P, so when the point J crosses the straight line M, the trajectory of P is a quadratic curve passing through A and B (the quadratic curve in the legend is an ellipse).

On the contrary, given the fixed points A, B and the moving point P on the conic, then the harnesses A(P) and B(P) are projectively corresponding.

Polar coordinate transformation and duality principle

To understand polar coordinate transformation, we must first understand the concepts of poles and polar lines. Both poles and epipolar lines are suitable for quadratic curves.

When point C is located on the conic, make any two chords DG and EF of the conic pass through C; Let the straight line DF intersect with EG in H and DE intersect with FG in I; Then, for tHIs quadratic curve, the polar line of c is hi, the polar line of hi is c, and the polar line and polar line are always mutual.

When point C is located on (or outside) the conic, it is relatively simple, so I won't repeat it here.

The duality principle is embodied here: if the points are * * * lines, then these points must be * * * points relative to the polar line of a conic; or vice versa, Dallas to the auditorium

Fixed point principle

Any projective transformation, whether it is a series of points, a light beam or a projective transformation on a conic, has fixed points. Finding fixed points is an important method to solve many drawing problems.

Look at the following figure: given two fixed points m and n on the quadratic curve f, the straight lines u and f intersect at p and q; A is the moving point on f, the straight line AN intersects with u at r, and the straight line MR intersects with f at a'; Then, from a to a' is a projective transformation, a and a' are projective correspondence, and the corresponding point of a is a'. If A approaches M, then a' approaches N, which means that the projective corresponding point of M is N, and it is easy to find that P and Q are the fixed points of this projective transformation, that is, the corresponding point of P is P and the corresponding point of Q is Q..

folding

The involution mentioned here refers to involution in projective transformation. If A to A' is a projective transformation, and the corresponding point of A' is A, then this projective transformation is involution.

There are only two basic forms of involution:

Reciprocal type: u u' = k (k ≠ 0);

Reciprocal type: u+u'=0.