§ 1: Central projection

Consider two non-parallel planes α and β and a point P that is not in any of both planes.
Let s be intersecting line of β and the plane through P parallel to α.
Let t be the intersecting line of α and β.
Let u be the intersecting line of α and the plane through P parallel to β.
For each point X in β that is not lying in s, let X' be the point of intersection of α and the line connecting P and X.

The mapping XX' is a bijection from β/s onto α/u, named central projection. We find:
i) If the points A, B, C in β/s are collinear (on l), then so are the image points A', B', C' (on l').
ii) If the lines a, b and c in β concur in a point L of β/s, then the image lines a', b' and c' concur in L'.
iii) If l and m are parallel lines in β that intersect s in distinct points, then l' and m' intersect in a point of u.
iv) If l and m are lines in β, parallel to s but not identical to s, or intersecting in a point of s, then l' and m' are parallel.
v) Let d(A,B) be the distance between A and B. Suppose that A, B and C are collinear in β/s. In general we have d(A,B) ≠ d(A',B') and d(A,B)/d(B,C) ≠ d(A',B')/d(B',C').
vi) Let ∠(l,m) be the smallest angle between l and m. Let A, B and C be non-collinear points in β/s. In general, we find ∠(AB,BC) ≠ ∠(A'B',B'C').
vii) The image of a circle c in β is a conic section in α: an ellipse if c and s don't intersect, a parabola if c touches s, and an hyperbola if c and s intersect in two points.

We provisionally define projective geometry as the study of those geometrical concepts that are invariant under central projection.
We have, among others, the following projective concepts:

a) points, lines, incidence of points and lines (a point A is incident with a line l if A lies on l); collinearity and concurrence, complete triangles and complete quadrangles;
b) separation and cross ratio
(let A, B, C, D be four points on a line l, then the pairs {A,B} and {C,D} separate if one of the points C and D lies between A and B, and the other doesn't;
the cross ratio (A,B;C,D) is the number (d(A,C)/d(A,D)):(d(B,C)/d(B,D)), provided with a minus sign if {A,B} and {C,D} separate), see section 9.

Concepts like distance and ratio of distances, measure of angles, ellipse, parabola and hyperbola, parallel, are not projective concepts (but the concepts conic and touching are).


O1 Verify all assertions above.

O2 When we use parallel projection instead of central projection, we replace the bundle of lines through P by a bundle of parallel lines that intersect α and β.
A geometrical concept that is invariant under parallel projection is called an affine concept. Which of the concepts mentioned above are zijn affine?

O3 Let l be a line in β that intersects s in S, and l' its image line in α. Let c be a circle that touches l'. Give the instructions for a continuous mapping from l onto c.

O4 Suppose that the circle c in β intersects the line s in A and B, so that the image curve is a hyperbola.
Which lines in β are corresponding to the asymptotes of this hyperbola?
Explain by reference to the situation in β that the lines in α with asymptotical direction intersect the hyperbola in at most one point.