§ 4: Perspective

If we see the picture of §1 through the eyes of a perspective drawer, we call point P the eye, plane α the scene,and plane β the tableau. The line s is called vanishing line or horizont.
Indeed, if two lines in β intersect at the horizont, the image lines in α are parallel. If we see α as a real projective plane, s is the complete original of the line at infinity in α.
In our thoughts, we can rotate the plane β around the line of intersection t until β coincides with α. If we call the image of X under this rotation X", then X"X ' is a bijection from the real projective plane α onto itself that preserves collinearity (see O12).
Turning down a plane β onto a plane α as above is important for the geometry of perspective drawing, descriptive geometry, because an object isn't determined by only one perspective drawing. We the construct perspective drawings with several tableaux through t on a single piece of paper α.

Remark: We call the mapping mentioned above, X"X ', a central collineation. We will elaborate this mapping in §16, and here follows an introduction.
The points of t are invariant and each pair of lines (l",l' ) intersects on t. We call t the axis of the central collineation.
Furthermore, the lines X"X ' all go through one point (the centre of the central collineation). See O13.
The central collineation is determined by its axis t, centre V, and one pair (P",P ') with P"P '. See O14.
To construct the horizont s" in the projection plane α (with an affine construction) we need only construct one point on s": because s" is parallel to t. See O15.
Finally we find in O16 the pointwise construction of a conic.

Remark: each central collineation is also the product of two central projections (from α to β with centre P1 , X"X , and from β to α with centre P2 , XX ' ; then the centre is the intersection point V of the line P1P2 and the plane α). You need not reckon with this remark when you attack the following problems.

O12 What is the image of the line at infinity under the mapping X"X ' ? And what is the complete original of the line at infinity?

O13 Prove the assertion above that the lines X"X ' go through one point with the theorem of Desargues.

O14 Of a central collineation we know the axis t, centre V, and one pair (P",P ') with P"P '. Construct with an arbitrary point Q" the image point Q'.

O15 Construct, with the same data as in O14, the original Q" of an arbitrary point Q' at infinity. Thereafter, draw s".

O16 Of central collineation we know the horizont s", the centre V, and one pair (P",P ') with P"P '.
P" lies on a circle c", and, of course, is collinear with V and P'.
Construct some points of the conic c'.
Distinguish three cases. Construct in each of the three cases as many points as is necessary to recognize the kind of the conic.