PROJECTIVE GEOMETRY COURSE

§ 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 *P _{1}* ,

*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.