Linear Algebra

Topic: Projective Geometry 337
Topic: Projective Geometry
There are geometries other than the familiar Euclidean one. One such geometry
arose in art, where it was observed that what a viewer sees is not necessarily
what is there. This is Leonardo da Vinci’s masterpiece The Last Supper.
What is there in the room, for instance where the ceiling meets the left and
right walls, are lines that are parallel. However, what a viewer sees is lines
that, if extended, would intersect. The intersection point is called the vanishing
point. This aspect of perspective is also familiar as the image of a long stretch
of railroad tracks that appear to converge at the horizon.
To depict the room, da Vinci has adopted a model of how we see, of how we
project the three dimensional scene to a two dimensional image. This model is
only a first approximation — it does not take into account that our retina is
curved and our lens bends the light, that we have binocular vision, or that our
brain’s processing greatly affects what we see — but nonetheless it is interesting,
both artistically and mathematically.
The projection is not orthogonal, it is a central projection from a single
point, to the plane of the canvas.
A
B
C
(It is not an orthogonal projection since the line from the viewer to C is not
orthogonal to the image plane.) As the picture suggests, the operation of central
projection preserves some geometric properties — lines project to lines.
However, it fails to preserve some others — equal length segments can project
to segments of unequal length; the length of AB is greater than the length of
BC because the segment projected to AB is closer to the viewer and closer
things look bigger. The study of the effects of central projections is projective
geometry. We will see how linear algebra can be used in this study.