The perspective image of an infinite checkerboard. It can be constructed starting from any triangle 
, where 
and 
form the near corner of the floor, and 
is the horizon (left figure). If 
is the corner tile, the lines 
and 
must be parallel to 
and 
respectively. This means that in the drawing they will meet 
and 
at the horizon, i.e., at point 
and point 
respectively (right figure). This property, of course, extends
to the two bunches of perpendicular lines forming the grid.
The adjacent tile 
(left figure) can then be determined by the following conditions:
1. The new vertices 
and 
lie on lines 
and 
respectively.
2. The diagonal 
meets the parallel line 
at the horizon 
.
3. The line 
passes through 
.
Similarly, the corner-neighbor 
of 
(right figure) can be easily constructed requiring
that:
1. Point 
lie on 
.
2. Point 
lie on the common diagonal 
of the two tiles.
3. Line 
pass through 
.
Iterating the above procedures will yield the complete picture. This construction shows how naturally projective geometry arises
from perspective design, since 
and 
can be interpreted as two coordinate axes in the real
projective plane with 
and 
their points
at infinity, joined by the line at infinity 
.
The Möbius net is the result of a projective transformation of the two-dimensional lattice. Unlike in affine geometry, the length proportions along the two perpendicular
directions are not preserved, whereas the cross ratio, which is invariant by central
projection, is. The "horizontal" sides of the projected tiles have different
lengths, but are related by the central projections from 
(left figure) and 
(right figure), so that
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(1)
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(2)
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