Perspective Projection - Transformations and Projections in Computer Graphics - page 80

The main idea was already proposed by Alberti and is illustrated in Figure 3.17 for
a circle. Start with a flat, nonperspective drawing of a curved object and place a regular
rectangular grid on it [part (a)]. Redraw the grid in perspective, with a vanishing point
[part (b)], and go over the two grids box by box. For each box, copy that part of the
object seen in the first grid and modify it according to the shape of the box in the
second grid. The final result (the circle in perspective) looks like an ellipse, but notice
how the left and right extreme points of the projected circle (i.e., the ellipse's major
axis) no longer lie on the central horizontal line but have moved below it.

Exercise 3.5: Explain why.

A variant of this method starts by locating key points on the curved object (points
that make it easy to draw the entire object), assigning them coordinates, and locating
them on the perspective grid. Figure 3.18 shows an example of a large digit 5 where
5 7 = 35 key points have been located. The digit is placed in a rectangle, and grid

lines are added and labeled 1 through 5 and “A” through “G,” resulting in a nonuniform
grid. This grid is then transformed in perspective (one-point or two-point) and the key
points located in the new grid, which makes it easy to draw the large 5 in perspective.

×

Exercise 3.6: Show the geometric construction that transfers the 35 key points to a
grid in one-point perspective.

G

F

E

D

C

B

A

1 2 3 4 5

Figure 3.18: A Large Digit “5.”

The great German painter Albrecht Durer showed how to extend Alberti's approach
to three-dimensional objects (Figure 3.19). Lay the object (a lute in the figure) on a
table behind a frame and attach a string with a pulley and a weight to the wall in front
of the frame. A wooden leaf is attached to the frame with hinges, and a sheet of blank
paper is mounted on the leaf. Now move the free end of the string to an arbitrary point
on the object and determine the point where the string intercepts the frame. (This
is done by two moveable wires or threads, as shown in the upper part of the figure.)