Alberti’s Method of Perspective Drawing
1. Position the objects in the composition in a rectangular window similar to that
seen in the Map View, in this case a Euclidean view of the tile floor;
2. Draw line m parallel to and to the left of the window;
3. Represent the size and spacing of the tiles on line m by points A, B, C, D and E;
4. Construct line l perpendicular to line m above the Map View;
5. Represent the size and spacing of the tiles on line l by point F, G, H, I, J, K and L;
6. Reproduce the spacing of points A-E on line l with point A’-E’ and;
7. Construct line n parallel to line m and position it to the left of point E’;
8. Construct point P and segments PA’, PB’, PC’, PD’, and PE’;
9. Find the intersections of line n with segments PA’, PB’, PC’, PD’, and PE’;
10. Construct parallels to line l through each point of intersection;
11. Construct a parallel to line l through point P. This is the Horizon Line for the
Perspective View;
12. Construct a point on the Horizon Line. This is the Vanishing Point for the
Perspective View;
13. Position the Vanishing Point as shown and construct segments from the vanishing
point to points F, G, H, I, J, K, and L;
14. Find the intersections of the line defining the sides of each tile in the Perspective
View;
15. Use the Polygon Interior option in the Construct menu in CD 6.1.3 to shade the
tiles in the Perspective View.
Sketching a House in 2-Point Perspective
Step 1: Using the Geometers Sketchpad, draw a horizon line and select two vanishing
points on the horizon line (see Figure 6.1.7). From each vanishing point, draw a line
containing the base of a wall of the house. Mark the point of intersection P1. The two
vanishing points and P1 determine the plane on which the house sits.
Step 2: Draw a segment P1-P2 perpendicular to the horizon line. This determines the
“front” edge of the house. Connect P2 to each vanishing point. This determines the
plane of two of the walls of the house. Draw two other lines parallel to P1-P2 on either
side of P1-P2, determining the ends of the wall planes, P3-P4 and P5-P6 (see Figure
6.1.8).
Step 3: Add hidden lines by extending additional rays from the vanishing points and label
intersection points P7 and P8. Connect these points to the other corners as indicated and
shade the faces to enhance the 3-dimensional appearance of the house (see Figure 6.1.9).
Step 4: Locate the centers (P9 and P10) of two opposite faces by finding the intersection
of the diagonals of each face. Construct lines through P9 and P10 parallel to the vertical
edges of the box. Select a point on the nearer of the two lines to be the peak of the roof,
labeling the point P11 (see Figure 6.1.10).
Step 5: Extend a line from P11 to Vanishing Point 1. Find the intersection of this line
with the other line drawn parallel to the vertical edges of the box. Label the intersection
P12. Connect P11 to P2 and P6 and P12 to P4 and P8 to make the roof (see Figure
6.1.11). Move P2 downward to achieve the perspective shown in Figure 6.1.11.
Step 6: Add additional shading, experiment with moving P2 and the two vanishing
points, and hide lines and labels as desired (see Figure 6.1.12).
Points and Lines
Points
 In the Euclidean plane, all points are ordinary in the sense that they represent
locations that may be specified using ordered pairs of real numbers.
 In the projective plane, there are two sorts of points, ordinary points and “ideal”
points (e.g., vanishing points. While artists and architects typically refer to ideal
points as “points at infinity,” mathematicians avoid that language because it
implies that infinity is a point or location, neither of which is true.
Lines
 Lines in the Euclidean plane consist entirely of ordinary points.
 Lines in the projective plane contain both ordinary and ideal points. Given a line
m, and any line n parallel to m is concurrent with m at the ideal point, Im. See
Figure 6.2.2
Parallel
 In the Euclidean plane, parallel lines have no points in common.
 In the projective plane, parallel lines have a single ideal point in common.
 Sets of parallel lines pointing in different directions have different ideal points.
The set of all ideal points is called the ideal line.
Definition 6.2.1
Ordinary points are locations in the Euclidean plane (E) and may be represented using
ordered pairs of real numbers.
Definition 6.2.2
Ideal points correspond to different directions in the Euclidean plane.
Definition 6.2.3
The set of all ideal points is called the ideal line (I) and corresponds to the set of all
possible directions in the Euclidean plane.
Definition 6.2.4
The real projective plane consists of all points in the union E  I.
Definition 6.2.5
A set of points, all collinear, is called a pencil of points. The line containing the given
points is called the axis of the pencil of points.
Definition 6.2.6
A set of lines, all concurrent, is called a pencil of lines. The point of concurrency is
called the center of the pencil of lines.
Example 6.2.1
In Figure 6.2.3, collinear points A, B, C, and D form a pencil of points with axis a.
Concurrent lines k, l, m, and n form a pencil of lines with center O.
Definition 6.2.7
A perspectivity is a one-to-one mapping from a
 Pencil of lines L to a pencil of points P if each line in L is incident with exactly
one point in P (see Figure 6.2.3); or a
 Pencil of points P to a pencil of points P’ if each line joining a point in P to its
corresponding points in P’ is incident with a fixed point O called the center of the
perspectivity (see Figure 6.2.4); or a

Pencil of lines O to a pencil of lines O’ if the intersection of each line on O with
its corresponding line in O’ lies on a fixed line called the axis of the perspectivity
(see Figure 6.2.5).
Definition 6.2.8
A complete quadrangle consists of a set of four points, no three collinear, and the six
lines determined by those points.
Definition 6.2.9
A complete quadrilateral consists of a set of four lines, no three concurrent, and the six
points determined by those lines.
Example 6.2.2
In Figure 6.2.6, points A, B, E, D and lines ABF, DEF, BEG, ADG, AEC, BDC constitute
a complete quadrangle. Lines ABF, DEF, BEG, ADG and points A, B, D, E, F, G
constitute a complete quadrilateral.
Theorem 6.2.1
Given any perspectivity P, an inverse perspectivity P’ exists such that
PP’ = P’P = I
Since all one-to-one mappings have inverses, and all perspectivities are one-to-one
mappings, all perspectivities have inverses.
Axioms of Projective Geometry
PG 1
PG2
PG3
PG4
PG5
There exists at least on line
Each line contains at least 3 points
Not all points are on the same line
Two distinct points determine a distinct line
Two distinct lines determine a distinct point
Table 6.2.1 Axioms of Projective Geometry