Multiple Vanishing Points

It is possible for objects in perspective to have more than one vanishing point.  When an objective has two vanishing points, it is said to appear in two-point perspective.  Similarly, three vanishing points correspond to three-point perspective.

When we view an object, we can usually make out which is the “front” of it.  We shall call the surface of this “front” the front plane.  The surface which the object is projected onto is called the projection plane.  Both the observer and the object rest on the ground plane.

Fig. 42 – Diagram showing the front plane,
projection plane and the surface plane.

In Fig. 42, the projection plane is taken to be the xz-plane.  The ground plane is the xy-plane.  The front plane is represented by the rectangle.  Hence, the front plane is always parallel to the projection plane, and perpendicular to the ground plane.  When this occurs, only one vanishing point is obtained when the object is viewed in perspective.  This is called one-point perspective. 

Fig. 43 – Rotating the front plane about the z-axis.

When the front plane is rotated about the z-axis, it is no longer parallel to the projection plane.  However, it is still perpendicular to the ground plane.  When the front plane is view in perspective, two vanishing points will be observed.  This is shown in Fig. 43.

Fig. 44 – Further rotating the front plane.

If the front plane is further rotated as shown in Fig. 44, it will no longer be perpendicular to the ground plane.  It is also not parallel to the projection plane.  The combination of these two factors will yield three vanishing points, when the front plane is view in perspective.

Consider a cube placed in the xyz coordinates axes.  First let the “front” of the cube be placed parallel to the projection plane.  In perspective, the cube will appear as shown in Fig. 45. 

Fig. 45 – A cube in one-point perspective.
The vanishing point is denoted by VP.

Next consider the cube with its “front” rotated about the z-axis.  The “front” will not be parallel to the projection plane, but will still be perpendicular to the ground plane.  In perspective, the cube will appear as shown in Fig. 46.

Fig. 46 – A cube in two-point perspective.

One can further orientate the cube such that its “front” is neither parallel to the projection plane, nor perpendicular to the ground plane.  The cube will then be viewed in three-point perspective, as shown in Fig. 47. 

Fig. 47 – A cube in three-point perspective.

The properties governing two- and three-point perspective can be generalized to more complicated objects.