Projection rays (projectors) emanate from a Center of Projection (COP) and intersect Projection Plane (PP). The COP for parallel projectors is at infinity. The length of a line on the projection plane is the same as the "true Length".

There are two different types of parallel projections:

If the direction of projection is perpendicular to the projection
plane then it is an **orthographic **projection. If
the direction of projection is not perpendicular to the
projection plane then it is an **oblique**
projection.

Look at the parallel projection of a point (x, y, z). (Note the left handed coordinate system). The projection plane is at z = 0. x, y are the orthographic projection values and xp, yp are the oblique projection values (at angle a with the projection plane)

Look at orthographic projection: it is simple, just discard the z coordinates. Engineering drawings frequently use front, side, top orthographic views of an object.

Here are three orthographic views of an object.

Orthographic projections that show more than 1 side of an
object are called **axonometric **orthographic
projections. The most common axonometric projection is an **isometric
**projection where the projection plane intersects each
coordinate axis in the model coordinate system at an equal
distance.

**Isometric Projection**

The projection plane intersects the x, y, z axes at equal distances and the projection plane Normal makes an equal angle with the three axes.

To form an orthographic projection xp = x, yp= y , zp = 0. To form different types e.g., Isometric, just manipulate object with 3D transformations.

**Oblique Projection**

The projectors are not perpendicular to the projection plane
but are parallel from the object to the projection plane The
projectors are defined by two angles A and d where:

A = angle of line (x,y,xp,yp) with projection plane,

d = angle of line (x, y, xp, yp) with x axis in projection plane

L = Length of Line (x,y,xp,yp).

Then:

cos d = (xp - x) / L ------> xp = x + Lcos d ,

sin d = (yp - y) / L ------> yp = y + Lsin d ,

tan A = z / L

Now define L1 = L / z ----> L= L1 z , so tan A = z / L = 1 / L1 ; xp = x + z(L1cos d) ; yp = y + z(L1sin d)

|1 0 0 0 | P = |0 1 0 0 | |L1cosq L1sinq 1 0 | |0 0 0 1 |

Now if A = 90° (projection line is perpendicular to PP) then tanA = infinity => L1 = 0, so have an rthographic projection.

Two special cases of oblique projection

A) A = 45° , tanA = 1 => L1 = 1 This is a **Cavalier **projection
such that all lines perpendicular to the projection planeare
projected with no change in length.

B) tanA = 2, A= 63.40°, L1 = 1 / 2

Lines which are perpendicular to the projection planeare
projected at 1 / 2 length . This is a **Cabinet**
projection.

Last changed September 30, 1998, G. Scott Owen, owen@siggraph.org