Parallel projections are very useful for engineers, and draftsmen to create working drawings, blueprints, schematics or working drawings of objects that preserve its shape and scale.

Points on an object are projected to the view plane along parallel lines (projectors).

The view plane (projection plane)
**ax + by + cz + d = 0** is intersected
with the projector drawn from the object point
along a fixed vector , i.e.,
All the points on the object are projected to the view plane
along parallel lines.
For example, in the figure shown below,
the projection of is
on the view plane
whose normal is .

The views formed by parallel projections varies according
to the angle that the direction of projection makes with
the projection plane.
If the projection is perpendicular to the
image plane, i.e., is along the same
direction as , then that projection is called
the *orthographic projection*.

The projection is oblique when the projection is not perpendicular to the image plane.

- Let
**ax+by+cz+d=0**be the projection plane - Let be the direction of projection
- Let be a point on the object to be projected
- Start at and travel along the line in direction
until plane
**ax+by+cz+d=0**is hit - It's easiest to use the parametric equation of the line
- At some value of
**t**, when the plane equation is satisfied, we are on the projection plane

- Solving for the unknown parameter value
provided (what does this mean?)

- Substituting this value of
**t**into the previous line equation for**x**,**y**, and**z**gives an expression for the projected point

- With some manipulation we can write this as a matrix equation

- Let
**z=0**be the projection plane with projector- Form the line equation
- Find , so that the projection matrix is
- When the
**p**and**q**values are equal, say both**1**, a projection results - It can be shown that ,

- Form the line equation
- Orthographic projection onto the projection plane
**z=0**is performed by the projection matrix

Find a matrix for parallel projections onto the plane

when

- (a)
- an orthographic projection is used
Substituting the values of in equation 6

Substituting the values of

**t**in equation 18, 19, 20 and solvingThe matrix for orthographic parallel projection is then given by

- (b)
- an oblique projection in direction is used