### Introduction to parallel projection

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.

### Parallel projection operator

• 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

### Parallel Projection Examples

• 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 ,
• Orthographic projection onto the projection plane z=0 is performed by the projection matrix

### Parallel Projection Excercise

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 solving

The matrix for orthographic parallel projection is then given by

(b)
an oblique projection in direction is used