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19. Projections

Dated: 14-05-2025

Projections

It can be defined as mapping of a point \(P(x, y, z)\) onto its image \(P^\prime(x^\prime, y^\prime, z^\prime)\) in the projection plane or view plane, which constitutes the display surface.
The mapping is determined by a projection line called the projector that passes through \(P\) and intersects the view plane.

Methods

There are 2 methods to do projection.

  • Parallel projection
  • Perspective projection

Taxonomy of Projection

cs602_i_19_1.png

Orthographic = Projectors are perpendicular to view plane.
Oblique = Projectors are not perpendicular to view plane.
Multiview = View plane is parallel to principal planes.
Axonometric = View plane is not parallel to principal planes.

Parallel Projection

Parallel projection methods are used by drafters and engineers to create working diagrams of objects which preserve their original scale and shape.
The complete representation of these details often require multiple views(projections) of the object onto different view planes.

The direction of projection is prescribed for all projections.
Orthographic projections are characterized by the fact that the direction of projection is perpendicular to the view plane.

Mathematical Description of a Parallel Projection

Projection rays (projectors) emanate from a Center of Projection(COP) and intersect Projection Plane (PP).
The COP for parallel projectors is at \(\infty\).
The length of a line1 on the projection plane is same as the true length.

Types of Parallel Projections

Orthographic Projection

If the direction of projection is perpendicular to the projection plane then it is an orthographic projection.

Axonometric Orthographic Projection

Orthographic projections that show more than one side of an object are called axonometric orthographic projections.

There are three axonometric projections

Isometric

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.

cs602_e_19_2.svg

\(x_i = y_i = z_i\)

Dimetric

The direction of projection makes equal angles with exactly two of the principal axes.

Trimetric

The direction of projection makes unequal angles with the three principal axes.

Oblique Projection

If the direction of projection is not perpendicular to the projection plane then it is an oblique projection.
In some applications packages, an oblique projection vector is specified with two angles, \(\alpha\) and \(\phi\).

cs602_e_19_3.svg

Blue lines are within projection plane. The line1 between \((x, y)\) and \((x, y, z)\) is perpendicular to the projection plane.

\[\cos(\phi) = \frac{x_p - x}{L}\]
\[\sin(\phi) = \frac{y_p - y}{L}\]
\[x_p = x + L \cos(\phi)\]
\[y_p = y + L \sin(\phi)\]
\[\tan(\theta) = \frac z L \implies L = z \cot(\theta)\]

The oblique projection equations are

\[x_p = x + z \cot(\theta) \cos(\phi)\]
\[y_p = y + z \cot(\theta) \sin(\phi)\]

The transformation matrix2 can be written as

\[ M = \begin{bmatrix} 1 & 0 & \cot(\theta)\cos(\phi) & 0 \\ 0 & 1 & \cot(\theta)\sin(\phi) & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \]

Two Special Cases of Oblique Projection

Cavalier

If \(\tan(\theta) = 1\) then the projection is cavalier.

Cabinet

If \(\tan(\theta) = 2\) then the projection is cabinet.

References

  1. Read more about lines. ↩↩

  2. Read more about matrices. ↩