CS602 Final Term Highlighted Handouts by Masters

Projections

For centuries, artists, engineers, designers, drafters, and architects have been facing difficulties and constraints imposed by the problem of representing a three-dimensional object or scene in a two-dimensional medium -- the problem of projection. The implementers of a computer graphics system face the same challenge.

Projection can be defined as a mapping of point P(x,y,z) onto its image P`(x`,y`,z`) 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.

There are two basic methods of projection

These methods are used to solve the basic problems of pictorial representations

We characterize each method and introduce the mathematical description of the projection process respectively.

Taxonomy of Projection

We can construct different projections according to the view that is desired. Following figure provides taxonomy of the families of perspective and parallel projections. Some projections have names – cavalier, cabinet, isometric, and so on. Other projections qualify the main type of projection – one principal vanishing–point perspective and so forth. Parallel projection methods are used by drafters and engineers to create working drawings of an object which preserves its scale and shape. The complete representation of these details often requires two or more views (projections) of the object onto different view planes. In parallel projection, image points are found as the intersection of the view plane with a projector drawn from the object point and having a fixed direction. The direction of projection is the prescribed direction for all projections. Orthographic projections are characterized by the fact that the direction of projection is perpendicular to the view plane. When the direction of projection is parallel to any of the principal axes, this produces the front, top, and side views of mechanical drawings (also referred to as multi view drawings). Axonometric projections are orthographic projections in which the direction of projection is not parallel to any of the three principal axes. Non orthographic parallel projections are called oblique parallel projection.

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 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:

Orthographic · Oblique · 1) Orthographic Projection

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