Introduction of Viewing  Transformations

• Projection is fundamentally a transformation or mapping of 3D objects upon 2D screen.
• Projection is mostly categorised in Parallel and Perspective projections depending upon where the rays from the object converge at the Centre of projection or not.
• We have Perspective projection, if the distance of Centre of projection from the projection plane is finite. This is termed as perspective because faraway objects look nearer and minor objects look bigger.
• Rays from the objects become parallel, when the distance of Centre of projection from the projection plane is infinite. This type of projection is termed as parallel projection.
• Parallel projection can be categorised as per to the angle which the direction of projection makes along with the projection plane.
• Whether the direction of projection of rays is perpendicular to the projection plane, we have an Orthographic projection, or else an Oblique projection.
• Orthographic which is perpendicular projection demonstrates only one face of a given object, which is only two dimensions: length as well as width, whereas Oblique projection demonstrates all the three dimensions, such as length, width and also height. Hence, an Oblique projection is one way to demonstrate all three dimensions of an object in a single view.
• The line perpendicular to the projection plane is foreshortened where projected line length is shorter than actual line length by the way of projection of rays, in Oblique projection. The direction of projection of rays finds out the amount of foreshortening.
• The verification in length of the projected line (because of the direction of projection of rays) which is measured in terms of foreshortening factor, f, that is expressed as the ratio of the projected length to its actual length.
• In Oblique projection, we have cavalier projection, if foreshortening factor f=1 and cabinet projection, if f=1/2.
• The plane of projection may be perpendicular axes or maybe not. If the plane of projection is perpendicular to the principal axes then we have multi-view projection or else axonometric projection.
• Depending upon the foreshortening factors, we have three diverse types of axonometric projections: as all foreshortening factors are equalled, Di-metric where any two foreshortening factors equal and Trimetric when all foreshortening factors unequal.
• In perspective projection, the parallel lines show to meet at a point which is a point at infinity. This point termed as vanishing point. A practical illustration is a long straight railroad track, when two parallel railroad tracks show to meet at infinity.
• A perspective projection can have mostly three-principal vanishing points when points at infinity with respect to x, y, and z-axes, respectively and at least one principle vanishing point.
• A particular point perspective transformation along with the Centre of projection along any of the coordinate axes yields a single or particular vanishing point, where two parallel lines show to meet at infinity.
• Two point perspective transformations are acquired by the concatenation of any two one-point perspective transformations. Consequently we can have two-point perspective transformations as: P_per-xy, P_per-yz, P_per-xz .
• Three point perspective transformations can be acquired by the composition of all the three one-point perspective transformations.