Projections in Computer Graphics

Representing an n-dimensional object into an n-1 dimension is known as projection. It is process of converting a 3D object into 2D object, we represent a 3D object on a 2D plane {(x,y,z)->(x,y)}. It is also defined as mapping or transforming of the object in projection plane or view plane. When geometric objects are formed by the intersection of lines with a plane, the plane is called the projection plane and the lines are called projections.    

Types of Projections:

1. Parallel projections 
2. Perspective projections 

Center of Projection: 

It is an arbitrary point from where the lines are drawn on each point of an object.

• If cop is located at a finite point in 3D space , Perspective projection is the result 
• If the cop is located at infinity, all the lines are parallel and the result is a parallel projection.

Parallel Projection:

 A parallel projection is formed by extending parallel lines from each vertex of object until they intersect plane of screen.  Parallel projection transforms object to the view plane along parallel lines. A projection is said to be parallel, if center of projection is at an infinite distance from the projected plane. A parallel projection preserves relative proportion of objects, accurate views of the various sides of an object are obtained with a parallel projection. The projection lines are parallel to each other and extended from the object and intersect the view plane. It preserves relative propositions of objects, and it is used in drafting to produce scale drawings of 3D objects. This is not a realistic representation, the point of intersection is the projection of the vertex.

Parallel projection is divided into two parts and these two parts sub divided into many.

Orthographic Projections:

In orthographic projection the direction of projection is normal to the projection of the plane. In orthographic lines are parallel to each other making an angle 90 with view plane. Orthographic parallel projections are done by projecting points along parallel lines that are perpendicular to the projection line. Orthographic projections are most often used to procedure the front, side, and top views of an object are called evaluations. Engineering and architectural drawings commonly employ these orthographic projections. Transformation equations for an orthographic parallel projection as straight forward. Some special orthographic parallel projections involve plan view, side elevations. We can also perform orthographic projections that display more than one phase of an object, such views are called monometric orthographic projections.

Oblique Projections:

Oblique projections are obtained by projectors along parallel lines that are not perpendicular to the projection plane. An oblique projection shows the front and top surfaces that include the three dimensions of height, width and depth. The front or principal surface of an object is parallel to the plane of projection. Effective in pictorial representation. 

• Isometric Projections: Orthographic projections that show more than one side of an object are called axonometric orthographic projections. The most common axonometric projection is an isometric projection. In this projection parallelism of lines are preserved but angles are not preserved.
• Dimetric projections: In these two projectors have equal angles with respect to two principal axis.
• Trimetric projections: The direction of projection makes unequal angle with their principal axis.

Cavalier Projections:

All lines perpendicular to the projection plane are projected with no change in length. If the projected line making an angle 45 degrees with the projected plane, as a result the line of the object length will not change.

Cabinet Projections:

 All lines perpendicular to the projection plane are projected to one half of their length. These gives a realistic appearance of object. It makes 63.4 degrees angle with the projection plane. Here lines perpendicular to the viewing surface are projected at half their actual length.

Perspective Projections:

• A perspective projection is the one produced by straight lines radiating from a common point and passing through point on the sphere to the plane of projection.
• Perspective projection is a geometric technique used to produce a three dimensional graphic image on a plane, corresponding to what person sees.
• Any set of parallel lines of object that are not parallel to the projection plane are projected into converging lines. A different set of parallel lines will have a separate vanishing point.
• Coordinate positions are transferred to the view plane along lines that converge to a point called projection reference point.
• The distance and angles are not preserved and parallel lines do not remain parallel. Instead, they all converge at a single point called center of projection there are 3 types of perspective projections.

Two characteristic of perspective are vanishing point and perspective force shortening. Due to fore shortening objects and lengths appear smaller from the center of projections. The projections are not parallel and we specify a center of projection cop.

Different types of perspective projections:

• One point perspective projections: In this, principal axis has a finite vanishing point. Perspective projection is simple to draw.

• Two point perspective projections: Exactly 2 principals have vanishing points. Perspective projection gives better impression of depth.

• Three point perspective projections: All the three principal axes have finite vanishing point. Perspective projection is most difficult to draw.

Perspective fore shortening:

The size of the perspective projection of the object varies inversely with distance of the object from the center of projection.