parallel and perspective projections

•          Parallel projections:

–        no shortening due to distance

–        several kinds, depending on orientation:

•          isometric, cavalier,…

•          Perspective projections:

–        shortening of objects in the distance

–        several kind, depending on orientation:

•          one, two, three vanishing points

Parallel Projection Matrix

•          Parallel projection onto z=0 plane:

x’=x, y’=y, w’=w

Matrix for this projection:

Perspective Projection Matrix

Projection onto plane z=0, with center of projection at z=-d:

Perspective projections pros and cons:

 Size varies inversely with distance - looks realistic – Distance and angles are not (in general) preserved – Parallel lines do not (in general) remain parallel

Parallel projection pros and cons: 

 Less realistic looking + Good for exact measurements + Parallel lines remain parallel – Angles not (in general) preserved

Parallel projections

 For parallel projections, we specify a direction of projection (DOP) instead of a COP. There are two types of parallel projections: w Orthographic projection — DOP perpendicular to PP w Oblique projection — DOP not perpendicular to PP There are two especially useful kinds of oblique projections: w Cavalier projection • DOP makes 45° angle with PP • Does not foreshorten lines perpendicular to PP w Cabinet projection • DOP makes 63.4° angle with PP • Fore shortens lines perpendicular to PP by one half

Perspective projections

Perspective  in the graphic arts is an approximate representation, on a flat surface (such as paper), of an image as it is seen by the eye. The two most characteristic features of perspective are that objects are smaller as their distance from the observer increases; and that they are foreshortened, meaning that an object's dimensions along the line of sight are shorter than its dimensions across the line of sight