Oblique Projections

Oblique Projections in Computer Graphics, its special cases, Cavalier Projections and Cabinet Projections. Also, some theory and graphical representations.

1. COMPUTER GRAPHICS Oblique Projections Shalvi Dhoundiyal (B.Sc. Hons. Computer Science) (III-year)

2. Plane geometric projections Parallel Perspective Orthographic Three-point Two-point Single-point Axonometric Oblique Cavalier Cabinet Isometric Dimetric Trimetric

3. Oblique Projections Oblique projection is a kind of parallel projection where projecting rays emerges parallelly from the surface of the object and incident at an angle other than 90 degrees on the plane. Only faces of the object parallel to the plane of projection are shown at their ture size and shape, i.e., angles and lengths are preserved for these faces only. y x z

4. Special Cases of Oblique projection When the angle between the oblique projectors and the plane of projection is : cot -1 (1) = 45 When the angle between the oblique projectors and the plane of projection is : cot -1 (1/2) = 63.43 ( Depending upon the angle between projectors and plane of projection. ) Cavalier Projection Cabinet Projection

5. Cavalier Projection In an oblique projection, when the angle between the oblique projectors and the plane of projection is cot -1 (1) , that is 45 . Foreshortening factors for all three principal directions are equal , hence, resulting in figure appears too thick .

6. Cabinet Projection Foreshortening factors for edges perpendicular to the plane of projection is one-half , hence, resulting figure appears to be more realistic . In an oblique projection, when the angle between the oblique projectors and the plane of projection is cot -1 (1/2) , that is 63.43 .

7. Transformation for an Oblique Projection Consider a unit vector [ 0 0 1 ] along the z-axis. Here, P 1 O and PP 2 are typical oblique projectors and z = 0 is the plane of projection. β = angle between projectors and plane of projection. Translating point P by -a in x-direction and - b in y-direction to point P1 at [ -a -b 1 ]. So, Transformation Matrix ( 2-Dimension ) 1 0 0 0 1 0 -a -b 1 [ T’ ] =

8. < < < y x z P [ 0 0 1 ] P 2 O P 1 a -a b -b < α β β < < < Direction of the Oblique Projection Matrix f

9. In 3-Dimensions, this 2-Dimensional transition is equivalent to shearing of the vector PO in the x and y direction. So, Transformation Matrix ( 3-Dimension ) [ T’’ ] = 1 0 0 0 0 1 0 0 -a -b 1 0 0 0 0 1 Here, Projection onto the z = 0 plane yields, [ T ] = 1 0 0 0 0 1 0 0 -a -b 0 0 0 0 0 1

10. a = f cos α b = f sin α f i s the projected length of the z-axis unit vector. α is the angle between the horizontal and the projected z-axis. the angle between projectors and plane of projection, β = cot -1 ( f ) So, Thus, transformation for an Oblique projection is : [ T ] = 1 0 0 0 0 1 0 0 - f cos α - f sin α 0 0 0 0 0 1 We have,

11. if f = 0, => β = 90 , this is an Orthographic Projection . Now, if f = 1, => β = cot -1 (1) = 45 , the edges perpendicular to the projection plane are not foreshortened. This is the condition for a Cavalier Projection . if f = 1/2, => β = cot -1 (1/2) = 63.435 , the edges perpendicular to the projection plane are foreshortened by one-half. This is the condition for a Cabinet Projection .

12. Most suited for... Illustrating circular objects or otherwise curved faces. Why? To prevents unwanted distortion, as it happens in most of the parallel projections, when one dimension is significantly larger than others, unless itself is parallel to plane of projection.

13. Thank You!