1. PROJECTIONS
KOYEL DATTA GUPTA

2. Classical Projections

3. Projection

4. Perspective vs Parallel
Computer graphics treats all projections the same and implements them with a single pipeline
Classical viewing developed different techniques for drawing each type of projection
Fundamental distinction is between parallel and perspective viewing even though mathematically parallel viewing is the limit of perspective viewing

5. Taxonomy of Planar Geometric Projections

6. Parallel Projection

7. Orthographic Projection
Projectors are orthogonal to projection surface
Arguably the simplest projection
Image plane is perpendicular to one of the coordinate axes;
Project onto plane by dropping that coordinate;
All rays are parallel.

8. Orthographic Projection
Projection plane parallel to principal face
Usually form front, top, side views

9. Advantages and Disadvantages
Preserves both distances and angles
Shapes preserved
Can be used for measurements
Building plans
Manuals
Cannot see what object really looks like because many surfaces hidden from view
Often we add the isometric

10. Axonometric Projections
Allow projection plane to move relative to object
classify by how many angles of
a corner of a projected cube are
the same
none: trimetric
two: dimetric
three: isometric
根据对立方体进行投影时一个角点处有多少个角相等进行分类
q 1
q 3
q 2

11. Types of Axonometric Projections

13. Advantages and Disadvantages
Lines are scaled (foreshortened) but can find scaling factors
Lines preserved but angles are not
Projection of a circle in a plane not parallel to the projection plane is an ellipse
Can see three principal faces of a box-like object
Some optical illusions possible
Parallel lines appear to diverge
Does not look real because far objects are scaled the same as near objects
Used in CAD applications

14. Oblique Projection
Arbitrary relationship between projectors and projection plane

18. Advantages and Disadvantages
Can pick the angles to emphasize a particular face
Architecture: plan oblique, elevation oblique
Angles in faces parallel to projection plane are preserved while we can still see “around” side

19. Perspective Projection

20. Perspective Projection
Projectors converge at center of projection
Naturally we see things in perspective
Objects appear smaller the farther away they are;
Rays from view point are not parallel.

21. Vanishing Points vanishing point
Parallel lines (not parallel to the projection plan) on the object converge at a single point in the projection (the vanishing point)
Drawing simple perspectives by hand uses these vanishing point(s)

22. Three-Point Perspective
No principal face parallel to projection plane
Three vanishing points for cube

23. Two-Point Perspective
On principal direction parallel to projection plane
Two vanishing points for cube

24. One-Point Perspective
One principal face parallel to projection plane
One vanishing point for cube

25. Advantages and Disadvantages
Objects further from viewer are projected smaller than the same sized objects closer to the viewer (diminution)
Looks realistic
Equal distances along a line are not projected into equal distances (nonuniform foreshortening)
Angles preserved only in planes parallel to the projection plane
More difficult to construct by hand than parallel projections (but not more difficult by computer)

26. Perspective projections
COP at origin
Looking in –z direction
Projection plane in front of origin at z=d

27. Foreshortening
By similar triangles in previous image, we see that and similarly for y.
Using the perspective matrix we get p’ =

28. Q1
Using the origin as the centre of projection, derive the perspective transformation onto the plane passing through the point R0(x0,y0,z0) and having normal vector N=n1i+n2j+n3k

29. A1 P(x,y,z) is projected onto P’(x’,y’,z’) x’=αx, y’= αy , z’= αz
n1x’+n2y’+n3z’=d (where d=n1x0+n2y0+n3z0)
α=d/(n1x+n2y+n3z) d
PerN,R0= d 0 0
0 0 d 0
n1 n2 n3 0

30. Q2
Find the perspective projection onto the view plane z=d where the centre of projection is the origin(0,0,0)

31. Q3
Derive the general perspective transformation onto a plane with reference point R0(x0,y0,z0), normal vector N=n1i+n2j+n3k, and using C(a,b,c) as the centre of projection

32. A3
PerN,R0’=TC. PerN,R0 .T-C

33. Q4
Derive parallel projection onto xy plane in the direction of projection V=ai+bj+ck

34. A4 x’-x=ka , y’-y=kb , z’-z=kc k=-z/c (z’=0 on xy plane) 1 0 -a/c
ParV= b/c