1. Computer Graphics
Projections

2. 3D Viewing Inherently more complex than 2D case.
Most display devices are only 2D
Need to use a projection to transform 3D object or scene to 2D display device.

3. Jetty at Margate England, 1898.
(Jack and Beverly Wilgus)

4. Graphics Cameras real pinhole camera: image inverted
eye
point
image
plane
computer graphics camera: convenient equivalent
stand at p and look through hole. anything in cone visible
shrink hole to a point, cone collapses to line (ray)
eye
point
center of
projection
image
plane

5. Transformations Screen coordinates Local coordinates Projection matrix
World to camera
matrix
Local to world matrix
28/04/2017

6. Projection: Mapping 3-D to 2-D
Scene models are in 3-D space and (but) images are 2-D
so need some way of projecting 3-D to 2-D
Recall, “fundamental challenge”
Planar Projection
Basic approach:
Define a plane in 3-D space
image plane (or film plane)
Project scene onto plane
Map to window viewport
Need to address two basic issues:
how to define plane
how to define mapping onto plane

7. Projections
Transforms points in a coordinate system of dimension n into points in one of less than n (ie 3D to 2D)
Projectors emanate from a centre of projection, pass through every point in the object and intersect a projection surface to form the 2D projection.

8. Projection: Essential Definitions (quick look)
Projectors
View plane (or film plane)
Direction of projection
Center of projection
Eye, viewer projection reference point

9. Planar Geometric Projections
Standard projections project onto a plane (the view plane) – as we’ve seen
Nonplanar projections needed for applications such as map construction
But how big is Greenland?

10. View Projection
We want to create a picture of the scene viewed from the camera
Two sorts of projection
Parallel projection
Perspective projection

11. Moving COP to Infinity as COP moves away, lines approach parallel
when COP at infinity, orthographic view

12. Projections. Two classes of projections : Perspective. Parallel.
Projections are core to the course.
Centre of
Projection. B B B
Centre of
Projection
at infinity B
Perspective

13. Perspective projection is useful for ‘non technical’ communications
Perspective renderings for marketing, etc. are readily obtained with computer-aided drawing (CAD) systems

14. Perspective Projections.
Defined by projection plane and centre of projection.
Visual effect is termed perspective foreshortening.
The size of the projection of an object varies inversely with distance from the centre of projection.
Similar to a camera - Looks realistic !
Not useful for metric information
Parallel lines do not in general project as parallel.
Angles only preserved on faces parallel to the projection plane.
Distances not preserved

15. Remember The Big Idea

16. Perspective Projections
A set of lines not parallel to the projection plane converge at a vanishing point.
Can be thought of in 3D as the projection of a point at infinity.
Homogeneous coordinate is 0 (x,y,0)

17. 1-Point Projection
Projection plane cuts 1 axis only.

18. One-point Perspective
One principal face is parallel to the projection plane

19. 1-Point Perspective
A painting (The Piazza of St. Mark, Venice) done by Canaletto in in one-point perspective

20. Two-point Perspective
One principal direction (i.e. axis) is parallel to the projection plane

21. 2-Point Perspective y z x
Projection plane

22. 2-Point Perspective Painting in two point perspective by Edward Hopper
The Mansard Roof (240 Kb); Watercolor on paper, 13 3/4 x 19 inches;
The Brooklyn Museum, New York

23. Three-point Perspective
Nothing parallel to the projection plane
Usually used when looking up at or down on buildings

24. 3-Point Perspective
Generally held to add little beyond 2-point perspective.
A painting (City Night, 1926) by Georgia O'Keefe, that is approximately in three-point perspective. y z x
Projection plane

25. 3 points

26. Perspective Projection – Simplest Case
Centre of projection at the origin,
Projection plane at z=d.
Projection
Plane. y
P(x,y,z) x
Pp(xp,yp,d) z d

27. Projection Calculations
x axis
y axis
z axis
P=(x, y, z)
(xprp, yprp, zprp)
(xp, yp, zp)
View Plane

28. Review: Basic Perspective Projection
similar triangles y
P(x,y,z)
P(x’,y’,z’) z
z’=d
homogeneous
coords

29. Perspective Projection
Important to understand the assumptions we made to derive this.

30. Perspective Projection

31. Types Of Projections
For anyone who did engineering or technical drawing

32. Parallel Projections
Specified by a direction to the centre of projection, rather than a point.
Centre of projection at infinity.
Orthographic
The normal to the projection plane is the same as the direction to the centre of projection.
Oblique
Directions are different.

33. Orthographic Projections
Most common orthographic
Projection :
Front-elevation,
Side-elevation,
Plan-elevation.
Angle of projection parallel to principal axis; projection plane is perpendicular to axis.
Commonly used in technical drawings

34. Orthographic Projection
Orthographic Projection onto a plane at z = 0.
xp = x , yp = y , z = 0.

35. Isometric Projection
Projection plane normal makes equal angles with each axis.
i.e normal is (dx,dy,dz), |dx| = |dy|=|dz|
Only 8 directions that satisfy this condition.

36. Isometric Projection All 3 axes equally foreshortenedz y
Projection
Plane
120º
All 3 axes equally foreshortened
measurements can be made
Hence the name iso-metric
Normal

37. Orthographic Derivation
scale, translate, reflect for new coord sys

38. Orthographic Derivation
scale, translate, reflect for new coord sys

39. Orthographic Derivation
scale, translate, reflect for new coord sys

40. Orthographic Derivation
scale, translate, reflect for new coord sys

41. Orthographic OpenGL glMatrixMode(GL_PROJECTION); glLoadIdentity();
glOrtho(left,right,bot,top,near,far);

42. Isometric Projections
Isometric projections have been used in computer games from the very early days of the industry up to today
Q*Bert
Sim City
Virtual Magic Kingdom

43. Oblique projections.
Projection plane normal differs from the direction of projection.
Usually the projection plane is normal to a principal axis.
Projection of a face parallel to this plane allows measurement of angles and distance.
Other faces can measure distance, but not angles.
Frequently used in textbooks : easy to draw !

44. Oblique Projection
Direction of projection is not perpendicular to the viewing plane
Most general parallel projection
Is this possible with a normal camera?

45. Oblique projection
Normal
Parallel to x axis y x
Projection
Plane z

46. Geometry of Oblique Projections
Point P=(0,0,1) maps to:
P’=(l.cosa, l.sina, 0) on xy plane,
and P(x,y,z) onto P’(xp,yp,0)
and

47. Orthographic Examples
How would you map an arbitrary bounding volume (nearxyz, farxyz) into the volume defined by (-1, -1, -1) and (1, 1, 1)?
(1) Scale and translate. First translate the center of AABB to 0, 0, 0; then scale by 2 / (near - far)

48. Quiz
I sat in the car, and realized the side mirror is 0.4m on my right and 0.3m in my front
I started my car and drove 5m forward, turned 30 degrees to right, moved 5m forward again, and turned 45 degrees to the right, and stopped
What is the position of the side mirror now, relative to where I was sitting in the beginning?