5 - Projective Geometry

Question 1

The coordinate system uses 5 main sets of coordinates

Answer 1

local,
world,
view,
clip
screen coordinates

Question 2

Projective Geometry

Answer 2

Projections transform points in n-space to m-space, where m < n. Transforming from 3D coordinates to 2D is known as projective transformation

Question 3

Perspective projection vs Parallel

Answer 3

Perspective :
where the distance from the centre of projection to the projection plane is infinite.

+ size varies inversely to distance - realistic
- distance and angles are not preserved
- parallel lines do not remain parallel

Parallel:
where the distance from the centre of projection to the projection plane is finite. (projection lines are parallel to each other)

+ good for exact measurements
+ parallel lines remain parallel
- distance and angles are not preserved
- less realistic

Question 4

For parallel projections, a Direction of Projection (DOP) may be specified instead of a COP.

There are 3 types of parallel projection:

Answer 4

orthographic, axonometric, and oblique.

Question 5

orthographic projection :

Answer 5

DOP is perpendicular to the projection plane.

Question 6

Perspective Projection types

Answer 6

Pinhole camera model
Frustum projection

Question 7

Pinhole camera model

Answer 7

assumes that light passes through a single point (the “pinhole”) before forming an image on the opposite side of a darkened box.

The distance between the aperture and the light-sensitive surface is called the focal length, f.

Question 8

Frustum projection model

Answer 8

uses a pyramid-shaped frustum to project a 3D scene onto a 2D screen.

Question 9

The Projection Matrix Pinhole Camera Model homogenous formula

Answer 9

|y1| |f/s1 0 o1 0 | |x1|
|y2| = | 0 f/s2 o2 0 | |x2|
|y3| | 0 0 1 0 | |x3|
| 1 |

Question 10

Frustum Definition

Answer 10

A frustum is defined by a near clipping distance n, a far clipping distance f and the left l, right r, top t and bottom b clipping planes – it generates the following matrix:

Question 11

Viewport Transformation

Answer 11

transforms from clip/canonical space (the normalised [-1, 1] × [-1, 1] space) to the viewport’s pixel coordinates, within [umin, umax] × [vmin, vmax].

Question 12

Viewport Transformation steps

Answer 12

This involves 2D translation and scaling:
⦁ Translate canonical volume to its corner
⦁ Scale to fit the size of the 3D viewport
⦁ Translate to the lower left corner