Lec 2 - Perspective & Polygonal Geometry

Question 1

How do you map from a 3D space to a 2D image?

Answer 1

The X, Y, Z points in the 3D world (a.k.a voxels) are mapped to x, y points in the 2D world by mapping rays of light moving to/ from the observer.

Question 2

What are vanishing points?

Answer 2

In 3D, parallel lines meet only at infinity, so the vanishing point can be thought of as the projection of a point at infinity.

Question 3

What is the axis vanishing point?

Answer 3

If the set of lines is parallel to one of the three principal axes, the vanishing point is called an axis vanishing point.

Question 4

What are some issues with spherical projection?

Answer 4

1. Doesn’t exactly match perspective projection

2. Only a problem for very wide fields of view

Question 5

What is perspective projection?

Answer 5

The right projection for putting a 3D scene on a flat surface. It is an abstraction of the pin-hole camera.

Question 6

What is parallel projection?

Answer 6

Parallel projection (also known as orthographic projection) is given by:
x = X
y = Y
That is, just drop Z coordinate!

Question 7

What is a polygon?

Answer 7

any plane figure bounded by straight

line segments

Question 8

What are the different types of polygons?

Answer 8

1. polygonal arcs (polylines)
2. polygonal boundaries (closed polylines)
3. filled polygons

Question 9

What are polygons used for?

Answer 9

as building blocks for approximating arbitrary curved arcs and regions

Question 10

How are polygons represented?

Answer 10

1. as a set of line segments
2. an ordered sequence of vertices using absolute or relative coordinates
   Note : Walking order convention often applies

Question 11

What are some types of polygons?

Answer 11

Convex
Concave
Non-Simple
Multiple boundary
Star

Question 12

What are convex polygons?

Answer 12

no internal angle is greater than 180

degrees

Question 13

What are concave polygons?

Answer 13

there are internal angles that can be

greater than 180 degrees

Question 14

How do you represent concave polygons?

Answer 14

as a conjunction of convex polygons (convex polygons have certain properties
that simplify geometric operations and tesselations

Question 15

How can you achieve higher performance using polygons?

Answer 15

Level-of-detail achieves higher performance, by adapting the number of polygons used.

Question 16

What is parallel projection also known as?

Answer 16

orthogonal projection

Question 17

What is virtual camera geometry?

Answer 17

The image projection surface is imagined to be in front of projection center.
Geometrically equivalent to perspective projection
Often more convenient to think about this projection

Question 18

What are the formulas for perspective projection?

Answer 18

Point P = (X,Y,Z) in 3D space has projection (x,y) in the image where
x/f = X/Z
y/f = Y/Z
f being the “focal distance” (sometimes f is called d).

Question 19

In which situations do the perspective formulas apply?

Answer 19

These formulas apply only for camera-centred coordinates, for which perspective projection has a particularly simple form.
For arbitrarily centred coordinate systems 3D transformations are necessary

Question 20

What are “one-point”, “two-point”, and “three-point”perspectives?

Answer 20

just special cases of perspective

projection, depending on how image plane lines up with significant planes in scene.

Question 21

What is spherical projection?

Answer 21

The projection used by the human eye. Lens is curved, though the centre of projection isn’t at centre of the lens (therefore not planar geometric projection

Question 22

What is the vertex split method?

Answer 22

Use an algorithm to decrease the number of polygons in a mesh by combining smaller polygons to form larger ones. This reduces the level of detail.