lec 4 - 2D transformations

Question 1

How are points represented?

Answer 1

Either in absolute position, relative to origin in Cartesian coordinates or polar coordinates.

Question 2

What do vectors represent?

Answer 2

relative displacement

Question 3

What do geometric transformations involve?

Answer 3

General geometric transformations concern object points rather than transformation of the entire coordinate frame,
They involve translations, where x0 = x + xt and y0 = y + yt and linear transformations v0 = Mv (where v0 is a vector and M is a matrix) such as scaling, shear, rotation, reflection.

Question 4

Why are translations and rotations difficult to combine?

Answer 4

Unfortunately, translation is treated differently (via addition) from scaling and rotation (via multiplication).This means translations and rotations cannot be combined easily.

Question 5

What are rigid body transformations?

Answer 5

Transformations that preserve distance : Rotation, translation, and reflection. That is, if two points are a certain distance apart, they will still be the same distance apart after undergoing one of these transformations (or indeed any combination of them).

Question 6

Why do we use the homogeneous coordinate system?

Answer 6

to be able to treat all three transformations (rotation, translation and scaling) in a consistent way, so that they can be combined easily i.e. using the same GPU unit to simplify the pipeline

Question 7

How do homogeneous coordinates work?

Answer 7

Homogeneous coordinates introduces a third dummy coordinate w, represented as (x. y, w). You imagine a third dimension z or (x/z, y/z), where a ray from the origin to point (x, y, z) in three-dimensions pierces the plane at z = 1.

Question 8

What are the benefits of using homogeneous coordinates?

Answer 8

This paradoxically suggests that there is some ideal point (x, y, 0).
They allow for many different homogeneous coordinate representations of the same point, e.g. (2, 3, 6) is the same as (4, 6, 12), etc.

Question 9

How do you homogenize?

Answer 9

In homogeneous coordinates the third dimension is often named w or (wx, wy, w) where w ≠ 0. You homogenize by dividing by w to get (x, y, 1).

Question 10

How does using homogeneous coordinates aid the translation operation?

Answer 10

The additional (redundant) dimension w allows translation to become linear and allows the translation to be written as a matrix product rather than a sum.

Question 11

What are affine transformations?

Answer 11

An affine transformation preserves

1. collinearity between points (ie. three points which lie on a line continue to be collinear after the transformation), and
2. ratios of distances between any two points.

Question 12

What are examples of affine transformations?

Answer 12

translation, rotation, scaling and shear are all affine transformations

Question 13

How can shears be useful?

Answer 13

many useful transforms can be made by appropriate combinations of shears this can have theoretical and computational benefits for example, rotations can be decomposed into shears and used for producing fast method of rotating, without need for using trogonometric function calls

Question 14

How can you use transformations to convert a circle to an ellipse?

Answer 14

We can obtain an ellipse by stretching and rotating the points of a unit circle. This can be achieved by

1. multiplying by a scaling matrix to achieve the ellipsoid shape (note different scaling in x and y directions for ellipse), followed by
2. multiplying by a rotation R to orient the point along the path of the ellipse.