Representations and Transformations

Question 1

With what shape do we draw figures etc.?

Answer 1

Triangels

Question 2

Why do we not use, e.g. quadralaterals to draw shapes?

Answer 2

Because quadralaterals have 4 points and these do not need to be on one plane (Ebene). But for Trinangles, these are always planar.

Question 3

How to we store triangles for an shape, so a connected set of triangles?

Answer 3

Question 4

local vs global coordiante systems

Answer 4

• Concept: Coordinate systems define a space for object placement and manipulation.
  Local vs. Global Coordinates:
• Local Coordinates: Define positions relative to an object’s own origin.
• Global Coordinates: Define positions relative to a global or world origin.

Question 5

What are the benefits of local cooridinate systems?

Answer 5

• Concept: Local coordinate systems allow nested relationships, where objects are defined relative to other objects.
• Example: A room may have furniture in its local coordinates, and each piece of furniture may have its own coordinates for items placed on it.
• Utility: Moving a parent object (like a table) automatically moves all nested child objects (like a book on the table).

Question 6

What types of transformations can i apply to an object?

Answer 6

• Translation: Moves an object.
• Rotation: Spins an object around an axis.
• Scale: Enlarges or shrinks an object.
• Shear: Skews an object along an axis.

Question 7

Translation

Answer 7

• Concept: Translation shifts every point of an object by the same vector.
• Formula: P′=P+T, where T is the translation vector.
• Effect: A simple addition operation that moves an object without altering its size, orientation, or shape.

Question 8

Scailing Transformation

Answer 8

Question 9

How can the Transformations: shear, scale and rotation be performed?

Answer 9

• Concept: Transformations can be constructed by specifying how they act on the standard basis vectors.
• Example: For a matrix M, each column represents the image of a standard basis vector after transformation.
• Application: Multiply the matrix M from the left with the Vektor. If you multiply from the right, you need to transfrom the matrix.

Question 10

Shear transformation

Answer 10

• Concept: Shearing shifts parts of an object to create a skew effect.
• Effect: Changes angles between axes but keeps them parallel.
• Shear Matrix: shifting the two axis in the direction of the shear axis.

Question 11

Shear matrix

Answer 11

Question 12

rotation matrix around the x-Axis

Answer 12

Question 13

rotation matrix around the y-Axis

Answer 13

Question 14

rotation matrix around the z-Axis

Answer 14

Question 15

What is an affine space

Answer 15

A subset of a vector space that includes translations and linear combinations

Question 16

What are affine coordinates?

Answer 16

Question 17

What are barycentric coordinates?

Answer 17

Question 18

Homogeneous Coordinates

Answer 18

Question 19

How do we apply several transformations?

Answer 19

By multiplying then one after the other from the left. Important is that the order in which the transformations are performed matters.

Question 20

How do we rotate around an arbitrary axis r by an angle theta assuming r has unit length.

Answer 20

• Rotation aroud an axis v by an angle theta
  1. generate an orthonomal basis
  2. transform into the new basis
  3. apply the rotation
  4. transform back into the original basis

Question 21

How do i compute an orthonomal basis?

Answer 21

Question 22

How do i use the orthonomal basis to compute a rotation around an arbitroty axis?

Answer 22

Question 23

What does the matrix for rotation around an arbitrary axis look like

Answer 23

Question 24

What does the matrix for rotation around an arbitrary axis look like if the angle alpha < 1

Answer 24

Question 25

How do i rotate around an axis r through an arbitrory point p by an angle alpha?

Answer 25

Question 26

what are nested local coordinate systems

Answer 26

Local coordiante systems are nestet, for example a table has a local coordiante system and on that table is a book that also has a local coordiante system, but its local coordinate system is inside the coordinate system of the table, so that every transformation i do in the coordinate system of the table applies to the coordinate system of the book. The local coordinate system of the book is nested inside the local coordiante system of the table.

Question 27

How do i compute the final location of a item in a nested coordinate system?

Answer 27

You need to transfrom if from the local coorindate system into the coordinate system of its host and so on and so on until you are in the world coordinates. So if G_1 is the matrix that transforms you into the world coordinates then G_1 = L_1 * L_2 * … L_j with L_j beeing the transfromation from your local system into the host system.