Lecture 9/10 3D transformation and projection Flashcards
What is the problem with gimble lock?
Numerically unstable: object might move unexpectedly in between keyframes
When does the gimbal lock occur?
When two axis align with each other
The pairing will be between the parent and middle axis… in XYZ combination the alignment is between X and Y
How many pairing types are there in a gimbal lock?
6:
XYZ
XZY
YXZ
YZX
ZXY
ZYX
Mimic the positive 3D co-ordinate directions on the left hand
Use right-handed coordinates unless otherwise noted
Mimic the positive 3D co-ordinate directions on the right hand
Use right-handed coordinates unless otherwise noted
Visualise a 3D matrix that rotates about the Z axis
This is the same as an anti-clockwise 2D rotation, Z co-ordinates remain the same.
Visualise a 3D matrix that rotates about the X axis
the rows with “1” preserve
the values of related components
Visualise a 3D matrix that rotates about the Y axis
the rows with “1” preserve
the values of related components
Visualise a shear matrix along the X axis
Visualise a 3D rotation axis about Z
Visualise a 3D rotation axis about Y
Visualise a 3D rotation axis about X
How many degrees of freedom a in 3D affine transformation
12 degrees of freedom
How do we generate an arbitrary rotation matrix?
Use an ordered combination of rotations
about the x-, y- and z-axes
Visualise a 3D homogeneous matrix
How many Euler angles are there?
6 … same axis is repeated twice … the first and last rotation are about the same axis
XYX
XZX
YXY
YZY
ZXZ
ZYZ
What are the 7 steps to achieve rotation about an arbitrary point?
What is the axis-angle rotation matrix?
What are the positives?
Axis–angle representation:
+ No gimbal lock
+ Minimal representation
What are the positives and negatives of Euler angles?
Euler Angles (3 ordered rotations):
+ Minimal parametrisation
+ Easy to think about rotations
– Gimbal lock
What are the positives and negatives of a 3×3 matrix (R) ?
3×3 matrix R:
+ No gimbal lock
– Not a minimal representation
What are 2 common projection types? (3D points to 2D)
Perspective projection (right)
- Pinhole camera model
Orthographic projection
- parallel lines stay parallel
- in CAD software
What is the intrinsic camera matrix that maps the 2D world onto the 2D plane?
What is the perspective camera equation?
What is the difference between perspective and projective projection?
In orthographic projection, the objects are projected onto a flat plane using parallel lines, rather than converging lines as in perspective projection.
