3D Rotations and Transformations

This lecture covers 3D rotations, extending the 2D rotation matrix to three dimensions. It explains the right-hand rule for axis orientation and defines elementary rotations about each axis. The lecture discusses combining rotations, Euler angles, and cardinal angles, emphasizing the importance of rotation order due to non-commutative properties. Methods to extract angles from a rotation matrix are presented, along with the gimbal lock problem and its impact on orientation. Finally, the concept of rotation error is introduced, summarizing the key points of 3D rotations.

What is the main focus of the lecture on 3D rotations?

• Developing 2D rotation matrices

• Exploring vector spaces

• Understanding linear transformations

• Extending 2D rotation matrices to 3D

What is a key property of the special orthogonal group in three dimensions?

• All vectors are zero vectors

• All vectors have different magnitudes

• All vectors are parallel

• All vectors are orthonormal

Which rule is used to determine orientation in 3D space?

• Index rule

• Left-hand rule

• Right-hand rule

• Thumb rule

What happens to the x-axis during a rotation purely about the x-axis?

• It becomes zero

• It rotates 90 degrees

• It remains unchanged

• It changes direction

How many combinations of rotations are there in total?

• 12

• 6

• 8

• 10

Why is the order of rotations important?

• Because rotations are always clockwise

• Because rotations are always counterclockwise

• Because matrices are commutative

• Because matrices are not commutative

Which function is used to extract the Phi angle from a rotation matrix?

• cos

• atan2

• tan

• sin

What is a major problem associated with gimbal lock?

• Loss of direction

• Loss of energy

• Loss of altitude

• Loss of speed

What does a rotation error of identity indicate?

• Rotation is infinite

• Rotation is not possible

• Desired rotation equals actual rotation

• Desired rotation is different from actual rotation