3d transform

Question 1

What is the purpose of 3D transformations in visual computing?

Answer 1

To manipulate objects in 3D space using translation, rotation, scaling, and coordinate system changes.

Question 3

What coordinate system is typically used in 3D graphics?

Answer 3

A right-handed coordinate system.

Question 5

What rule defines positive rotation direction in 3D?

Answer 5

The right-hand rule: thumb points along axis, fingers curl in direction of positive rotation.

Question 7

What is the rotation matrix around the X-axis?

Answer 7

[[1, 0, 0], [0, cosθ, -sinθ], [0, sinθ, cosθ]]

Question 9

What is the rotation matrix around the Y-axis?

Answer 9

[[cosθ, 0, sinθ], [0, 1, 0], [-sinθ, 0, cosθ]]

Question 11

What is the rotation matrix around the Z-axis?

Answer 11

[[cosθ, -sinθ, 0], [sinθ, cosθ, 0], [0, 0, 1]]

Question 13

What are Euler angles used for in 3D graphics?

Answer 13

To represent 3D orientation using three sequential axis rotations.

Question 15

What is an example of an Euler angle rotation order?

Answer 15

R = Rz(γ) · Ry(β) · Rx(α)

Question 17

Why does the order of Euler rotations matter?

Answer 17

Because matrix multiplication is not commutative — different orders yield different results.

Question 19

What is gimbal lock?

Answer 19

A loss of one degree of freedom in rotation when two axes align in Euler angle sequences.

Question 21

What transformation matrix form is used for combining rotation and translation in 3D?

Answer 21

A 4×4 homogeneous transformation matrix.

Question 23

What is the homogeneous coordinate form of a 3D point?

Answer 23

[x, y, z, 1]

Question 25

What does a translation matrix in 3D look like?

Answer 25

[[1, 0, 0, tx], [0, 1, 0, ty], [0, 0, 1, tz], [0, 0, 0, 1]]

Question 27

What is a 3D affine transformation?

Answer 27

A transformation combining linear effects (rotation, scale, shear) with translation.

Question 29

How many degrees of freedom does a 3D affine transform have?

Answer 29

12 degrees of freedom.

Question 31

What is the structure of a 3D affine transformation matrix?

Answer 31

A 4×4 matrix with a 3×3 linear part and a 3×1 translation column.

Question 33

What is a compound transformation?

Answer 33

A sequence of transformations applied in a specific order using matrix multiplication.

Question 35

Why is order important in compound transformations?

Answer 35

Because transformations are not commutative — order changes the result.

Question 37

What is the formula for applying multiple transformations to a point?

Answer 37

p' = T3 · T2 · T1 · p

Question 39

What are local and global coordinate systems?

Answer 39

Local refers to coordinates relative to an object; global refers to the overall world space.

Question 41

What is hierarchical transformation?

Answer 41

Combining transforms from parent to child objects, such as base → arm → hand.

Question 43

What is an example use case of hierarchical transformations?

Answer 43

Robotics, where joint transforms combine to position the end effector.

Question 45

Why use homogeneous coordinates in 3D transformations?

Answer 45

They allow translations to be expressed as matrix operations, enabling unified transformation handling.