Rotations
- Dilen Shah
- Jul 3, 2021
- 3 min read
Updated: Jul 7, 2021
From my previous posts of Matrices, we saw that a transformed matrix in Maya stores an object’s orientation information in the upper 3 x3 portions of the transformation matrix, we also learned that in order to be able to edit and query individual transformation components.
Euler Rotations
Maya stores each component separately and stores a rotation as a set of Euler angles. Euler angles are a set of three values describing rotation around each axis. Each of these three Euler angles could be represented as a rotation matrix. When calculating a transformation matrix via multiplies these three matrices together to rotate an object. The order in which these matrices are multiplied is determined by the objects rotation order (xyz, yzx, zxy, xzy, yxz, zyx).
Previously, I mentioned that a matrix is essentially a coordinate system, which means that each of the three rotation matrices can be thought of as their own separate coordinate system that can only be rotated about their associated axis of rotation. By creating a sequence of matrix multipliers, we are in essence creating a small hierarchy of transformations in the default X, Y, Z rotation order, the Z rotation matrix is evaluated first. Any rotation on this coordinate system is propagated to the Y rotation coordinate system and then any rotation on the way coordinate system is propagated to the X coordinate system.
Since each of these coordinate systems can only be rotated about their associated axis of rotation, there’s a chance that by rotating the middle rotation matrix, we make the outer two rotation axis lineup.
This is called gimbal lock and causes the transformation to lose a degree of freedom. Gimbal lock can become an issue when animating rotations on assets like shoulders, camera rigs, and airplanes. So generally you would want to pick a rotation order that allows you to rotate, in a case of a shoulder rig and whatever the most common deformation or animation is going to be, and X is always going to be evaluated last so we can always twist the arm no matter what orientation the arm is in.
So here I have a sample image to represent the three rotation matrices that composes a transformation matrix so you can have a little representation here for the Z-axis, Y-axis, and X-axis in that particular order and this represents the default X, Y, Z rotation order. So what this demonstrates is that since each of this axis are a separate coordinate system, when we rotate the Y rotation, it’s going to also rotate the child coordinate system, which is the x-axis.
Quaternion Rotations
A quaternion is another way to represent an orientation. A quaternion is a mathematical construct containing four values X, Y, Z, and W just like a vector with a homogeneous coordinate.
These values, however, do not have the same meaning or use as a directional vector. These values can be thought of as defining an axis in 3D space and a rotation about that axis. Now the X, Y, and Z values do not specify an axis vector directly. The axis vector of a quaternion specified as follows. The rotation about this axis is calculated with the W component and is found with this equation. Given the W equation, we can solve for Theta and use it to calculate the axis vector.
Most of the time you do not need to work with these equations unless you’re doing some intense R&D work. So what’s the point of a quaternion then? Quaternions are useful because they avoid gimbal lock and they allow us to interpolate between arbitrary rotations. Why then do we use Euler angles and not Quaternions for animation? Euler angles are a lot more intuitive to work with and quaternions. Try telling some animators that they have to animate using the X, Y, Z, and W components of a quaternion instead of the X, Y, Z Euler angles that everyone is used to. Quaternions also have some useful properties that we can use to create some interesting tools. For example, the DOT product between two quaternions can be used to determine how close to Orientations are to each other. Quaternions are not an essential skill to master in 3D graphics, but they do allow for the creation of some interesting tools.
If you still want to learn more on the quaternion rotations, here is a really useful link where Grant Sanderson explains it in really easy terms to follow along.
Thank you to many people who helped me along my way to understand 3D Math myself. Won’t be here if they did not help me learn and share their knowledge. Resources and images were either done by me or found online from Wikipedia and other related websites.
Until next time.
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