From 43701fb086a550bea04f60215d51bd9a93b0b455 Mon Sep 17 00:00:00 2001 From: Evan Pratten Date: Fri, 3 Dec 2021 13:12:09 -0500 Subject: [PATCH] one section left --- content/blog/2021-12-03-quaternions.md | 118 ++++++++++++++++++ .../images/posts/quaternions/dual_quats.jpg | Bin 0 -> 107849 bytes static/images/posts/quaternions/quat_mul.jpg | Bin 0 -> 47142 bytes static/images/posts/quaternions/quat_vec3.png | Bin 0 -> 285273 bytes 4 files changed, 118 insertions(+) create mode 100644 static/images/posts/quaternions/dual_quats.jpg create mode 100644 static/images/posts/quaternions/quat_mul.jpg create mode 100644 static/images/posts/quaternions/quat_vec3.png diff --git a/content/blog/2021-12-03-quaternions.md b/content/blog/2021-12-03-quaternions.md index 81d80b5..1332e4c 100644 --- a/content/blog/2021-12-03-quaternions.md +++ b/content/blog/2021-12-03-quaternions.md @@ -53,4 +53,122 @@ let line = LineSegment2D { }; ``` +## An intro to quaternions +> In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space.
+> \[[Wikipedia](https://en.wikipedia.org/wiki/Quaternion)\] + +Quaternions are essentially 4-dimensional numbers consisting of one real, and three imaginary components. They are expressed with the components $w$, $i$, $j$, and $k$. So far, this sounds pretty scary, but it is a lot simpler to deal with when you remember mathematicians like overcomplicating their variable names :wink: + +The following is a slightly modified version of the quaternion expression, made to express the concept in a more computer-oriented manner: + +$$ +q = w + x \mathbf{i} + y \mathbf{j} + z \mathbf{k} +$$ + +$i$, $j$, and $k$ can be interpreted as unit-vectors pointing along the three spatial axes. + +You may notice the quaternion expression is split into two distinct parts: the real and vector parts: + +$$ +q = \overbrace{w}^{\text{real}} + \overbrace{x \mathbf{i} + y \mathbf{j} + z \mathbf{k}}^{\text{vector}} +$$ + +The real part is a single real number, and the vector part is simply a 3-dimensional vector ($x,y,z$) with some pesky imaginary basis vectors attached to the components ($i,j,k$). From a programming standpoint, we can simply ignore the imaginary numbers, and treat a quaternion as a structure similar to the following: + +```rust +// Don't actually implement a quaternion this way +// This is just to show how I imagine them when working with them +struct Quaternion { + pub real: f32, + pub vector: Vector3 +} +``` + +An important fact to keep in mind is that quaternions can and will be found in their normalized form, where all the coefficients (real and vector) are values between $-1$ and $1$. + +
+ +
+ +## Quaternions are transformations + +For the next few sections, there is a single important detail: + +Quaternions are transformations + +Like everything else in this post, this statement is not true for all applications of quaternions, but is true for the two core uses in graphics programming: rotations and translations. + +You can essentially treat a quaternion as an action applied to a 3-dimensional vector. For example, a quaternion that rotates $90^{\circ}$ around the $z$ axis, applied to the vector $\big[\begin{smallmatrix}1 \\\\ 0 \\\\ 0\end{smallmatrix}\big]$ would produce: $\big[\begin{smallmatrix}0 \\\\ -1 \\\\ 0\end{smallmatrix}\big]$. Similairly, a quaternion could move a vector to a new origin (turning it into a line segment). We will get to the second transformation in a bit. + +### Transforms: expressed mathematically + +To transform any vector by a quaternion ($q$ in this case), you must first turn the vector into a pure quaternion (in this case, $k$): + +$$ +\begin{aligned} +v &= \begin{bmatrix} 1 \\\\ 2 \\\\ 3 \end{bmatrix} \\\\ +k &= 0 + 1 \mathbf{i} + 2 \mathbf{j} + 3 \mathbf {k} +\end{aligned} +$$ + +As you can see, a pure quaternion is a quaternion with a real part of $0$ and a vector part equal to the vector it is being made of. + +Next, you need the conjugate of the quaternion (which is called $q^\*$). The following are both the expressions for a quaternion, and a conjugate quaternion: + +$$ +\begin{aligned} +q &= w + x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \\\\ +q^\* &= w - x \mathbf{i} - y \mathbf{j} - z \mathbf{k} +\end{aligned} +$$ + +With all the required parts, the transformed vector ($p$) is equal to: + +$$ +p = q^\*kq +$$ + +NOTE: Quaternion multiplicaiton is [noncommutative](https://en.wikipedia.org/wiki/Commutative_property), meaning that the order you multiply things matters. This is not normal multiplication. + +
+ +
+ +### Expressing a rotation with a quaternion + +An extremely intuitive way to work with rotations in general, and especially in the quaternion world, is via something called an axis-angle. + +The name is quite self-explanitory, but to describe it better, an axis-angle is made up of a 3-dimensional unit vector pointing in an arbitrary direction, and a single angle (in radians) describing the clockwise twist around that axis. + +Conveniently, as I pointed out earlier, quaternions also work in the same way, with a $w$ value (essentially the angle), and a vector part (essentially the axis). In rust, we can construct a quaternion from an axis-angle as follows: + +```rust +// In this example, we will describe a rotation of +// 90 degrees around the positive Z axis +let axis = Vector3::z_axis(); +let angle = std::f32::consts::FRAC_PI_2; + +let quaternion = UnitQuaternion::from_axis_angle(&axis, angle); +``` + +Now, if we wanted to rotate the vector $\big[\begin{smallmatrix}1 \\\\ 2 \\\\ 3\end{smallmatrix}\big]$ by `quaternion`, we could simply write: + +```rust +let rotated_vector = quaternion.transform_vector(&Vector3::new(1.0, 2.0, 3.0)); +``` + +### Expressing a translation and a rotation with a quaternion + +## Further "reading" + +So, do you know what you are doing when it comes to quaternions now? + +No? + +Well, I can't say I'm surprised. Now that you have learned the basics of quaternions, I recommend working through [Grant Sanderson](https://www.3blue1brown.com/)'s videos on the topic: + + + +

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