Day 14 - nalgebra

Relevancy: 1.9 stable

The nalgebra crate provides a wide set of mathematical primitives for linear algebra, computer physics, graphic engines etc. I'm not going to dive deep into the underlying math, there are a lot of tutorials and courses, some of them specifically targeted at the programmers. My goal for today is just a brief showcase of what we can do in Rust with nalgebra.

Basic vector and matrix operations

let v = Vector2::new(0.0f64, 1.0);
    // 90 degrees clockwise
    //  0, 1
    // -1, 0
    let rot = Matrix2::new(0.0f64, -1.0, 1.0, 0.0);
    println!("{:?}", rot * v);
$ cargo run
Vec2 { x: 1, y: 0 }

In nalgebra there are several statically sized vector and square matrix types (for dimensions up to 6). The standard mathematical operators are overloaded, so all allowed kinds of vector/matrix multiplication should just work. In the example above we defined the rotation matrix ourselves, but there is a nice shortcut: the RotN type.

let angle = FRAC_PI_2;
    let rot = Rotation2::new(Vector1::new(angle));
    println!("{:?}", rot * v);

The output is the same but this time we tell Rust what to do, not how to do it. Note that we need to wrap the angle in a single-element vector.

We can use vectors to translate (move) points.

let point = Point2::new(4.0f64, 4.0);
    println!("Translate from {:?} to {:?}",

A number of other operations are also exposed as top-level functions, such as transform(), rotate() along with their inverse counterparts.

Dot and cross product

nalgebra::translate(&v, &point));

    let v1 = Vector3::new(2.0f64, 2.0, 0.0);
    let v2 = Vector3::new(2.0f64, -2.0, 0.0);
    if nalgebra::approx_eq(&0.0f64, &nalgebra::dot(&v1, &v2)) {
        println!("v1 is orthogonal to v2");

The output is:

$ cargo run
v1 is orthogonal to v2
Vec3 { x: 0, y: 0, z: -8 }
Vec3 { x: 0, y: 0, z: 8 }

Dot product can be used to check if two vectors are orthogonal to each other. That happens if their dot product is equal to 0. As floating point comparisons are sometimes suprising, we should use the approx_eq() function.

The cross product of two vectors is always perpendicular (sometimes we say normal) to both of them. As you can see from the example it is also not commutative. Normal vectors are very important in computer graphics for calculating light and shading of the scene.

Dynamic vectors

All of the nalgebra types we've seen so far have their higher-dimensional variants up to Vec6/Mat6 etc. But what if we want to go further? Very high number of dimensions is common for example in digital signal processing. In nalgebra there is a DVec type for that purpose.

println!("{:?}", nalgebra::cross(&v2, &v1));

    const SIZE: usize = 512;
    let sine = DVector::from_fn(SIZE, |i: usize| {
        let t = i as f64 / 16.0f64;
    draw(&sine, Path::new("out_sine.png"));

    let window = DVector::from_fn(SIZE, |i: usize| {
        0.54f64 - 0.46 * (PI * 2.0 * (i as f64) / (SIZE - 1) as f64).cos()
    draw(&window, Path::new("out_window.png"));

We can use the from_fn() mwthod to create a vector by generating each element in a closure. The window variable is a Hamming window; such window functions are a common preprocessing step in DSP.

The draw() function borrows a DVec and a Path and plots a simple representation of the vector to a PNG file using the image crate. The code is on GitHub for anybody interested. Here's the output of all three steps of the above code:

sine, window, windowed sine

See also