What is the difference between sequential rotations and using ortho projection in CetZ?

Raul_Durand · January 16, 2025, 4:52pm

Is there any fundamental difference between using rotations and ortho projection?

Is

    ortho(x:-90deg, y:-90deg, z:60deg) { ... }

the same as

    rotate(x: -90deg)
    rotate(y: -90deg)
    rotate(z: 60deg)

?

Also, are the rotations in the ortho projection alwas applied in that order, starting from x to z?

Y.D.X · January 17, 2025, 10:24am

I have never used ortho before, but I think the source code agrees with your conjecture.

github.com

cetz-package/cetz/blob/c76142ac9cfc723ee9b41135c78c14ccbe15e19d/src/draw/projection.typ#L128-L129


      
          #let ortho(x: 35.264deg, y: 45deg, z: 0deg, sorted: true, cull-face: none, reset-transform: false, body, name: none) = group(name: name, ctx => {
            _projection(body, ortho-matrix(x, y, z), ortho-projection-matrix,

github.com

cetz-package/cetz/blob/c76142ac9cfc723ee9b41135c78c14ccbe15e19d/src/draw/projection.typ#L9-L15


      
          // Get an orthographic view matrix for 3 angles
          #let ortho-matrix(x, y, z) = matrix.mul-mat(
            matrix.ident(),
            matrix.transform-rotate-x(x),
            matrix.transform-rotate-y(y),
            matrix.transform-rotate-z(z),
          )

github.com

cetz-package/cetz/blob/c76142ac9cfc723ee9b41135c78c14ccbe15e19d/src/draw/transformations.typ#L65


      
          /// rect((-1,-1), (1,1))
          /// // Rotate on y-axis
          /// rotate(y: 80deg)
          /// circle((0,0))
          /// ```
          ///
          /// - ..angles (angle): A single angle as a positional argument to rotate on the z-axis by.
          ///   Named arguments of `x`, `y` or `z` can be given to rotate on their respective axis.
          ///   You can give named arguments of `yaw`, `pitch` or `roll`, too.
          /// - origin (none,coordinate): Origin to rotate around, or (0, 0, 0) if set to `none`.
          #let rotate(..angles, origin: none) = {
            assert(angles.pos().len() == 1 or angles.named().len() > 0,
              message: "Rotate takes a single z-angle or angles " +
                       "(x, y, z or yaw, pitch, roll) as named arguments, got: " + repr(angles))
          
            let named = angles.named()
            let names = named.keys()
          
            let mat = if angles.pos().len() == 1 {
              matrix.transform-rotate-z(angles.pos().at(0))
            } else if names.all(n => n in ("x", "y", "z")) {

github.com

cetz-package/cetz/blob/c76142ac9cfc723ee9b41135c78c14ccbe15e19d/src/draw/transformations.typ#L76-L78


      
          matrix.transform-rotate-xyz(named.at("x", default: 0deg),
                                      named.at("y", default: 0deg),
                                      named.at("z", default: 0deg))

github.com

cetz-package/cetz/blob/c76142ac9cfc723ee9b41135c78c14ccbe15e19d/src/matrix.typ#L200-L214


      
          /// Returns a $4 \times 4$ rotation matrix - euler angles
          ///
          /// Calculates the product of the three rotation matrices
          /// $R = Rz(z) Ry(y) Rx(x)$
          ///
          /// - x (angle): Rotation about x
          /// - y (angle): Rotation about y
          /// - z (angle): Rotation about z
          /// -> matrix
          #let transform-rotate-xyz(x, y, z) = {
            ((cos(y)*cos(z), sin(x)*sin(y)*cos(z) - cos(x)*sin(z), cos(x)*sin(y)*cos(z) + sin(x)*sin(z), 0),
             (cos(y)*sin(z), sin(x)*sin(y)*sin(z) + cos(x)*cos(z), cos(x)*sin(y)*sin(z) - sin(x)*cos(z), 0),
             (-sin(y), sin(x)*cos(y), cos(x)*cos(y), 0),
             (0,0,0,1))
          }