Transforms and matrices

Transforms and matrices

For basic transformations of the drawing space, use scale(sx, sy), rotate(a), and translate(tx, ty).

translate() shifts the current axes by the specified amounts in x and y. It's relative and cumulative, rather than absolute:

origin()
for i in range(0, step=30, length=6)
    sethue(HSV(i, 1, 1)) # from Colors
    setopacity(0.5)
    circle(0, 0, 40, :fillpreserve)
    setcolor("black")
    strokepath()
    translate(50, 0)
end

translate

scale(x, y) or scale(n) scales the current workspace by the specified amounts. Again, it's relative to the current scale, not to the document's original.

origin()
for i in range(0, step=30, length=6)
    sethue(HSV(i, 1, 1)) # from Colors
    circle(0, 0, 90, :fillpreserve)
    setcolor("black")
    strokepath()
    scale(0.8, 0.8)
end

scale

rotate() rotates the current workspace by the specified amount about the current 0/0 point. It's relative to the previous rotation, not to the document's original.

origin()
for i in 1:8
    randomhue()
    squircle(Point(40, 0), 20, 30, :fillpreserve)
    sethue("black")
    strokepath()
    rotate(π/4)
end

rotate

To return home after many changes, you can use setmatrix([1, 0, 0, 1, 0, 0]) to reset the matrix to the default. origin() resets the matrix then moves the origin to the center of the page.

rescale() is a convenient utility function for linear interpolation, also called a "lerp".

Luxor.scale — Function.
scale(x, y)

Scale workspace by x and y.

Example:

scale(0.2, 0.3)
scale(f)

Scale workspace by f in both x and y.

Luxor.rotate — Function.
rotate(a::Float64)

Rotate workspace by a radians clockwise (from positive x-axis to positive y-axis).

Luxor.translate — Function.
translate(point)
translate(x::Real, y::Real)

Translate the workspace to x and y or to pt.

Scaling of lines

Line thicknesses are not scaled. For example, with a line thickness set by setline(1), lines drawn before and after scale(2) will be the same thickness. If you want line thicknesses to respond to the current scale, so that after say setline(1), lines change thickness after calls to scale(n), you could define your own strokeraw() function that calls the cairo_stroke primitive directly:

import Cairo
function strokeraw()
    ccall((:cairo_stroke, Cairo._jl_libcairo), Nothing, (Ptr{Nothing},), Luxor.get_current_cr().ptr)
end

Matrices and transformations

In Luxor, there's always a current matrix. It's a six element array:

\[\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{bmatrix}\]

which is usually handled in Julia/Cairo/Luxor as a simple vector/array:

julia> getmatrix()
6-element Array{Float64,1}:
   1.0
   0.0
   0.0
   1.0
   0.0
   0.0

transform(a) transforms the current workspace by 'multiplying' the current matrix with matrix a. For example, transform([1, 0, xskew, 1, 50, 0]) skews the current matrix by xskew radians and moves it 50 in x and 0 in y.

function boxtext(p, t)
    sethue("grey30")
    box(p, 30, 50, :fill)
    sethue("white")
    textcentered(t, p)
end

for i in 0:5
    xskew = tand(i * 5.0)
    transform([1, 0, xskew, 1, 50, 0])
    boxtext(O, string(round(rad2deg(xskew), digits=1), "°"))
end

transform

getmatrix() gets the current matrix, setmatrix(a) sets the matrix to array a.

Luxor.getmatrix — Function.
getmatrix()

Get the current matrix. Returns an array of six float64 numbers:

  • xx component of the affine transformation

  • yx component of the affine transformation

  • xy component of the affine transformation

  • yy component of the affine transformation

  • x0 translation component of the affine transformation

  • y0 translation component of the affine transformation

Some basic matrix transforms:

  • translate

transform([1, 0, 0, 1, dx, dy]) shifts by dx, dy

  • scale

transform([fx 0 0 fy 0 0]) scales by fx and fy

  • rotate

transform([cos(a), -sin(a), sin(a), cos(a), 0, 0]) rotates around to a radians

rotate around O: [c -s s c 0 0]
  • shear

transform([1 0 a 1 0 0]) shears in x direction by a

  • shear in y: [1 B 0 1 0 0]

  • x-skew

transform([1, 0, tan(a), 1, 0, 0]) skews in x by a

  • y-skew

transform([1, tan(a), 0, 1, 0, 0]) skews in y by a

  • flip

transform([fx, 0, 0, fy, centerx * (1 - fx), centery * (fy-1)]) flips with center at centerx/centery

  • reflect

transform([1 0 0 -1 0 0]) reflects in xaxis

transform([-1 0 0 1 0 0]) reflects in yaxis

When a drawing is first created, the matrix looks like this:

getmatrix() = [1.0, 0.0, 0.0, 1.0, 0.0, 0.0]

When the origin is moved to 400/400, it looks like this:

getmatrix() = [1.0, 0.0, 0.0, 1.0, 400.0, 400.0]

To reset the matrix to the original:

setmatrix([1.0, 0.0, 0.0, 1.0, 0.0, 0.0])
Luxor.setmatrix — Function.
setmatrix(m::Array)

Change the current matrix to matrix m. Use getmatrix() to get the current matrix.

Luxor.transform — Function.
transform(a::Array)

Modify the current matrix by multiplying it by matrix a.

For example, to skew the current state by 45 degrees in x and move by 20 in y direction:

transform([1, 0, tand(45), 1, 0, 20])

Use getmatrix() to get the current matrix.

Luxor.crossproduct — Function.
crossproduct(p1::Point, p2::Point)

This is the perp dot product, really, not the crossproduct proper (which is 3D):

Luxor.blendmatrix — Function.
blendmatrix(b::Blend, m)

Set the matrix of a blend.

To apply a sequence of matrix transforms to a blend:

A = [1 0 0 1 0 0]
Aj = cairotojuliamatrix(A)
Sj = scalingmatrix(2, .2) * Aj
Tj = translationmatrix(10, 0) * Sj
A1 = juliatocairomatrix(Tj)
blendmatrix(b, As)
Luxor.rotationmatrix — Function.
rotationmatrix(a)

Return a 3x3 Julia matrix that will apply a rotation through a radians.

Luxor.scalingmatrix — Function.
scalingmatrix(sx, sy)

Return a 3x3 Julia matrix that will apply a scaling by sx and sy.

Luxor.translationmatrix — Function.
translationmatrix(x, y)

Return a 3x3 Julia matrix that will apply a translation in x and y.

Use the getscale(), gettranslation(), and getrotation() functions to find the current values of the current matrix. These can also find the values of arbitrary 3x3 matrices.

Luxor.getscale — Function.
getscale(R::Matrix)
getscale()

Get the current scale of a Julia 3x3 matrix, or the current Luxor scale.

Returns a tuple of x and y values.

Luxor.gettranslation — Function.
gettranslation(R::Matrix)
gettranslation()

Get the current translation of a Julia 3x3 matrix, or the current Luxor translation.

Returns a tuple of x and y values.

Luxor.getrotation — Function.
getrotation(R::Matrix)
getrotation()

Get the rotation of a Julia 3x3 matrix, or the current Luxor rotation.

\[\begin{bmatrix} a & b & tx \\ c & d & ty \\ 0 & 0 & 1 \\ \end{bmatrix}\]

The rotation angle is atan(-b, a) or atan(c, d).

You can convert between the 6-element and 3x3 versions of a transformation matrix using the functions cairotojuliamatrix() and juliatocairomatrix().

Luxor.cairotojuliamatrix — Function.
cairotojuliamatrix(c)

Return a 3x3 Julia matrix that's the equivalent of the six-element matrix in c.

Luxor.juliatocairomatrix — Function.
juliatocairomatrix(c)

Return a six-element matrix that's the equivalent of the 3x3 Julia matrix in c.