CSG operations

Boolean operations

Basic operations

union(a::AbstractGeometry...)

julia> s = union(cube(50), sphere(50)); # equivalently: cube(50) ∪ sphere(50)

N-ary union may be performed in this way:

julia> s = union(([0,10i]+square(5) for i in 1:5)...); # equivalently: ⋃(([0,10i]+square(5) for i in 1:5)...)

intersect(a::AbstractGeometry...)

julia> s = intersect(cube(50), sphere(50)); # equivalently: cube(50) ∩ sphere(50)

setdiff(a::AbstractGeometry, b::AbstractGeometry)

julia> s = setdiff(cube(50), [50,0,0]+sphere(50)); # equivalently: cube(50) - sphere(50)

complement(x::AbstractGeometry)
~x

Complements are provided as a shortcut to simplify “subtractive” operations, i.e. chains of intersections and differences. See Three-dimensional embeddings of two-dimensional objects.

Rewriting rules

The CSG tree as constructed by the user is subjected to immediate rewriting (this is performed by the union, intersect etc. functions), before any meshing function is called.

The following rewriting rules are used:

• associative operations (union, intersect, hull, minkowski) are regrouped:

julia> display(union(union(square(1), circle(1)), polygon([])))
union
├─ Square
├─ Circle
└─ ShapeMesh # 0 polygon(s), 0 vertices

• single-operand unions and intersects are removed:

julia> display(union(square(1)))
Square

• symbolic complements are replaced by the appropriate values:

julia> display(intersect(square(1), ~circle(1), polygon([])))
intersection
├─ difference
│  ├─ Square
│  └─ Circle
└─ ShapeMesh # 0 polygon(s), 0 vertices

• empty unions and intersections are removed:

julia> display(union(square(1),union(),circle(1)))
union
├─ Square
└─ Circle

Convex hull

hull(s::AbstractGeometry...)
hull(s::AbstractGeometry | StaticVector...)

julia> s = hull(cube(50), [50,0,0]+sphere(50));

In the case of mixed dimensions, two-dimensional objects are understood as included in the horizontal plane, unless they have been subjected to a three-dimensional affine transformation; in that case, this transformation is applied to their vertices. See Three-dimensional embeddings of two-dimensional objects.

Minkowski sum

minkowski(s1::AbstractGeometry, s2::AbstractGeometry)

julia> c = cube(10);

julia> s1 = minkowski(square(50), circle(20));

julia> s2 = minkowski(c, cube(20,1,1));

julia> s3 = minkowski(c, polygon([[0,0],[0,30],[30,0]]));

The Minkowski sum between a polygon and a circle of radius r is the same as the offset of the polygon by this radius:

Minkowski sum between volumes is allowed; e.g. the Minkowski sum of two axis-aligned parallelepipeds is again a parallelepiped:

Minkowski sum between a volume and a polygon is also allowed; here the polygon is a triangle in the horizontal plane:

Slicing and projection

Slicing and projection convert a volume to a shape. These transformations are only defined with respect to horizontal planes, since these are the only planes in which canonical (x,y) coordinates are defined.

To use another plane, say the image of the horizontal plane by a rotation R, apply the inverse rotation of R to the object to bring the situation back to the horizontal plane.

slice

slice(z, s...)
slice(s...)

julia> s = slice()*setdiff(sphere(20),sphere(18));

project

project(s...)

julia> s = project()*setdiff(sphere(20),sphere(18));

Intersection with half-space

half(direction, s...; origin = 0)
half(direction; origin = 0) * s

julia> s = half(:bottom)*setdiff(sphere(20),sphere(18));