Modeling operations

shell(g, t)

Hollow the feature g into a thin-walled shell of wall thickness 2t.

Shelling is difference(offset(g, t), offset(g, -t)): it grows the solid by t and erodes it by t, so what remains is a skin of thickness 2t following the original surface. The SDF is $|s_g(x)| - t$. For a sphere the volume is closed-form ($4\pi/3\cdot[(R + t)^{3} - (R - t)^{3}]$); other shapes shell exactly wherever the boolean kernel handles the two offsets.

Arguments

• g: the solid to hollow.
• t: half the wall thickness (the wall is 2t thick).

Example

julia> volume(shell(sphere(1.0), 0.2))
5.09356888902025 m^3

julia> vtk(shell(sphere(2.0), 0.2))
VTKView(640×480, inline)

offset(g, d; join=:rounded)

Offset the boundary of feature g by the signed distance d.

Positive d grows the solid (a Minkowski dilation), negative d erodes it. The SDF is simply $s_g(x) - d$, so it stays exact for primitives of the same family: a sphere of radius R becomes R + d, a box of side s becomes s + 2d, a cylinder (r, h) becomes (r + d, h + 2d). The general case is a faceted BRep of the offset field.

Arguments

• g: the feature to offset.
• d: the offset distance (positive grows, negative shrinks).

Keyword arguments

• join = :rounded: the corner treatment — :rounded (circular arcs), :miter (sharp, extended), or :bevel (flat 45° cuts).

Example

julia> volume(offset(sphere(1.0), 0.5))
14.137166941154067 m^3

julia> vtk(offset(sphere(2.0), 0.3))
VTKView(640×480, inline)

dilate(g, r)

Grow a solid outward by a distance r (a Minkowski dilation).

Theory

Dilating a solid by $r$ means "take every point within $r$ of the surface". For a signed distance field this is simply $s(x) - r$: subtracting a constant from the field moves the zero level set outward by exactly that constant. A sphere grows $R \to R + r$, and a box grows a rounded-corner box of the same half-thickness $h + r$ (the corners round because they are exactly the points $r$ from the original corner). It is the same "offset outward" operation as offset, exposed under the mathematical name.

Arguments

• g — an implicit geometry (Feature or ImplicitGeometry).
• r — the dilation distance (a length).

Returns

A new ImplicitGeometry offset outward by r. The inverse operation is a negative r, which shrinks the solid. dilate works at the SDF level, so wrap features in to_sdf(...).

Example

julia> bounds(dilate(to_sdf(box(2.0, 2.0, 2.0)), 0.5))
([-1.5, -1.5, -1.5], [1.5, 1.5, 1.5])

loft(bottom, top, height)
loft(profiles::AbstractVector, heights::AbstractVector)

Loft a solid between two cross-sections (or skin it through many).

A loft morphs one 2D profile into another as z sweeps from $-\mathrm{height}/2$ (bottom) to +height/2 (top). For circle→circle the result is an exact cone or frustum with an analytic volume; other pairs morph through rounded intermediates. The multi-profile form skins a solid through N ≥ 2 closed cross-sections at the given heights.

Arguments

• bottom / top: the two cross-section sketches.
• height: the loft height.
• profiles / heights: (multi-profile form) the cross-sections and their heights.

Example

julia> volume(loft(circle_2d(1.0), circle_2d(2.0), 3.0))
21.991148575128552 m^3

julia> vtk(loft(circle_2d(1.0), circle_2d(2.0), 3.0))
VTKView(640×480, inline)

sweep(profile, path)

Sweep a 2D profile along a path curve, holding it perpendicular to the path.

Sweeping is extrusion along a curve instead of a straight line. The profile is carried along path (a Curve3D such as a spline_path or helix) and the ends are capped with flat disks. The volume is Pappus's area(profile) · path length.

Arguments

• profile: the 2D profile to sweep.
• path: the sweep path (a Curve3D).

Example

julia> volume(sweep(circle_2d(0.5), spline_path([[0.0, 0, 0], [2.0, 0, 0]])))
1.5707963267948966 m^3

julia> vtk(sweep(circle_2d(0.5), spline_path([[0.0, 0, 0], [2.0, 0, 0]])))
VTKView(640×480, inline)

mirror(g, axis)

Mirror the feature g across the plane perpendicular to axis.

axis is a coordinate axis (1 = x, 2 = y, 3 = z) or a datum_plane. The mirror is volume-preserving (it only reflects the shape). In the SDF the query point is reflected before evaluating the child field.

Arguments

• g: the feature to mirror.
• axis: the coordinate axis whose normal plane is the mirror (1, 2, or 3), or a datum_plane.

Example

julia> bounds(evaluate(mirror(sphere(1.0), 1), SDFBackend())) == ([-1.0, -1.0, -1.0], [1.0, 1.0, 1.0])
true

julia> vtk(mirror(sphere(1.0), 1))
VTKView(640×480, inline)

linear_pattern(g, axis, spacing, n)

Repeat the feature g n times along a coordinate axis, spacing apart.

This is the "rectangular pattern" of a part — n equally-spaced copies of g translated along the axis (an integer 1/2/3, or a datum_axis). The SDF is the minimum over the copies, so the copies stay differentiable.

Arguments

• g: the feature to pattern.
• axis: the pattern axis (1 = x, 2 = y, 3 = z) or a datum_axis.
• spacing: the distance between adjacent copies.
• n: the number of copies.

Example

julia> bounds(evaluate(linear_pattern(sphere(0.5), 1, 2.0, 3), SDFBackend())) == ([-0.5, -0.5, -0.5], [4.5, 0.5, 0.5])
true

julia> vtk(linear_pattern(sphere(0.5), 1, 2.0, 3))
VTKView(640×480, inline)

linear_pattern(a::Assembly, name, dir, spacing, n) -> Assembly

Return the assembly with n copies of the named component translated by k·spacing·dir. The original keeps its name; copies are named <name>_2 … <name>_n.

circular_pattern(g, n; axis=3)

Repeat the feature g n times around a rotation axis.

This is the "polar pattern" of a part — n equally-spaced copies of g rotated about the axis (an integer 1/2/3, or a datum_axis). The SDF is the minimum over the rotated copies.

A copy centred on the axis is unchanged by the rotation, so a visible ring needs the copy translated off the axis first.

Arguments

• g: the feature to pattern.
• n: the number of copies.

Keyword arguments

• axis = 3: the rotation axis (1 = x, 2 = y, 3 = z) or a datum_axis.

Example

julia> ring = circular_pattern(translate(sphere(0.5), 1.0, 0, 0), 4);

julia> field(evaluate(ring, SDFBackend()), [0.0, 0.0, 0.0])
0.5

julia> vtk(ring)
VTKView(640×480, inline)

circular_pattern(a::Assembly, name, axis; n) -> Assembly

Return the assembly with n copies of the named component rotated by 2πk/n about the axis (a DatumAxis). Copies are named <name>_2 … <name>_n.

fillet(g, radius; edges=nothing)

Round the convex edges and corners of g by dilating it with a ball of radius.

The whole-body form is a Minkowski dilation ($s_g(x) - r$): it both rounds convex corners and grows the shape by r (exact for spheres and boxes). When edges is given, only those named edges are blended by the exact rolling-ball fillet, which removes material.

Arguments

• g: the feature to fillet.
• radius: the fillet radius.

Keyword arguments

• edges = nothing: a topological edge name (or vector of names, e.g. "Edge3") to fillet selectively, or nothing for the whole body.

Example

julia> volume(fillet(sphere(1.0), 0.3)) == volume(sphere(1.3))
true

julia> vtk(fillet(sphere(1.0), 0.3))
VTKView(640×480, inline)

chamfer(g, distance; edges=nothing)

Cut the convex edges of g with a flat 45° plane of setback distance.

The whole-body form is the L1-metric (Manhattan) analogue of fillet: a dilation by distance in the L1 metric, so it grows the shape with flat 45° edges (exact for an axis-aligned box). When edges is given, only those named edges are cut, which removes material.

Arguments

• g: the feature to chamfer.
• distance: the chamfer setback.

Keyword arguments

• edges = nothing: a topological edge name (or vector of names) to chamfer selectively, or nothing for the whole body.

Example

julia> volume(chamfer(box(2.0, 2.0, 2.0), 0.2))
13.290666666666668 m^3

julia> vtk(chamfer(box(2.0, 2.0, 2.0), 0.2))
VTKView(640×480, inline)

draft(g::Feature, angle; axis::Int=3, ref::Real=0.0)

Taper the cross-section of g along the pull axis by angle (radians).

Drafting is the machining operation that gives molded parts their release angle: each cross-section is scaled by a factor that grows linearly along the pull axis, narrowing (positive angle) toward +axis. ref is the height at which the cross-section is left undistorted (the mold's neutral plane).

Arguments

• g: the feature to draft.
• angle: the taper angle in radians.

Keyword arguments

• axis = 3: the pull axis (1 = x, 2 = y, 3 = z).
• ref = 0.0: the height where the cross-section is undistorted.

Example

julia> bounds(evaluate(draft(box(2.0, 2.0, 2.0), 0.2), SDFBackend())) == ([-1.2027100355086726, -1.2027100355086726, -1.0], [1.2027100355086726, 1.2027100355086726, 1.0])
true

julia> vtk(draft(box(2.0, 2.0, 2.0), 0.2))
VTKView(640×480, inline)