Predicates

This section lists predicates that can be used to check properties of geometric objects, both of themselves and relative to other geometric objects.

One important note to make is that these predicates are not necessarily exact. For example, rather than checking if a point p is exactly in a sphere of radius r centered at c, we check if norm(p-c) ≈ r with an absolute tolerance depending on the point type, so p might be slightly outside the sphere but still be considered as being inside. This absolute tolerance can be adjusted in specific scopes as discussed in the Tolerances section.

Robust predicates are often expensive to apply and approximations typically suffice. If needed, consider ExactPredicates.jl or AdaptivePredicates.jl.

isparametrized

Meshes.isparametrized — Function

isparametrized(object)

Tells whether or not the geometric object is parametrized, i.e. can be called as object(u₁, u₂, ..., uₙ) with local coordinates (u₁, u₂, ..., uₙ) ∈ [0,1]ⁿ where n is the parametric dimension.

See also paramdim.

Meshes.paramdim — Function

paramdim(geometry)

Return the number of parametric dimensions of the geometry. For example, a sphere embedded in 3D has 2 parametric dimensions (polar and azimuthal angles).

See also isparametrized.

paramdim(polytope)

Return the parametric dimension or rank of the polytope.

paramdim(connectivity)

Return the parametric dimension of the connectivity.

paramdim(domain)

Return the number of parametric dimensions of the domain as the number of parametric dimensions of its elements.

isperiodic

Meshes.isperiodic — Function

isperiodic(topology)

Tells whether or not the topology is periodic along each parametric dimension.

isperiodic(geometry)

Tells whether or not the geometry is periodic along each parametric dimension.

isperiodic(grid)

Tells whether or not the grid is periodic along each parametric dimension.

issimplex

Meshes.issimplex — Function

issimplex(geometry)

Tells whether or not the geometry is a simplex.

issimplex(connectivity)

Tells whether or not the connectivity is a simplex.

isclosed

Meshes.isclosed — Function

isclosed(chain)

Tells whether or not the chain is closed.

A closed chain is also known as a ring.

isconvex

Meshes.isconvex — Function

isconvex(geometry)

Tells whether or not the geometry is convex.

issimple

Meshes.issimple — Function

issimple(polygon)

Tells whether or not the polygon is simple. See https://en.wikipedia.org/wiki/Simple_polygon.

issimple(chain)

Tells whether or not the chain is simple.

A chain is simple when all its segments only intersect at end points.

hasholes

Meshes.hasholes — Function

hasholes(geometry)

Tells whether or not the geometry contains holes.

point₁ ≤ point₂

Base.:< — Method

<(x, y)

Less-than comparison operator. Falls back to isless. Because of the behavior of floating-point NaN values, this operator implements a partial order.

Implementation

New types with a canonical partial order should implement this function for two arguments of the new type. Types with a canonical total order should implement isless instead.

See also isunordered.

Examples

julia> 'a' < 'b'
true

julia> "abc" < "abd"
true

julia> 5 < 3
false

Base.:> — Method

>(x, y)

Greater-than comparison operator. Falls back to y < x.

Implementation

Generally, new types should implement < instead of this function, and rely on the fallback definition >(x, y) = y < x.

Examples

julia> 'a' > 'b'
false

julia> 7 > 3 > 1
true

julia> "abc" > "abd"
false

julia> 5 > 3
true

Base.:≤ — Method

<=(x, y)
≤(x,y)

Less-than-or-equals comparison operator. Falls back to (x < y) | (x == y).

Examples

julia> 'a' <= 'b'
true

julia> 7 ≤ 7 ≤ 9
true

julia> "abc" ≤ "abc"
true

julia> 5 <= 3
false

Base.:≥ — Method

>=(x, y)
≥(x,y)

Greater-than-or-equals comparison operator. Falls back to y <= x.

Examples

julia> 'a' >= 'b'
false

julia> 7 ≥ 7 ≥ 3
true

julia> "abc" ≥ "abc"
true

julia> 5 >= 3
true

point₁ ⪯ point₂

Meshes.:≺ — Method

≺(A::Point, B::Point)

The product order of points A and B (\prec).

A ≺ B if aᵢ < bᵢ for all coordinates aᵢ and bᵢ.

See https://en.wikipedia.org/wiki/Partially_ordered_set#Orders_on_the_Cartesian_product_of_partially_ordered_sets

Meshes.:≻ — Method

≻(A::Point, B::Point)

The product order of points A and B (\succ).

A ≻ B if aᵢ > bᵢ for all coordinates aᵢ and bᵢ.

See https://en.wikipedia.org/wiki/Partially_ordered_set#Orders_on_the_Cartesian_product_of_partially_ordered_sets

Meshes.:⪯ — Method

⪯(A::Point, B::Point)

The product order of points A and B (\preceq).

A ⪯ B if aᵢ ≤ bᵢ for all coordinates aᵢ and bᵢ.

See https://en.wikipedia.org/wiki/Partially_ordered_set#Orders_on_the_Cartesian_product_of_partially_ordered_sets

Meshes.:⪰ — Method

⪰(A::Point, B::Point)

The product order of points A and B (\succeq).

A ⪰ B if aᵢ ≥ bᵢ for all coordinates aᵢ and bᵢ.

See https://en.wikipedia.org/wiki/Partially_ordered_set#Orders_on_the_Cartesian_product_of_partially_ordered_sets

point ∈ geometry

Base.in — Method

point ∈ geometry

Tells whether or not the point is in the geometry.

geometry₁ ⊆ geometry₂

Base.issubset — Method

geometry₁ ⊆ geometry₂

Tells whether or not geometry₁ is contained in geometry₂.

intersects

Meshes.intersects — Function

intersects(geometry₁, geometry₂)

Tells whether or not geometry₁ and geometry₂ intersect.

References

Gilbert, E., Johnson, D., Keerthi, S. 1988. A fast Procedure for Computing the Distance Between Complex Objects in Three-Dimensional Space

Notes

The fallback algorithm works with any geometry that has a well-defined supportfun.

Meshes.supportfun — Function

supportfun(geometry, direction)

Support function of geometry for given direction.

References

Gilbert, E., Johnson, D., Keerthi, S. 1988. A fast Procedure for Computing the Distance Between Complex Objects in Three-Dimensional Space

outer = [(0,0),(1,0),(1,1),(0,1)]
hole1 = [(0.2,0.2),(0.4,0.2),(0.4,0.4),(0.2,0.4)]
hole2 = [(0.6,0.2),(0.8,0.2),(0.8,0.4),(0.6,0.4)]
poly  = PolyArea([outer, hole1, hole2])
ball1 = Ball((0.5,0.5), 0.05)
ball2 = Ball((0.3,0.3), 0.05)
ball3 = Ball((0.7,0.3), 0.05)
ball4 = Ball((0.3,0.3), 0.15)

intersects(poly, ball1)

true

intersects(poly, ball2)

true

intersects(poly, ball3)

true

intersects(poly, ball4)

true

iscollinear

Meshes.iscollinear — Function

iscollinear(A, B, C)

Tells whether or not the points A, B and C are collinear.

iscoplanar

Meshes.iscoplanar — Function

iscoplanar(A, B, C, D)

Tells whether or not the points A, B, C and D are coplanar.