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Triangulation Statistics

DelaunayTriangulation.IndividualTriangleStatistics — Type
IndividualTriangleStatistics{T}

Struct storing statistics of a single triangle.

Fields

  • area::T: The area of the triangle.
  • lengths::NTuple{3,T}: The lengths of the edges of the triangle, given in sorted order.
  • circumcenter::NTuple{2,T}: The circumcenter of the triangle.
  • circumradius::T: The circumradius of the triangle.
  • angles::NTuple{3, T}: The angles of the triangle, given in sorted order.
  • radius_edge_ratio::T: The ratio of the circumradius to the shortest edge length.
  • edge_midpoints::NTuple{3,NTuple{2,T}}: The midpoints of the edges of the triangle.
  • aspect_ratio::T: The ratio of the inradius to the circumradius.
  • inradius::T: The inradius of the triangle.
  • perimeter::T: The perimeter of the triangle.
  • centroid::NTuple{2,T}: The centroid of the triangle.
  • offcenter::NTuple{2,T}: The offcenter of the triangle with radius-edge ratio cutoff β=1. See this paper.
  • sink::NTuple{2,T}: The sink of the triangle relative to the parent triangulation. See this paper.

Constructors

The constructor is 

IndividualTriangleStatistics(p, q, r, sink = (NaN, NaN))

where p, q, and r are the coordinates of the triangle given in counter-clockwise order. sink is the triangle's sink. This must be provided separately since it is only computed relative to a triangulation, and so requires vertices rather than coordinates; see triangle_sink.

Extended help

The relevant functions used for computing these statistics are

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DelaunayTriangulation.triangle_area — Function
triangle_area(ℓ₁², ℓ₂², ℓ₃²) -> Number

Compute the area of a triangle given the squares of its edge lengths. If there are precision issues that cause the area to be negative, then the area is set to zero.

See also squared_triangle_area.

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triangle_area(p, q, r) -> Number

Computes the area of the triangle with coordinates p, q, r.

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DelaunayTriangulation.triangle_circumradius — Function
triangle_circumradius(A, ℓmin², ℓmed², ℓmax²) -> Number

Computes the circumradius of a triangle with area A and squared edge lengths ℓmin² ≤ ℓmed² ≤ ℓmax². The circumradius is given by

\[r = \dfrac{\ell_{\min}\ell_{\text{med}}\ell_{\max}}{4A}.\]

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triangle_circumradius(p, q, r) -> Number

Computes the circumradius of the triangle with coordinates (p, q, r).

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DelaunayTriangulation.triangle_perimeter — Function
triangle_perimeter(ℓmin::Number, ℓmed::Number, ℓmax::Number) -> Number

Computes the perimeter of a triangle with edge lengths ℓmin ≤ ℓmed ≤ ℓmax. The perimeter is given by

\[P = \ell_{\min} + \ell_{\text{med}} + \ell_{\max}.\]

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triangle_perimeter(p, q, r) -> Number

Computes the perimeter of the triangle with coordinates (p, q, r).

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DelaunayTriangulation.triangle_inradius — Function
triangle_inradius(A, perimeter) -> Number

Computes the inradius of a triangle with area A and perimeter perimeter. The inradius is given by

\[r_i = \dfrac{2A}{P},\]

where $P$ is the perimeter.

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triangle_inradius(p, q, r) -> Number

Computes the inradius of the triangle with coordinates (p, q, r).

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DelaunayTriangulation.triangle_aspect_ratio — Function
triangle_aspect_ratio(inradius::Number, circumradius::Number) -> Number

Computes the aspect ratio of a triangle with inradius inradius and circumradius circumradius. The aspect ratio is given by

\[\tau = \dfrac{r_i}{r},\]

where $r_i$ is the inradius and $r$ is the circumradius.

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triangle_aspect_ratio(p, q, r) -> Number

Computes the aspect ratio of the triangle with coordinates (p, q, r).

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DelaunayTriangulation.triangle_radius_edge_ratio — Function
triangle_radius_edge_ratio(circumradius::Number, ℓmin::Number) -> Number

Computes the radius-edge ratio of a triangle with circumradius circumradius and minimum edge length ℓmin, given by

\[\rho = \dfrac{r}{\ell_{\min}},\]

where $r$ is the circumradius and $\ell_{\min}$ is the shortest edge length.

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triangle_radius_edge_ratio(p, q, r) -> Number

Computes the radius-edge ratio of the triangle with coordinates (p, q, r).

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DelaunayTriangulation.triangle_angles — Function
triangle_angles(p, q, r) -> (Number, Number, Number)

Computes the angles of a triangle with vertices p, q, and r. The formula for, say, the angle at p is given by

\[\theta_1 = \arctan\left(\dfrac{2A}{\left(p - q\right)\cdot\left(p - r\right)}\right),\]

where A is the area of the triangle. The angles are returned in sorted order.

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DelaunayTriangulation.triangle_lengths — Function
triangle_lengths(p, q, r) -> (Number, Number, Number)

Computes the lengths of the edges of the triangle with coordinates p, q, r. The lengths are returned in sorted order.

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DelaunayTriangulation.triangle_circumcenter — Function
triangle_circumcenter(p, q, r, A=triangle_area(p, q, r)) -> (Number, Number)

Computes the circumcenter of the triangle with coordinates (p, q, r). The circumcenter is given by

\[c_x = r_x + \dfrac{d_{11}d_{22} - d_{12}d_{21}}{4A}, \quad c_y = r_y + \dfrac{e_{11}e_{22} - e_{12}e_{21}}{4A},\]

where $d_{11} = \|p - r\|_2^2$, $d_{12} = p_y - r_y$, $d_{21} = \|q - r\|_2^2$, $d_{22} = q_y - r_y$, $e_{11} = p_x - r_x$ $e_{12} = d_{11}$, $e_{21} = q_x - r_x$, and $e_{22} = d_{21}$.

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triangle_circumcenter(tri::Triangulation, T) -> (Number, Number)

Computes the circumcenter of the triangle T in the triangulation tri.

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DelaunayTriangulation.triangle_offcenter — Function
triangle_offcenter(p, q, r, c₁=triangle_circumcenter(p, q, r), β=1.0) -> (Number, Number)

Computes the off-center of the triangle (p, q, r).

Arguments

  • p, q, r: The coordinates of the triangle, given in counter-clockwise order.
  • c₁=triangle_circumcenter(p, q, r): The circumcenter of the triangle.
  • β=1.0: The radius-edge ratio cutoff.

Output

  • cx: The x-coordinate of the off-center.
  • cy: The y-coordinate of the off-center.
Difference in definitions

In the original this paper, the off-center is defined to instead be the circumcenter if it the triangle pqc₁ has radius-edge ratio less than β. Here, we just let the off-center be the point c so that pqc has radius-edge ratio of exactly β.

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DelaunayTriangulation.triangle_sink — Function
triangle_sink(p, q, r, tri::Triangulation) -> (Number, Number)

Computes the sink of each triangle in tri. See this paper for more information.

Extended help

Sinks were introduced in this paper. For a given triangle T, the sink of T is defined as follows:

  1. If c, the circumcenter of T, is in the interior of T, then the sink of T is T.
  2. If T is a boundary triangle, then the sink of T is T.
  3. If neither 1 or 2, then the sink is defined as the sink of the triangle V, where V is the triangle adjoining the edge of T which intersects the line mc, where m is the centroid of T.

In cases where the triangulation has holes, this definition can lead to loops. In such a case, we just pick one of the triangles in the loop as the sink triangle.

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DelaunayTriangulation.TriangulationStatistics — Type
TriangulationStatistics{T,V,I}

A struct containing statistics about a triangulation.

Fields

  • num_vertices::I: The number of vertices in the triangulation.
  • num_solid_vertices::I: The number of solid vertices in the triangulation.
  • num_ghost_vertices::I: The number of ghost vertices in the triangulation.
  • num_edges::I: The number of edges in the triangulation.
  • num_solid_edges::I: The number of solid edges in the triangulation.
  • num_ghost_edges::I: The number of ghost edges in the triangulation.
  • num_triangles::I: The number of triangles in the triangulation.
  • num_solid_triangles::I: The number of solid triangles in the triangulation.
  • num_ghost_triangles::I: The number of ghost triangles in the triangulation.
  • num_boundary_segments::I: The number of boundary segments in the triangulation.
  • num_interior_segments::I: The number of interior segments in the triangulation.
  • num_segments::I: The number of segments in the triangulation.
  • num_convex_hull_vertices::I: The number of vertices on the convex hull of the triangulation.
  • smallest_angle::V: The smallest angle in the triangulation.
  • largest_angle::V: The largest angle in the triangulation.
  • smallest_area::V: The smallest area of a triangle in the triangulation.
  • largest_area::V: The largest area of a triangle in the triangulation.
  • smallest_radius_edge_ratio::V: The smallest radius-edge ratio of a triangle in the triangulation.
  • largest_radius_edge_ratio::V: The largest radius-edge ratio of a triangle in the triangulation.
  • area::V: The total area of the triangulation.
  • individual_statistics::Dict{T,IndividualTriangleStatistics{V}}: A map from triangles in the triangulation to their individual statistics. See IndividualTriangleStatistics.

Constructors

To construct these statistics, use statistics, which you call as statistics(tri::Triangulation).

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