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TriangleLink`
TriangleLink`

TriangleTriangulate

TriangleTriangulate[expr,settings]

triangulates a Triangle expression using settings and returns the result in a new Triangle expression.

Details and Options

  • To use TriangleTriangulate, you first need to load it using Needs["TriangleLink`"].
  • The settings given to TriangleTriangulate are a string of different commands:
  • "p"triangulate a planar straight line graph (PLC)
    "q"quality mesh generation with no angles smaller than 20 degrees; an alternate minimum angle may be specified after the "q"
    "a"impose a maximum triangle area constraint; a fixed area constraint (that applies to every triangle) may be specified after the "a"
    "A"assign a regional attribute to each triangle that identifies what segment-bounded region it belongs to
    "c"enclose the convex hull with segments
    "D"conforming Delaunay: if all triangles in the mesh are to be Delaunay, not just constrained Delaunay, or if you want to ensure that all Voronoi vertices lie within the triangulation
    "r"refine a previously generated mesh
    "Y"prohibit the insertion of Steiner points on the mesh boundary; if specified twice ("YY"), it prohibits the insertion of Steiner points on any segment, including internal segments
    "S"specify the maximum number of added Steiner points
    "o2"generate second-order subparametric elements with six nodes each
    "C"check the consistency of the final mesh
    "Q"quiet: no terminal output except error
  • TriangleTriangulate has the following options:
  • "TriangleRefinement" Nonefunction to refine a triangle

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

First, load the package:

In[1]:=1
✖
https://wolfram.com/xid/0iwilmq127916721ou0-fxs7lh

This creates an instance of a Triangle expression:

In[2]:=2
✖
https://wolfram.com/xid/0iwilmq127916721ou0-o1426k
Out[2]=2

This sets up points and segments to use:

In[3]:=3
✖
https://wolfram.com/xid/0iwilmq127916721ou0-m4mik5

This sets the points and facets in the Triangle instance:

In[4]:=4
✖
https://wolfram.com/xid/0iwilmq127916721ou0-njb7qi

This carries out the triangulation, returning a new Triangle instance:

In[5]:=5
✖
https://wolfram.com/xid/0iwilmq127916721ou0-zylv5s
Out[5]=5

This extracts the points and elements from the triangulation:

In[6]:=6
✖
https://wolfram.com/xid/0iwilmq127916721ou0-vyvwtg

With the following support function, you can visualize the triangles:

In[7]:=7
✖
https://wolfram.com/xid/0iwilmq127916721ou0-zl8tzc
In[8]:=8
✖
https://wolfram.com/xid/0iwilmq127916721ou0-oo6mjk
Out[8]=8

Options  (1)Common values & functionality for each option

"TriangleRefinement"  (1)

First, load the package:

In[1]:=1
✖
https://wolfram.com/xid/0iwilmq127916721ou0-z5cchz

This creates an instance of a Triangle expression:

In[2]:=2
✖
https://wolfram.com/xid/0iwilmq127916721ou0-66d747
Out[2]=2

This sets up points and segments to use:

In[3]:=3
✖
https://wolfram.com/xid/0iwilmq127916721ou0-l6hmwy

This sets the points and facets in the Triangle instance:

In[4]:=4
✖
https://wolfram.com/xid/0iwilmq127916721ou0-6o8a0u

This carries out the triangulation, returning a new Triangle instance:

In[5]:=5
✖
https://wolfram.com/xid/0iwilmq127916721ou0-i8enpi
Out[5]=5

This extracts the points and elements from the triangulation:

In[6]:=6
✖
https://wolfram.com/xid/0iwilmq127916721ou0-hqh1dz

With the following support function, you can visualize the triangles:

In[7]:=7
✖
https://wolfram.com/xid/0iwilmq127916721ou0-m0ro2h
In[8]:=8
✖
https://wolfram.com/xid/0iwilmq127916721ou0-zzr3vl
Out[8]=8

This sets up a compiled function that returns True if a triangle should be refined and False otherwise:

In[9]:=9
✖
https://wolfram.com/xid/0iwilmq127916721ou0-vvqbsi

This carries out the triangulation with the refinement function, returning a new Triangle instance:

In[10]:=10
✖
https://wolfram.com/xid/0iwilmq127916721ou0-ym9gaq
Out[10]=10

This extracts the points and elements from the triangulation:

In[11]:=11
✖
https://wolfram.com/xid/0iwilmq127916721ou0-2y6u3q

Visualize the triangles:

In[12]:=12
✖
https://wolfram.com/xid/0iwilmq127916721ou0-hgmqx4
Out[12]=12

Neat Examples  (1)Surprising or curious use cases

Use a black-and-white image as a refinement driver. Set an image, create a distance function, and visualize the distance function:

In[1]:=1
✖
https://wolfram.com/xid/0iwilmq127916721ou0-yj81es
In[2]:=2
✖
https://wolfram.com/xid/0iwilmq127916721ou0-mnsz6i
In[3]:=3
✖
https://wolfram.com/xid/0iwilmq127916721ou0-or4rub
Out[3]=3

Create an InterpolatingFunction from the distance function:

In[4]:=4
✖
https://wolfram.com/xid/0iwilmq127916721ou0-vcenyz
Out[4]=4

Create and populate a Triangle instance:

In[5]:=5
✖
https://wolfram.com/xid/0iwilmq127916721ou0-25bouh

Set up a compiled function that calls the InterpolatingFunction:

In[6]:=6
✖
https://wolfram.com/xid/0iwilmq127916721ou0-iqhren

This carries out the triangulation with the refinement function, returning a new Triangle instance:

In[7]:=7
✖
https://wolfram.com/xid/0iwilmq127916721ou0-64t9r7
Out[7]=7

This extracts the points and elements from the triangulation:

In[8]:=8
✖
https://wolfram.com/xid/0iwilmq127916721ou0-dosgxu

Visualize the triangles:

In[9]:=9
✖
https://wolfram.com/xid/0iwilmq127916721ou0-rbqeb
Out[9]=9
Wolfram Research (2014), TriangleTriangulate, Wolfram Language function, https://reference.wolfram.com/language/TriangleLink/ref/TriangleTriangulate.html.
✖
Wolfram Research (2014), TriangleTriangulate, Wolfram Language function, https://reference.wolfram.com/language/TriangleLink/ref/TriangleTriangulate.html.

Text

Wolfram Research (2014), TriangleTriangulate, Wolfram Language function, https://reference.wolfram.com/language/TriangleLink/ref/TriangleTriangulate.html.

✖
Wolfram Research (2014), TriangleTriangulate, Wolfram Language function, https://reference.wolfram.com/language/TriangleLink/ref/TriangleTriangulate.html.

CMS

Wolfram Language. 2014. "TriangleTriangulate." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/TriangleLink/ref/TriangleTriangulate.html.

✖
Wolfram Language. 2014. "TriangleTriangulate." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/TriangleLink/ref/TriangleTriangulate.html.

APA

Wolfram Language. (2014). TriangleTriangulate. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/TriangleLink/ref/TriangleTriangulate.html

✖
Wolfram Language. (2014). TriangleTriangulate. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/TriangleLink/ref/TriangleTriangulate.html

BibTeX

@misc{reference.wolfram_2025_triangletriangulate, author="Wolfram Research", title="{TriangleTriangulate}", year="2014", howpublished="\url{https://reference.wolfram.com/language/TriangleLink/ref/TriangleTriangulate.html}", note=[Accessed: 16-May-2025 ]}

✖
@misc{reference.wolfram_2025_triangletriangulate, author="Wolfram Research", title="{TriangleTriangulate}", year="2014", howpublished="\url{https://reference.wolfram.com/language/TriangleLink/ref/TriangleTriangulate.html}", note=[Accessed: 16-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_triangletriangulate, organization={Wolfram Research}, title={TriangleTriangulate}, year={2014}, url={https://reference.wolfram.com/language/TriangleLink/ref/TriangleTriangulate.html}, note=[Accessed: 16-May-2025 ]}

✖
@online{reference.wolfram_2025_triangletriangulate, organization={Wolfram Research}, title={TriangleTriangulate}, year={2014}, url={https://reference.wolfram.com/language/TriangleLink/ref/TriangleTriangulate.html}, note=[Accessed: 16-May-2025 ]}