--- title: "TriangleTriangulate" language: "en" type: "Symbol" summary: "TriangleTriangulate[expr, settings] triangulates a Triangle expression using settings and returns the result in a new Triangle expression." keywords: - triangle - triangle mesh - mesh - triangulate - delaunay - delaunay mesh - refinement - mesh refinement - boundary mesh - area mesh - grid canonical_url: "https://reference.wolfram.com/language/TriangleLink/ref/TriangleTriangulate.html" source: "Wolfram Language Documentation" related_guides: - title: "TriangleLink" link: "https://reference.wolfram.com/language/TriangleLink/guide/TriangleLink.en.md" related_functions: - title: "TriangleCreate" link: "https://reference.wolfram.com/language/TriangleLink/ref/TriangleCreate.en.md" - title: "TriangleSetPoints" link: "https://reference.wolfram.com/language/TriangleLink/ref/TriangleSetPoints.en.md" - title: "TriangleGetPoints" link: "https://reference.wolfram.com/language/TriangleLink/ref/TriangleGetPoints.en.md" - title: "TriangleSetSegments" link: "https://reference.wolfram.com/language/TriangleLink/ref/TriangleSetSegments.en.md" - title: "TriangleGetSegments" link: "https://reference.wolfram.com/language/TriangleLink/ref/TriangleGetSegments.en.md" related_tutorials: - title: "TriangleLink User Guide" link: "https://reference.wolfram.com/language/TriangleLink/tutorial/Overview.en.md" --- ## 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" [`None`](https://reference.wolfram.com/language/ref/None.en.md) function to refine a triangle --- ## Examples (3) ### Basic Examples (1) First, load the package: ```wl In[1]:= Needs["TriangleLink`"] ``` This creates an instance of a Triangle expression: ```wl In[2]:= inst = TriangleCreate[] Out[2]= TriangleExpression[1] ``` This sets up points and segments to use: ```wl In[3]:= pts = {{0., 0.}, {2., 0.}, {2., 2.}, {0., 1.}}; segments = {{1, 2}, {2, 3}, {3, 4}, {4, 1}}; ``` This sets the points and facets in the Triangle instance: ```wl In[4]:= TriangleSetPoints[inst, pts]; TriangleSetSegments[inst, segments]; ``` This carries out the triangulation, returning a new Triangle instance: ```wl In[5]:= outInst = TriangleTriangulate[ inst, "pqa0.025"] Out[5]= TriangleExpression[2] ``` This extracts the points and elements from the triangulation: ```wl In[6]:= coords = TriangleGetPoints[outInst]; meshElements = TriangleGetElements[outInst]; ``` With the following support function, you can visualize the triangles: ```wl In[7]:= TriangleWireframe[i_] := Line[ Flatten[ i[[All, #]]& /@ {{1, 2}, {2, 3}, {3, 1}}, 1]] In[8]:= Graphics[GraphicsComplex[coords, TriangleWireframe[meshElements]]] Out[8]= [image] ``` ### Options (1) #### "TriangleRefinement" (1) First, load the package: ```wl In[1]:= Needs["TriangleLink`"] ``` This creates an instance of a Triangle expression: ```wl In[2]:= inst = TriangleCreate[] Out[2]= TriangleExpression[3] ``` This sets up points and segments to use: ```wl In[3]:= pts = {{0., 0.}, {2., 0.}, {2., 2.}, {0., 1.}}; segments = {{1, 2}, {2, 3}, {3, 4}, {4, 1}}; ``` This sets the points and facets in the Triangle instance: ```wl In[4]:= TriangleSetPoints[inst, pts]; TriangleSetSegments[inst, segments]; ``` This carries out the triangulation, returning a new Triangle instance: ```wl In[5]:= outInst = TriangleTriangulate[ inst, "pq"] Out[5]= TriangleExpression[4] ``` This extracts the points and elements from the triangulation: ```wl In[6]:= coords = TriangleGetPoints[outInst]; meshElements = TriangleGetElements[outInst]; ``` With the following support function, you can visualize the triangles: ```wl In[7]:= TriangleWireframe[i_] := Line[ Flatten[ i[[All, #]]& /@ {{1, 2}, {2, 3}, {3, 1}}, 1]] In[8]:= Graphics[GraphicsComplex[coords, TriangleWireframe[meshElements]]] Out[8]= [image] ``` This sets up a compiled function that returns ``True`` if a triangle should be refined and ``False`` otherwise: ```wl In[9]:= cf = Compile[{{coordinates, _Real, 2}, {area, _Real, 0}}, Block[{c = coordinates, res, dxoa, dyoa, dxda, dyda, dxod, dyod, oalen, dalen, odlen, maxlen}, dxoa = c[[1, 1]] - c[[3, 1]]; dyoa = c[[1, 2]] - c[[3, 2]]; dxda = c[[2, 1]] - c[[3, 1]]; dyda = c[[2, 2]] - c[[3, 2]]; dxod = c[[1, 1]] - c[[2, 1]]; dyod = c[[1, 2]] - c[[2, 2]]; oalen = dxoa ^ 2 + dyoa ^ 2; dalen = dxda ^ 2 + dyda ^ 2; odlen = dxod ^ 2 + dyod ^ 2; maxlen = If[dalen > oalen, dalen, oalen]; maxlen = If[oalen > maxlen, oalen, maxlen]; res = If[maxlen > 0.025 * (c[[1, 1]] ^ 2 + c[[1, 2]] ^ 2) + 0.02, True, False]; res] ]; ``` This carries out the triangulation with the refinement function, returning a new Triangle instance: ```wl In[10]:= outInst2 = TriangleTriangulate[inst, "pq", "TriangleRefinement" -> cf] Out[10]= TriangleExpression[5] ``` This extracts the points and elements from the triangulation: ```wl In[11]:= coords = TriangleGetPoints[outInst2]; meshElements = TriangleGetElements[outInst2]; ``` Visualize the triangles: ```wl In[12]:= Graphics[GraphicsComplex[coords, TriangleWireframe[meshElements]]] Out[12]= [image] ``` ### Neat Examples (1) Use a black-and-white image as a refinement driver. Set an image, create a distance function, and visualize the distance function: ```wl In[1]:= shape = [image]; In[2]:= dist = DistanceTransform[Image[1 - ImageData[EdgeDetect[shape]]]]; In[3]:= ImageAdjust@dist Out[3]= [image] ``` Create an ``InterpolatingFunction`` from the distance function: ```wl In[4]:= data = Transpose[Reverse[-ImageData[dist] * (2 * ImageData[shape] - 1)]]; range = Transpose[{{1, 1}, Dimensions[data]}]; ifun = ListInterpolation[data, range, InterpolationOrder -> 1] Out[4]= InterpolatingFunction[{{1., 240.}, {1., 231.}}, <>] ``` Create and populate a Triangle instance: ```wl In[5]:= inst = TriangleCreate[]; TriangleSetPoints[inst, Flatten[Outer[List, Sequence@@range], 1]]; TriangleSetSegments[inst, {{1, 3}, {3, 4}, {4, 2}, {2, 1}}]; ``` Set up a compiled function that calls the ``InterpolatingFunction`` : ```wl In[6]:= cf = Compile[{{c, _Real, 2}, {area, _Real, 0}}, Block[{com}, com = Total[c] / 3; If[area > Max[Abs[ifun[com[[1]], com[[2]]]], 2.5], True, False] ] ]; ``` This carries out the triangulation with the refinement function, returning a new Triangle instance: ```wl In[7]:= outInst = TriangleTriangulate[inst, "pq", "TriangleRefinement" -> cf] Out[7]= TriangleExpression[7] ``` This extracts the points and elements from the triangulation: ```wl In[8]:= coords = TriangleGetPoints[outInst]; meshElements = TriangleGetElements[outInst]; ``` Visualize the triangles: ```wl In[9]:= TriangleWireframe[i_] := Line[Flatten[ i[[All, #]]& /@ {{1, 2}, {2, 3}, {3, 1}}, 1]] Graphics[GraphicsComplex[coords, TriangleWireframe[meshElements]], ImageSize -> Medium] Out[9]= [image] ``` ## See Also * [TriangleCreate](https://reference.wolfram.com/language/TriangleLink/ref/TriangleCreate.en.md) * [TriangleSetPoints](https://reference.wolfram.com/language/TriangleLink/ref/TriangleSetPoints.en.md) * [TriangleGetPoints](https://reference.wolfram.com/language/TriangleLink/ref/TriangleGetPoints.en.md) * [TriangleSetSegments](https://reference.wolfram.com/language/TriangleLink/ref/TriangleSetSegments.en.md) * [TriangleGetSegments](https://reference.wolfram.com/language/TriangleLink/ref/TriangleGetSegments.en.md) ## Tech Notes * [*TriangleLink* User Guide](https://reference.wolfram.com/language/TriangleLink/tutorial/Overview.en.md) ## Related Guides * [TriangleLink](https://reference.wolfram.com/language/TriangleLink/guide/TriangleLink.en.md)