Delaunay Triangulation Libraries 2D/2.5D

Fade is a fast C++ Delaunay triangulation library and it comes in two flavors: First, Fade 2D is a planar Constrained Delaunay triangulation, see the above red 2D cyclist-shape. Second, Fade 2.5D creates elevated Delaunay triangulations from 2.5D point clouds like the above terrain, for instance. Both software versions are equipped with additional robust algorithms from the meshing field. Moreover, ready-made C++ examples demonstrate the application of these algorithms:

Fade 2D: Constrained Delaunay Triangulation

  • Delaunay triangulation for 2D points, multithreaded
  • Constrained Delaunay triangulation for segments and polygons
  • Polygon clipping, boolean operations on polygons
  • Voronoi diagram with fast nearest-neighbor search
  • Fast segment intersection tester
  • Delaunay Mesh Generator and Grid Mesher
  • Free student/research license included. Commercial licenses with dedicated programming support are available

Fade 2.5D: Delaunay Triangulation for Elevated Point Clouds

The Fade 2.5D software contains all features of Fade2D. In addition points have a z-coordinate and the software provides additional high-end algorithms for point clouds:

  • You can reduce a 2.5D point cloud with Fade’s CloudPrepare tool
  • Then create a lifted Delaunay triangulation (a TIN) from it
  • Afterwards you can align its edges to valleys and ridges
  • Then eliminate noise with weighted Laplacian smoothing
  • Or use the Cookie Cutter tool to cut out polygonal pieces.
  • Use Cut-and-Fill for earthwork volume computations.
  • If required make height queries for arbitrary (x,y) coordinates.
  • Or compute ISO-contours from the terrain.

“To clarify, 2.5D allows exactly one z-value per (x,y)-coordinate pair and as a result it is suitable for surfaces which have an intersection-free projection to the (x,y)-plane like terrains. “

Delaunay Terrain triangulation (2.5D TIN), 550 000 triangles
Terrain (2.5D TIN), 550,000 triangles computed in 0.11 seconds
ISO contours of a terrain consisting of  550 000 triangles.
Terrain triangulation 2.5D, ISO contours of 550,000 triangles at 30 different levels computed in 0.38 seconds.

Constrained Delaunay Triangulation

The Fade software can insert constraint edges into a Delaunay triangulation and thus make it a Constrained Delaunay triangulation. Moreover, the constraint edges can be subdivided into sub-segments to achieve well-shaped adjacent triangles.

C++ Delaunay triangulation of points without constraint edges
Delaunay triangulation of points without constraint edges
C++ Constrained Delaunay triangulation library
Constrained Delaunay triangulation with constraint edges
Conforming Delaunay triangulation -  subdivision for constraint segments to achieve better triangle quality.
Subdivided constraint edges for better triangle quality using the makeDelaunay() method
Zones (Shapes)
Zones define areas of interest. They support Polygon Clipping, or more general, they support the boolean set operations union, intersection, difference and symmetric difference.

Fade 2.5D can insert breaklines at their elevation or it can adapt them to surface-level before insertion. It can subdivide the segments automatically to better fit the surface.

Breaklines (constraint edges)
Polygonal breaklines (blue polygon), projected (red polygon) and inserted into a 2.5D triangulation. The left image shows the original long segments. In contrast the right image shows automatically subdivided segments.

Delaunay Mesh Generator and Grid Mesher

The Mesh Generator software creates quality triangles inside a given shape. You can claim a quality feature like the minimum interior triangle angle. But you can also precisely control the mesh generation procedure according to parameters like the maximum edge length or a bound on the size-difference of adjacent triangles, for example.

Quality meshing inside a polygon.
A Delaunay Meshing function creates a high quality triangle mesh

Performance of Fade

The C++ library can run multiple threads on different cores and is very fast: It triangulates one million points (2 million triangles) in 0.17 seconds on a Core i7 6800K, for instance. Have a look at this comparison of single- and multithreaded runs with large point-sets.

Delaunay Benchmark single- and multithreaded
Delaunay benchmark: 1 million points (2 million triangles) in 0.17 s multithreaded or in 0.42 s single-threaded

34 replies on “Delaunay Triangulation Libraries 2D/2.5D”

Hello, I have a question. Can Fade2D make a concave triangulation? How to do? It seems always get a convex polygon.

“run-time grows only linearly with the number of input points” means that you developed a new algorithm in the asymptotic runtime complexity class O(n). Have you published any paper about it or will it be kept secret ?

Imagine some O(n) algorithm that computes a certain result within a day and imagine that the final output must be sorted which takes just a second. Is the overall algorithm still O(n) or is it O(n log n)? As a scientist you must write O(n log n) but nobody would say that such a software product has a superlinear time consumption.

The same is true for Fade2D: The superlinear fraction is insignificant and we observe approximately 0.6, 6.0 and 60.5 seconds for 1, 10 and 100 million points. This justifies the statement “the practical runtime grows only linearly” while the correct big-O notation would be misleading. The algorithm has not been published, I focus on software now.


Thanks for your free Delaunay triangulation library, it’s very fast in my research. But I didn’t see the detailed algorithm for Delaunay triangulation(DT), is that L-S? Watson? F-P? or other algorithms?
Thank you!

Hello. I used the pre-compiled Fade2D library for Ubuntu 15.04 in my project that is compiled under Ubuntu 16.04. I added the library into usr/local/lib and I did all necessary steps to the command ldconfig. But I got an error in compilation concerning the license :

../share/include/include_fade2d/License.h:39: undefined reference to `GEOM_FADE2D::setLic(std::__cxx11::basic_string<char, std::char_traits, std::allocator > const&, std::__cxx11::basic_string<char, std::char_traits, std::allocator > const&, std::__cxx11::basic_string<char, std::char_traits, std::allocator > const&, std::__cxx11::basic_string<char, std::char_traits, std::allocator > const&, std::__cxx11::basic_string<char, std::char_traits, std::allocator > const&)’

Do you have any idea to overcome this problem ?
Thank you.

Hello Dang,

Ubuntu 16.04 is too new for the Fade version you use. You can already download Fade v1.41 beta (it is stable but I always wait for some feedback before I label a version as a release version). Fade v1.41 beta contains Ubuntu15.10 binaries and they should work with Ubuntu 16.04. And just for the case that they do not, I have recompiled them again on a Ubuntu 16.04 machine for you:

[file removed, you can use the current version instead]

It would be helpful if you could leave some feedback on this new version here.

Best regards

Dear Cui Lejing,

The .poly-Format is currently not supported. For which program do you need it as input? If useful I can put that on my todo list. Currently you can use the *.obj file format and, if not directly supported by your software, look for an obj-to-poly converter tool.


Another option is to write your own output function. Have a look at FaceList for that purpose.

Hello Bernhard,
would it be possible to get a 64 bit version compiled against the current Windows platform SDK 1.4.1 (compatible with Visual Studio 2017?). Many thanks!

Hello, my name is Shunji Kikuchi in Japan.
(1) std::vector
I have much interest in Constrained Delaunay Triangle methods.
So I execute example3, but crashed.
I find the following methods have some problem.
– dt.insert(vInputPoints)
– ConstraintGraph2* pCG=dt.createConstraint(vSegments,CIS_CONSTRAINED_DELAUNAY);
I avoid this problem by using for loop without using std::vector.

(2) Question
By avoid the above problem, I make a my sample program.
Fade seems to add additional points in CDT. Is it correct ?
I want to make CDT diagram of just given xy-points.
Are there any option/parameter to specify this.

Thank you.

Hello Shunji Kikuchi,

Example3 is simple and well tested. I doubt that it crashes due to a bug but rather due to some incompatible linking or something like that. Please contact me via email to send details:
* Have you modified the source code of Example3?
* Which compiler do you use?
* Against which binary to you link?
I will support you also if you have no license, it is very important that the library is stable in all supported environments.

When you use CIS_CONSTRAINED_DELAUNAY then no additional points are inserted *unless* the constraint segments cross each other. In this case they are subdivided at their intersection point. It is always a good idea to visualize the result to see what’s going on. Use Fade_2D::show(“”) to write a Postscript file.

Hello I’m Shunji Kikuchi in Japan.

(1)Fade2 add additional xy-points
Yesterday, I sent this problem to you.
Maybe, this problem happens by my usage of “applyConstraintsAndZones”.
As I reported, “createConstraint(segA, CIS_CONSTRAINED_DELAUNAY)”
is crushed for multi-segments.
In order to avoid this problem, I make the following codes.

std::vector segA; segA.clear();
for (int i0 = 0; i0 getCoords(pntA);
for (int k0 = 1; k0 < (int)pntA.size(); k0++) {
Point2 v1 = Point2(pntA[k0-1].x(), pntA[k0-1].y());
Point2 v2 = Point2(pntA[k0].x(), pntA[k0].y());
segA.push_back(Segment2(v1, v2));
ConstraintGraph2* pCG = m_fade2D.createConstraint(
/* or
ConstraintGraph2* pCG = m_fade2D.createConstraint(

But these ways does not work well.

Thank you.

Dear Shunji Kikuchi,

thank you for your email. I have tried to compile and run
example3 with Visual Studio 2017. These were my steps:

1. Download Fade v1.63 and unzip
2. Change to fadeRelease\examples_2D\vs2017_exampleProject
3. Open examples_2D.sln with Visual Studio 2017
4. Select x64 Release and build
5. Open a command line window, go to fadeRelease\x64
6. Launch examples_2D.exe and select ‘3’

That works exactly as it should. And it should also work on
your side if you use the original VS2017 project that is
provided with the download.

I guess that your project settings are wrong and that you have
a version mismatch i.e., you link with a library that has been
made for another compiler version. My questions:

1. Does at least the above (original project) work for you?

2. What setting does your VS2017 project use in Configuration
Properties -> General -> Platform Toolset ? It should be
Visual Studio 2017 (v141).

3. What string do you use in Linker -> Additional Dependencies?

If that does not solve the problem, please provide a minimal
test project that reproduces the behavior: One *.cpp file with
a Visual Studio project so that I can compile it exactly like
you do. Thank you.

Best regards
Bernhard Kornberger
Geom Software

I would like to use Fade2D on Debian Stretch for purely academic purposes. However, the supplied versions are not working. Could you compile it on Debian stretch?

Thank you

Hi, have you tried the code in the examples2D/ directory and did you edit the Makefile to find out if one of the provided versions is compatible?

#DISTRO :=../lib_centos6.4_${ARCHITECTURE}
#DISTRO :=../lib_ubuntu14.04_${ARCHITECTURE}
#DISTRO :=../lib_ubuntu16.04_${ARCHITECTURE}
#DISTRO :=../lib_ubuntu17.04_${ARCHITECTURE}
DISTRO :=../lib_ubuntu18.04_${ARCHITECTURE}
#DISTRO :=../lib_APPLE
#DISTRO :=../lib_raspberry_armv6l
#DISTRO :=../lib_raspberry_armv7l

I use Fade2D with Visual Studio 2017 on Windows 10 system. I linked the fade2D_Win32_v141_Debug.lib in my project (MFC project). I don’t call any API from the Fade2D module, just link the lib and launch my MFC project.
I got a lot of memory leaks in the output (about 700 lines). Here are some of these lines:
{2123} normal block at 0x00CA18D8, 8 bytes long.
Data: 40 FE C8 00 00 00 00 00
{2122} normal block at 0x00C996B8, 24 bytes long.
Data: 20 8F C9 00 90 8E C9 00 F0 9A C9 00 01 01 CD CD
{2121} normal block at 0x00C8FE38, 68 bytes long.
Data: > 3E 00 00 00 E7 03 00 00 D8 18 CA 00 B8 96 C9 00
{2120} normal block at 0x00CA25F0, 1 bytes long.
Data: CD
{2095} normal block at 0x00C9DE30, 32 bytes long.
Data: 46 61 64 65 20 32 44 2C 20 44 45 42 55 47 2C 20
{2094} normal block at 0x00CA0E58, 8 bytes long.
Data: D0 17 42 0F 00 00 00 00
{2091} normal block at 0x00CA0FA8, 8 bytes long.

Any idea?
Thank you


Thank you for this report. Very recently I have got the first report about this warning but I was not able to reproduce the warnings on my machine. I have also tested the library with Valgrind and it reports

==31458== HEAP SUMMARY:
==31458== in use at exit: 0 bytes in 0 blocks
==31458== total heap usage: 239 allocs, 239 frees, 104,002 bytes allocated
==31458== All heap blocks were freed -- no leaks are possible
==31458== For counts of detected and suppressed errors, rerun with: -v
==31458== ERROR SUMMARY: 0 errors from 0 contexts (suppressed: 0 from 0)

It would be very helpful if you could send me a minimal VS2017 project that I can compile on my machine to debug this issue. Thank you!

Best regards

I use fade2d in ue4,when I use dt->getTrianglePointers() or pZone0->getTriangles() in beginplay ,the beginplay method will raised error when closed method.error is about free memory.

can you tell me how to loop all Triangle don’t use dt->getTrianglePointers() or pZone0->getTriangles()


Both methods work perfectly, they have been tested countless times in industrial environments. Please make sure that the Fade_2D instance is still alive at the time you use the triangles. Because with the destructor of the Fade_2D class the memory of all vertices and triangles is also released.

If this hint does not solve your problem, please send me a *minimal* example that I can compile and test.

test project on

please use win64,click play button multi time(not every time error happen)

Fade_2D dt;
Point2 p0(0.0, 0.0);
Point2 p1(1.0, 0.0);
Point2 p2(0.5, 2.0);
Point2 p3(0.5, 0.5);

// Insert the points
std::vector vTriangles;

I use above code in ue4 beginplay method,when I clicked play button dt.getTrianglePointers(vTriangles) crash my app some times (not every time) .the beginplay method raised error when closed method.error is about free memory.
I have post minimal ue4 project on github that only have one actor Source\ue4_fade2d_test\TestActor.cpp and beginplay is above simple code. Project please use win64 ,click play button multi time ,the error will happen(not every time)

By the way: I had just a case where a user has used
header files from one Fade version and binaries from
another Fade version. This can cause trouble due to
different memory footprints of classes in the headers
and in the binaries. When this is also the case in
your project please remove all Fade binaries from
your development folder and download a fresh copy
of Fade.


Your code looks correct. But the related functions of Fade are basic functions that are certainly correct.I don’t know UE4, but since the error doesn’t occur every time, I assume a memory problem that was caused elsewhere, but is only having an effect here. Another possibility is that you have mixed different VisualStudio versions, that also rarely goes well. The below error message is also not really related to Fade2D but to something inside UE4.

UE4Editor-Core.dll!rml::internal::freeSmallObject(void * object) Line 2522 C++
[Inline Frame] UE4Editor-Core.dll!rml::internal::internalPoolFree(rml::internal::MemoryPool * memPool, void *) Line 2621 C++
[Inline Frame] UE4Editor-Core.dll!rml::internal::internalFree(void *) Line 2644 C++
UE4Editor-Core.dll!scalable_free(void * object) Line 2933 C++
UE4Editor-Core.dll!FMemory::Free(void * Original) Line 80 C++

is not related to Fade2D.

Hi !
I’ve tried to test your library and it generates errors frequently at createZone_cookieCutter with message “Dt2_cat_cutt.cpp Line 944” Expression:vChildConstraintSegments.size() == 2)
Can you solve it ?

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