When you triangulate the vertices of a polygon, the edges of the polygon are not necessarily automatically edges of the Delaunay triangulation. But you can insert the edges as constraint edges (sometimes called ‘breaklines’). This is demonstrated for 2D below and in later examples you will see that it works also for 2.5D triangle meshes, see 2.5D Terrain Triangulation.

## Create a Delaunay triangulation

A very simple Delaunay triangulation without constraint edges is created:

// * 1 * Generate some input points std::vector<Point2> vInputPoints; vInputPoints.push_back(Point2(-100,-100)); vInputPoints.push_back(Point2(+100,+100)); vInputPoints.push_back(Point2(-50,-70)); vInputPoints.push_back(Point2(-50,-30)); vInputPoints.push_back(Point2(50,70)); vInputPoints.push_back(Point2(50,30)); // * 2 * Triangulate the points and show Fade_2D dt; dt.insert(vInputPoints); dt.show("example3_noConstraints.ps",true);

## Insert an Edge (‘Breakline’)

Now we want to change the pure Delaunay triangulation from above by inserting an edge from bottom left to top right. For this purpose we prepare a vector containing one or more `Segment2`

objects and call `createConstraint()`

.

std::vector<Segment2> vSegments; vSegments.push_back(Segment2(vInputPoints[0],vInputPoints[1])); ConstraintGraph2* pCG; pCG=dt.createConstraint(vSegments,CIS_CONSTRAINED_DELAUNAY);

The used and only recommended constraint-insertion-strategy is `CIS_CONSTRAINED_DELAUNAY`

. It inserts a constraint edge **without** subdivision. More precisely subdivision takes place in only the two cases where it can’t be avoided: When the constraint edge hits an existing vertex or when it intersects another constraint edge.

## Conforming Delaunay Edges

Triangles next to long constraint edges can have a bad shape i.e., extremely large and small interior angles. Thus conforming triangulations are often more desirable:

std::vector<Segment2> vSegments; vSegments.push_back(Segment2(vInputPoints[0],vInputPoints[1])); ConstraintGraph2* pCG; pCG=dt.createConstraint(vSegments,CIS_CONSTRAINED_DELAUNAY); double minLen(0.1); pCG->makeDelaunay(minLen);

The `makeDelaunay(double minLen)`

method method is used to subdivide the constraint-edges recursively until the adjacent triangles satisfy the empty-circle condition. The `minLen`

parameter is thought to prevent excessive subdivision in narrow settings.

Did you read this article because you want to triangulate a polygon? Don’t stop reading, the next article introduces the Zone concept, which is also useful for this purpose.

Now that you know how to create constraint graphs you are ready use polygonal zones.

## 4 replies on “Constraint Edges – Example3”

Dear Dr. Kornberger,

If there is a random polygon. How I can triangulate it?

Just insert all points into a object of Fade_2D or do I need to add all boundaries as Segment2 and apply

ConstraintGraph2* pCG = dt.createConstraint(vSegments, CIS_CONSTRAINED_DELAUNAY);

dt.applyConstraintsAndZones();

In both ways I didn’t get a proper result. Looking forward to your help!

Dear Boyan,

When you insert just the points you will get the Delaunay triangulation of the points. But the edges of your polygon are not necessarily part of the Delaunay triangulation. You must enforce these edges with createConstraint. But a Constrained Delaunay triangulation is always convex, this is why you see additional triangles when you triangulate a non-convex polygon. If you want to extract only the triangles inside the polygon you must create an INSIDE ZONE. See the next example (example4.cpp). Does that answer your question?

Hi,

Can I somehow delete an already created constraint segment? If so, how?

So far, all I could do was create an entirely new triangulation, and add the edited constraints there.

Thanks in advance!

Hi,

I’m sorry, removing constraints is not supported but it’s on the wishlist now. In case you need it for a commercial project please let me know so that I can put priority into this task.