Constraint Edges – Example3

When you triangulate just the vertices of a polygon, it is not guaranteed that its edges are edges of the resulting Delaunay triagulation. But you can enforce edges in a triangulation as demonstrated in the below example where constraint edges are inserted into a 2D triangulation. Similarly you can insert breaklines into 2.5D triangle meshes, see Breaklines, ISO Contours and the Cookie Cutter.

Create a Delaunay triangulation

// * 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);

The above code creates a simple Delaunay triangulation of 6 input points but without Constraint Edges. After that it draws the result, see the right image.

Insert a Constraint Edge

Next we modify the Delaunay triangulation from above: Assume that we want to establish an edge from the bottom left to the top right: For this purpose we prepare a vector containing one or more Segment2 objects. Then we call createConstraint().

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

The above createConstraint() command inserts the constraint edge without subdivision. More precisely two cases exist that would enforce subdivision: When the constraint edge hits an existing vertex or when it intersects another constraint edge.

“This example uses CIS_CONSTRAINED_DELAUNAY. You will notice that there are also other constraint insertion strategies. However, these are deprecated in favor of newer techniques and are only included in the API for backward compatibility.“

Conforming Edges

Triangles next to long Constraint Edges can have a bad shape as shown in the above image. 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 in the above code makes it possible. It subdivides the constraint-edges recursively until the adjacent triangles satisfy the empty-circle condition. It’s minLen parameter prevents 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 useful for this purpose.

That’s all you need to know about constraint edges. But more options exist in 2.5D, you might also want to look at Breaklines, ISO Contours and the Cookie Cutter.

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