Jason Davies

Line Simplification

In cartography, line simplification algorithms are often used to render high-resolution geographic features at an appropriate output resolution. They can be applied to both polylines and polygons.

Visvalingam鈥檚 Algorithm: Avoiding Intersections

Many geometry algorithms require polygons to be simple, i.e. without self-intersections. The following map of the Norwegian coastline demonstrates occasional intersections generated by Visvalingam鈥檚 algorithm, highlighted in red.

Data: N2000 Kartdata (scale: 1:2,000,000).

Intersections can be avoided using a simple modification.

Visvalingam鈥檚 algorithm gives each point in a line an importance weighting, so that least important points are removed first. The importance is commonly derived from the area of the triangle formed by each point and its immediate neighbours.

In the following example, the triangle areas have been replaced with ranks for simplicity, starting with 0 for the smallest area.

On the left, the unmodified Visvalingam algorithm results in an intersection when point 0 is removed. On the right, the modified algorithm has promoted this point to the next-largest effective area, hence two points have rank 0, and the intersection is avoided when they are both removed at once.

Explanation

We begin by assuming that there are no intersections between any segments. Each time we remove a point from the heap, we ensure that the assumption still holds. It鈥檚 somewhat easier to explain via the pseudocode below, with modifications to the standard Visvalingam algorithm highlighted:

// points is a doubly linked list
simplify(points)
  heap = minHeap(compareArea)
  maxArea = 0
  intersecting = []

  for (point in points)
    point.area = area(point)
    heap.add(point)

  while (point = heap.pop())
    if (point.area < maxArea) point.area = maxArea
    else maxArea = point.area

    // Check that the new segment doesn鈥檛 intersect with
    // any existing segments, except for the point鈥檚
    // immediate neighbours.
    if (intersect(heap, point.previous, point.next))
      intersecting.push(point)
      continue

    // Reattempt to process points that previously would
    // have caused intersections when removed.
    while (i = intersecting.pop()) heap.push(i)

    remove(point) // remove from doubly linked list
    update(point.previous, heap)
    update(point.next, heap)

update(point, heap)
  point.area = area(point)
  heap.update(point)

Performance

Checking for intersections between the new segment and virtually all existing segments quickly becomes impractical for non-trivial numbers of segments.

The number of intersection tests can be reduced drastically using a spatial index, e.g. an R*-tree is used in the current implementation (RBush) with excellent results.

Future Work

The current implementation is 2D-only, but I鈥檇 like to add support for spherical arc intersections. This could potentially end up as part of TopoJSON so that problematic self-intersecting polygons are avoided when drawn with D3.

Further Reading

• Mike Bostock has written a nice illustrated explanation of how Visvalingam鈥檚 algorithm works.
• Quadratic Koch Island Simplification: a stress test for topology preservation.
• The Douglas鈥揚eucker algorithm: sufficiency conditions for non-self-intersections
• MapShaper.org: A Map Generalization Web Service

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