Add vertices along geodesic edges (densify)
Adds vertices along each edge of a line or polygon so no piece is longer than a set length, each on the true geodesic, so maps that draw straight segments still follow the Earth's curve.
The line now has 9 vertices, none more than 178.688 km apart.
| Latitude | Longitude |
|---|---|
| 39.8561000 deg | -104.6737000 deg |
| 40.2522447 deg | -102.6439138 deg |
| 40.6123597 deg | -100.5914119 deg |
| 40.9356009 deg | -98.5178976 deg |
| 41.2211897 deg | -96.4252630 deg |
| 41.4684219 deg | -94.3155787 deg |
| 41.6766746 deg | -92.1910802 deg |
| 41.8454138 deg | -90.0541506 deg |
| 41.9742000 deg | -87.9073000 deg |
- Vertices given
- Longest piece
- Total length
Provenance
- Computed by
- geometry.shape.densify 1.0.0, core 0.1.0
- Model
- Each edge's geodesic on WGS 84 (Karney inverse) is cut into n = ⌈length / longest piece⌉ equal pieces, with the new vertices placed by the direct problem
- Accuracy
- Every new vertex lies on its edge's geodesic to a nanometer; the pieces of an edge are equal to a nanometer
- Notes
- None
- Cites
- Karney, C. F. F., Journal of Geodesy, Algorithms for geodesics
Something look off?
How we got thisFormula, worked example, sources, and proof
Model: Each edge's geodesic on WGS 84 (Karney inverse) is cut into n = ⌈length / longest piece⌉ equal pieces, with the new vertices placed by the direct problem
Show your work
Longest piece
each edge's length / ⌈length / longest piece⌉1 edges, 1,429.503 km in all= 178.688 kmVertices now
the given vertices plus the pieces' new ends2 given= 9
The same steps an agent gets from the MCP server with explain: true.
Accuracy: Every new vertex lies on its edge's geodesic to a nanometer; the pieces of an edge are equal to a nanometer
When to use this: Use this before anything that treats a line as straight between its points. A route drawn on a map, a shape reprojected, a buffer or an overlay computed in a plane, a corridor checked against terrain: all of them chord the gaps, and over a long leg the chord can sit far from the geodesic the leg actually means. Adding vertices no further apart than a chosen distance brings the chords back onto the curve. It is also what makes two tracks sampled differently comparable, since a discrete Fréchet distance depends on how each was sampled.
Limitations: This adds vertices; it never moves or removes them, so the original points all survive and the result is longer to carry and slower to draw. It is the opposite of simplification, not a smoother: the shape is unchanged, only better described. The maximum length is a bound and pieces come out shorter, since an edge is divided into equal parts — an edge of 120 km at a 40 km maximum becomes three pieces of exactly 40 km, but one of 121 km becomes four of 30.25 km. There is a ceiling on how many vertices a result may have, and a very small maximum over a long route is refused rather than silently truncated.
Worked example: Denver to Chicago, in pieces of at most 200 km. Source: Karney (2013) inverse and direct on WGS 84. Checked against his own geographiclib over fifteen cases — lines, polygons, a meridian, the equator, the antimeridian and high latitudes — agreeing on every vertex count, on the lengths to 1.6e-13 relative, and placing the interior vertices within 2.1e-14 deg. It is golden test vector v001, and every build checks the tool still gives its answer within its tolerance.
You enter
- Longest piece
- 200 km
- Vertices
- 39.8561, -104.6737 41.9742, -87.9073
You get
- Vertices now
- 9
- Vertices given
- 2
- Longest piece
- 178.688 km
- Total length
- 1,429.503 km
Review: Not yet independently reviewed by a GIS professional.
Last verified: 2026-09-18, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.
Status: version 1.0.0, core 0.1.0. See this tool in the verification report.
Checked against: 21 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.
Sources
- Algorithms for geodesics, Karney, C. F. F., Journal of Geodesy, Vol. 87, No. 1. pp. 43-55 (the direct and inverse problems).