One-layer analysis operations

The input

Five Natural Earth countries, each carrying a name, an eu flag (four members, one not) and a pop_est. The eu attribute is what the grouping operations below group by.

Dissolve

Merges features that share a value and removes the boundaries between them — here the eu attribute, so the four members become one shape and the United Kingdom stays its own. The members are not all adjacent, so their shape is one feature in several pieces: Ireland is an island, and dissolving does not pretend otherwise. Leave the grouping empty to merge the whole layer into one.

The merged shape keeps more than the attribute it was grouped by. feature_count always comes along — how many things went into this shape — and any attribute can be carried through by summarising it: total and average for numbers, lowest and highest, number of values, or list the values (with your own separator, optionally deduplicated and sorted). Here the populations are added up and the names listed, so the EU row reports what four countries hold between them. Without a summary an attribute is simply dropped, because a merged shape has no single value for it.

Result attributes: eu feature_count pop_est_total name_list

Statistics

The same grouping and the same attribute summaries as dissolve, without the geometry: a table rather than a layer. Nothing is drawn, which is the point — no expensive geometry work when the answer is a number.

Result attributes: eu feature_count pop_est_total pop_est_average

Centroid

One point per feature, at its centre of area. A centroid can fall outside its own shape — a crescent’s does — which is what the label point is for.

Result attributes: name eu pop_est

Label point

One point per feature, guaranteed inside it and as far from the edges as possible. Where you would put the name.

Result attributes: name eu pop_est

Buffer

A zone at a fixed distance on the ground — 50 km here. Measured in metres, not degrees, so it is the same width at every latitude.

Result attributes: name eu pop_est

Convex hull

The smallest shape with no dents that still contains everything — an elastic band round the layer.

Result attributes: feature_count

Simplify

Fewer points, same map. A shared border is simplified once and both neighbours keep identical points, so no slivers open up between them.

Result attributes: name eu pop_est

Cartogram

Each country resized so its area shows its population rather than its ground area: Ireland shrinks, the United Kingdom and France grow. The default method stretches one sheet, so neighbours stay attached — Belgium and the Netherlands do not come apart, and no gaps open along the borders.

Why the third picture: the first two are Web Mercator, which inflates area by 1/cos²(latitude) — within this frame alone that draws the United Kingdom about a third larger against France than it is. The cartogram itself is right (measured on the sphere, every country here comes out at 5485 m² per person, France 3.6% larger than the United Kingdom, exactly its population lead), but in Mercator the United Kingdom still *looks* the bigger of the two. A cartogram’s only claim is that area is the value, so the projection it is drawn in has to keep areas — which is what the equal-area version shows, and why the cartogram demo map opens in Equal Earth.

Result attributes: name eu pop_est

Voronoi

Defined on points: one cell per point, covering everything closer to it than to any other. Hand it polygons or lines and each feature is first reduced to one point — a polygon’s area centroid, or the point halfway along a line — and the cells are built from those, so a layer of countries gives one cell per country.

Result attributes: name eu pop_est

Delaunay

The mirror image of Voronoi, and defined on points in the same way: triangles joining the points that are neighbours. A polygon or line input is reduced to one point per feature first — a polygon’s area centroid, or the point halfway along a line — and the triangles join those. Every input point ends up a corner of at least one triangle, which is why the distinct corners number exactly as many as the inputs. It is not merely a step towards Voronoi: a triangulation is the usual way to turn spot heights into a terrain surface (a TIN, and the contours and slopes read off it), to interpolate between measurements, and to ask which points are neighbours without choosing a distance — the neighbour graph spatial statistics builds its weights from. What comes back here is only the triangles and the indices of the points at their corners, with no attributes carried over and no height, so in this tool it answers the neighbour-and-spacing question rather than the surface one.

Result attributes: triangle point_1 point_2 point_3