S2 is a library for spherical geometry that aims to have the same robustness, flexibility, and performance as the very best planar geometry libraries.

Spherical Geometry

Let’s break down the elements of this goal. First, why spherical geometry? (By the way, the name “S2” is derived from the mathematical notation for the unit sphere, .)

Traditional cartography is based on map projections, which are simply functions that map points on the Earth’s surface to points on a planar map. Map projections create distortions due to the fact that the shape of the Earth is not very close to the shape of a plane. For example, the well-known Mercator projection is discontinuous along the 180 degree meridian, has large scale distortions at high latitudes, and cannot represent the north and south poles at all. Other projections make different compromises, but no planar projection does a good job of representing the entire surface of the Earth.

S2 approaches this problem by working exclusively with spherical projections. As the name implies, spherical projections map points on the Earth’s surface to a perfect mathematical sphere. Such mappings still create some distortion, of course, because the Earth is not quite spherical–but as it turns out, the Earth is much closer to being a sphere than a plane. With spherical projections, it is possible to approximate the entire Earth’s surface with a maximum distortion of 0.56%. Perhaps more importantly, spherical projections preserve the correct topology of the Earth – there are no singularities or discontinuities to deal with.

Why not project onto an ellipsoid? (The Earth isn’t quite ellipsoidal either, but it is even closer to being an ellipsoid than a sphere.) The answer relates to the other goals stated above, namely performance and robustness. Ellipsoidal operations are still orders of magnitude slower than the corresponding operations on a sphere. Furthermore, robust geometric algorithms require the implementation of exact geometric predicates that are not subject to numerical errors. While this is fairly straightforward for planar geometry, and somewhat harder for spherical geometry, it is not known how to implement all of the necessary predicates for ellipsoidal geometry.

Robustness

Which brings up the question, what do we mean by “robust”?

In the S2 library, the core operations are designed to be 100% robust. This means that each operation makes strict mathematical guarantees about its output, and is implemented in such a way that it meets those guarantees for all possible valid inputs. For example, if you compute the intersection of two polygons, not only is the output guaranteed to be topologically correct (up to the creation of degeneracies), but it is also guaranteed that the boundary of the output stays within a user-specified tolerance of true, mathematically exact result.

Robustness is very important when building higher-level algorithms, since unexpected results from low-level operations can be very difficult to handle. S2 achieves this goal using a combination of techniques from computational geometry, including conservative error bounds, exact geometric predicates, and snap rounding.

S2 attempts to be precise both in terms of mathematical definitions (e.g., whether regions include their boundaries, and how degeneracies are handled) and numerical accuracy (e.g., minimizing cancellation error).

Flexibility

S2 is organized as a toolkit that gives clients as much control as possible. For example, S2 makes it easy to implement your own geometry subtypes that store data in a format of your choice (the S2Shape interface). Similarly, most operations are designed to give clients fine control over the semantics, e.g. whether polygon boundaries are considered to be open or closed (or semi-open). S2 has APIs at several different levels to accommodate various uses of the library.

Performance

S2 is designed to have good performance on large geographic datasets. Most operations are accelerated using an in-memory edge index data structure (S2ShapeIndex). For example if you have a million polygons, finding the polygon(s) that contain a given point typically takes a few hundred nanoseconds. Similarly it is fast to find objects that are near each other, such as finding all the places of business near a given road, or all the roads near a given location.

Many operations on indexed data also have output-sensitive running times. For example, if you intersect the Pacific Ocean with a small rectangle near Seattle, the running time essentially depends on the complexity of the Pacific Ocean near Seattle, not the complexity of the Pacific Ocean overall.

Scope

The S2 library provides the following:

  • Representations of angles, intervals, latitude-longitude points, unit vectors, and so on, and various operations on these types.

  • Geometric shapes over the unit sphere, such as spherical caps (“discs”), latitude-longitude rectangles, polylines, and polygons.

  • Robust constructive operations (e.g., union) and boolean predicates (e.g., containment) for arbitrary collections of points, polylines, and polygons.

  • Fast in-memory indexing of collections of points, polylines, and polygons.

  • Algorithms for measuring distances and finding nearby objects.

  • Robust algorithms for snapping and simplifying geometry (with accuracy and topology guarantees).

  • A collection of efficient yet exact mathematical predicates for testing relationships among geometric objects.

  • Support for spatial indexing, including the ability to approximate regions as collections of discrete “S2 cells”. This feature makes it easy to build large distributed spatial indexes.

On the other hand, the following are outside the scope of S2:

  • Planar geometry. (There are many fine existing planar geometry libraries to choose from.)

  • Conversions to/from common GIS formats. (To read such formats, use an external library such as OGR.)

Layered Design

The S2 library is structured as a toolkit with various layers. At the lowest level, it provides a set of operations on points, edges, and simple shapes such as rectangles and discs. For example, this includes classes such as S2Point, S2CellId, S1Angle, S1Interval, S2LatLng, S2Region. It also includes the exact predicates defined in s2predicates.h and s2edge_crossings.h, utility functions such as those in s2edge_distances.h, and classes related to the S2Cell hierarchy (a hierarchical decomposition of the sphere) such as S2CellId, S2Region, and S2RegionCoverer.

The next layer provides flexible support for polygonal geometry, including point sets and polylines. It is built around the following core classes:

  • S2Builder: a robust tool for constructing, and simplifying geometry, built on the framework of snap rounding.

  • S2ShapeIndex: an indexed collection of points, polylines, and/or polygons, possibly overlapping.

  • S2BooleanOperation: evaluates boolean operations and predicates for arbitrary collections of polygonal geometry.

  • S2ClosestEdgeQuery: measures distances and finds closest edges for arbitrary collections of polygonal geometry.

These are only the most important classes, but what they have in common is that (1) they work with arbitrary collections of points, polylines, and polygons; (2) they allow clients to control the underlying geometry representation (via the S2Shape interface), and (3) they give clients fine control over the semantics of operations (e.g., how to handle degeneracies, whether polygons are open or closed, whether and how the output should be snapped, etc).

The final layer consists of convenience classes, such as S2Polygon, S2Loop, and S2Polyline. These types have wide interfaces and convenience methods that implement many of the operations above (e.g., intersection, distances), however they don’t offer the same flexibility in terms of data representation and control over semantics (e.g., polygon boundaries are always semi-open).

For example, you can measure the distance from a point to a polygon as follows:

S2Point p = ...;
S2Polygon a = ...;
S1Angle dist = a.GetDistance(p);

whereas with the S2ShapeIndex layer you would write

S2Point p = ...;
S2Polygon a = ...;
S2ShapeIndex index;
index.Add(absl::make_unique<S2Polygon::Shape>(&a));
S2ClosestEdgeQuery query(&index);
S1ChordAngle dist = query.GetDistance(p);

The first case is simpler, but the second case supports many more options (e.g. measuring the distance to a collection of geometry, or finding the objects within a certain distance).

Design Choices

This section summarizes some of the major design choices made by the library.

Spherical Geodesic Edges

In S2, all edges are “spherical geodesics”, i.e. shortest paths on the sphere. For example, the edge between two points 100 meters apart on opposite side of the North pole goes directly through the North pole. In contrast, the same edge in the standard “plate carrée” projection (i.e., raw latitude/longitude coordinates) would follow a line of constant latitude, which in this case happens to be a semicircular path that always stays exactly 50 meters away from the North pole. (Lines of latitude except for the equator are never shortest paths.)

With geodesic edges there are no special cases near the poles or the 180 degree meridian; for example, if two points 10km apart are separated by the 180 degree meridian, the edge between them simply crosses the meridian rather than going all the way around the other side of the Earth.

Orientation Matters

In S2, polygon loops contain the region on their left. This differs from planar libraries that might define loops as being “clockwise” or “counter-clockwise”, or that accept loops of either orientation. None of these concepts really applies to polygons drawn on a sphere; for example, consider a loop that goes around the equator. Is it clockwise or counter-clockwise? (For those familiar with the term, the concept of “winding number” does not exist on the sphere.)

For these reasons, S2 follows the “interior is on the left” rule. (This is the same as the counter-clockwise rule for small loops, but differs for very large loops that enclose more than half of the sphere.)

Unit Vectors vs. Latitude/Longitude Pairs

In S2, points are represented internally as unit-length vectors (points on the surface of a three-dimensional unit sphere) as opposed to traditional (latitude, longitude) pairs. This is for two reasons:

  • Unit vectors are much more efficient when working with geodesic edges. Using (latitude, longitude) pairs would require constantly evaluating trigonometric functions (sin, cos, etc), which is slow even on modern CPU architectures.

  • Unit vectors allow exact geometric predicates to be evaluated efficiently. To do this with (latitude, longitude) pairs, you would essentially need to convert them to the unit vector representation first.

For precision and efficiency, coordinates are representated as double-precision numbers. Robustness is achieved through the use of the techniques mentioned previously: conservative error bounds (which measure the error in certain calculations), exact geometric predicates (which determine the true mathematical relationship among geometric objects), and snap rounding (a technique that allows rounding results to finite precision without creating topological errors).

Classes vs. Functions

Most algorithms in S2 are implemented as classes rather than functions. So for example, rather than having a GetDistance function, there is an S2ClosestEdgeQuery class with a GetDistance method. This design has two advantages:

  • It allows caching and reuse of data structures. For example, if an algorithm needs to make thousands of distance queries, it can declare one S2ClosestEdgeQuery object and call its methods thousands of times. This provides the opportunity to avoid recomputing data that can be used from one query to the next.

  • It provides a convenient way to specify options (via a nested Options class), and to avoid respecifying those options many times when repeated operations need to be performed.

Geocentric vs. Geodetic Coordinates

Geocentric and geodetic coordinates are two methods for defining a (latitude, longitude) coordinate system over the Earth’s surface. The question sometimes arises: which system does the S2 library use?

The answer is: neither (or both). Points in S2 are modeled as lying on a perfect sphere (just as points in a traditional planar geometry library are modeled as lying on a perfect plane). How points are mapped onto the sphere is outside the scope of S2; it works equally well with geocentric or geodetic coordinates. (Actual geographic datasets use geodetic coordinates exclusively.)

Authors

The S2 library was written primarily by Eric Veach. Other significant contributors include Jesse Rosenstock, Eric Engle (Java port lead), Robert Snedegar (Go port lead), and Julien Basch.