Geographic projections, spherical shapes and spherical trigonometry.
Map projections are sometimes implemented as point transformations. For instance, spherical Mercator:
mercator(num x, num y) => [x, log(tan(pi / 4 + y / 2))];
This is a reasonable mathematical approach if your geometry consists of continuous, infinite point sets. Yet computers do not have infinite memory, so we must instead work with discrete geometry such as polygons and polylines!
Discrete geometry makes the challenge of projecting from the sphere to the plane much harder. The edges of a spherical polygon are geodesics (segments of great circles), not straight lines. Projected to the plane, geodesics are curves in all map projections except geoGnomonic, and thus accurate projection requires interpolation along each arc. D3 uses adaptive sampling inspired by a popular line simplification method to balance accuracy and performance.
The projection of polygons and polylines must also deal with the topological differences between the sphere and the plane. Some projections require cutting geometry that crosses the antimeridian, while others require clipping geometry to a great circle.
Spherical polygons also require a winding order convention to determine which side of the polygon is the inside: the exterior ring for polygons smaller than a hemisphere must be clockwise, while the exterior ring for polygons larger than a hemisphere must be anticlockwise. Interior rings representing holes must use the opposite winding order of their exterior ring. This winding order convention is also used by TopoJSON and ESRI shapefiles; however, it is the opposite convention of GeoJSON’s RFC 7946. (Also note that standard GeoJSON WGS84 uses planar equirectangular coordinates, not spherical coordinates, and thus may require stitching to remove antimeridian cuts.)
D3’s approach affords great expressiveness: you can choose the right projection, and the right aspect, for your data. D3 supports a wide variety of common and unusual map projections. For more, see Part 2 of The Toolmaker’s Guide.