Appendix B — Misreadings

The book makes one claim: grid type is relative to a frame. Some readings of it miss that claim, and some readers reach past the book for an authority that seems to settle the matter first. This appendix collects those responses as they turn up, with a short answer to each. It will grow.

B.1 “Wikipedia already defines a regular grid”

It does, and its definitions agree with this book. They just leave the frame unstated, because they assume one.

The Wikipedia article Regular grid (read 15 October 2026) defines a regular grid as a tessellation of n-dimensional Euclidean space by congruent parallelotopes. A rectilinear grid is a tessellation by rectangles or boxes that are not all congruent. A curvilinear grid has the same combinatorial structure as a regular grid but quadrilateral cells instead of rectangles. Every one of these is a statement about the shape of cells in a space, and the space is fixed in advance: Euclidean, with Cartesian coordinates. The article never mentions geographic coordinates, map projections or coordinate reference systems.

Earth data does not come with a single Euclidean space. Longitude and latitude are not Euclidean coordinates, and each map projection supplies a different plane. Applied to Earth data, the article’s definitions give different answers depending on which plane you draw the cells in. The NSIDC 25 km grid is a tessellation by congruent squares in EPSG:3412, which makes it regular by Wikipedia’s definition. Drawn in longitude and latitude, the same cells are general quadrilaterals, which makes it curvilinear by the same article’s definition (Section 7.2). Wikipedia does not decide between those answers. It cannot, because it never asks which plane.

The article’s own example of a rectilinear grid makes the point. Its example is logarithmic graph paper. The lines on log paper are unevenly spaced in x and evenly spaced in log(x), so the paper is rectilinear in one frame and regular in another. That is the claim of this book, in the article’s own example.

Better authorities. The article has two references: a MathWorld entry and Thompson, Soni and Weatherill’s Handbook of Grid Generation. The grid-generation literature the handbook represents defines a structured grid as a mapping. The grid is uniform in a computational (index) space, and a coordinate transformation carries it into the physical space where it may be curvilinear. Grid type is a property of that mapping and the space it lands in, not of the cells alone. The sources this book already relies on take the same view:

• Balaji’s grid standard calls the first three rungs logically rectangular: “The coordinate space may continue to be physically curvilinear; yet, in index space, grid cells will be rectilinear boxes.”
• ESMF classifies uniform, rectilinear and curvilinear grids as subtypes of logically rectangular grids, with the coordinates stored separately from the index structure.
• The CF conventions let one dataset carry native 1D projected or rotated coordinates alongside 2D longitude and latitude, tied together by a grid_mapping variable. CF thereby records that one grid can be 1D and regular in one frame and 2D in another (Section 7.5).

So the disagreement is not with Wikipedia. It is with reading a definition written for one Euclidean space as if it settled a question about several frames.

B.2 “Projected data is flat, longitude and latitude need the ellipsoid”

A common rule of thumb splits data in two. Longitude and latitude are angles, so distances and areas need spherical or ellipsoidal methods. Projected data is in metres, so planar measures are fine. The rule is fine as a warning about angles, but it draws the line in the wrong place.

Every projected coordinate is a position on the ellipsoid. The projection only changes how it is written down, and each projection stretches the surface by a different amount in different places. Planar measures on projected coordinates are only as good as the projection’s scale factor where you measure. In EPSG:3412, the NSIDC south polar stereographic frame, the scale is true at 70 S. A 25 km map cell covers about 665 square km on the ground at the pole and about 520 at 50 S, where the areal scale factor is 1.21. That is a 21 percent error from a regular grid in metres. World Mercator (EPSG:3395) overstates area by a factor of 4 at 60 degrees. Even UTM, which is built to keep distortion small, is off in area by 0.5 percent six degrees from its central meridian and by 2.4 percent at twelve degrees, which is what happens when one zone is stretched to cover a whole region. An equal-area frame such as EASE-Grid 2.0 (EPSG:6933) gets area exactly right everywhere, and in exchange it distorts distance by a factor of 1.7 along the parallel at 60 N.

The other half of the rule is backwards for regular grids. A grid that is regular in longitude and latitude has cells bounded by meridians and parallels, and the area of such a cell has a closed form: R^2 * dlon * (sin(lat2) - sin(lat1)) on the sphere, with an equivalent closed form on the ellipsoid. One number per row covers the whole grid, and it is exact. Ellipsoidal methods sound like an extra burden, but here they are the simple case.

So the useful question is not “is it projected?” but “what does this frame preserve, and where?”. That is the same kind of question as “regular in what?”. Whether a grid is regular depends on the frame, and so does whether a planar measure means anything on the ground.

B.3 “So 2D longitude and latitude arrays are wrong”

No. For a genuinely curvilinear grid, such as the NEMO ORCA tripolar grid (Section 7.7) or a satellite swath (Section 7.8), the arrays are the grid, and nothing smaller describes it. Storing longitude and latitude next to a regular grid’s native coordinates is also fine, and CF encourages it. The trouble starts when the 2D arrays are the only thing stored and the frame that made the grid regular is lost.

B.4 “A longitude and latitude grid is curvilinear because the Earth is curved”

A grid with uniform steps in longitude and latitude is regular in the longitude and latitude frame. Its cells shrink toward the poles on the ground, and its lines are curves when drawn in most projections, but that is the frame doing its work, not the grid changing type. The curvature of the Earth is a separate question about the surface the frame sits on. That question is about wrapping, poles and area, and Chapter 8 takes it up.

B.5 “Regular means the cells are all the same size”

Only in the frame. A regular longitude and latitude grid has cells whose ground area varies with the cosine of latitude, so a 1 degree cell at 80 degrees covers less than a fifth of the area of one at the equator. In the other direction, NASA’s L3 bin scheme has cells of nearly equal area and is not regular in any frame; it is not even indexed by row and column (Section 8.3). Equal area is a property of cells on the Earth. Regularity is a property of positions in a frame.