equiaffine area
A general affine map scrambles ordinary area — stretch a region twice as wide and its area doubles. But if you restrict the allowed maps to those that neither enlarge nor shrink area, a new invariant appears: area itself becomes meaningful again. Equiaffine geometry is affine geometry narrowed to the area-preserving maps, and equiaffine area is the notion of area that this narrowed geometry can measure.
Here is the mechanism. An affine map (x, y) -> A times (x, y) plus a shift multiplies every area by the absolute value of the determinant of A. So area is not an affine invariant in general, because the determinant can be any nonzero number. Restrict to affine maps whose matrix has determinant equal to plus or minus 1 — the equiaffine or unimodular affine group — and now every map leaves area exactly as it was. Under this smaller group, the area of a region, and equally the ratio of areas of two regions, becomes a genuine invariant, even though length and angle are still meaningless. This is the precise sense in which the simpler statement 'affine maps preserve ratios of areas' is true: a general affine map scales every area by the same factor, so the ratio of two areas is unchanged even when each area separately is not.
Equiaffine area is the natural measuring stick for problems where shape is irrelevant but the amount of region matters and you have no notion of unit length — for instance comparing the area of an inscribed triangle to that of its surrounding ellipse, a ratio that is the same for every ellipse because any ellipse is the affine image of a circle. It sits just below ordinary affine geometry in the tower, adding back the single invariant of area. The caution is not to confuse it with Euclidean area: equiaffine area has no fixed units, it only compares regions, so it tells you that one region is twice another, never that a region is '5 square centimetres'.
The largest triangle inscribed in a circle takes up a fixed fraction of the disk's area. Apply any affine map and the circle becomes an ellipse, the triangle becomes another triangle, but because affine maps scale all areas by the same factor, the inscribed triangle still occupies that same fraction of the ellipse. The ratio is an equiaffine invariant; the individual areas are not.
A general affine map preserves ratios of areas; the unimodular subgroup preserves area outright.
Equiaffine area measures only relative size, not absolute units: it has no notion of a square centimetre, so it can say one region is twice another but never assigns a region a numerical area in fixed units.