Editing Equivariant map (section)

===Elementary geometry===
[[File:Triangle.Centroid.svg|thumb|The centroid of a triangle (where the three red segments meet) is equivariant under [[affine transformation]]s: the centroid of a transformed triangle is the same point as the transformation of the centroid of the triangle.]]
In the geometry of [[triangle]]s, the [[area]] and [[perimeter]] of a triangle are invariants under [[Euclidean transformation]]s: translating, rotating, or reflecting a triangle does not change its area or perimeter. However, [[triangle center]]s such as the [[centroid]], [[circumcenter]], [[incenter]] and [[orthocenter]] are not invariant, because moving a triangle will also cause its centers to move. Instead, these centers are equivariant: applying any Euclidean [[Congruence (geometry)|congruence]] (a combination of a translation and rotation) to a triangle, and then constructing its center, produces the same point as constructing the center first, and then applying the same congruence to the center. More generally, all triangle centers are also equivariant under [[Similarity (geometry)|similarity transformations]] (combinations of translation, rotation, reflection, and scaling),<ref>{{citation
 | last = Kimberling | first = Clark | authorlink = Clark Kimberling
 | issue = 3
 | journal = [[Mathematics Magazine]]
 | jstor = 2690608
 | mr = 1573021
 | pages = 163–187
 | title = Central Points and Central Lines in the Plane of a Triangle
 | volume = 67
 | year = 1994
 | doi=10.2307/2690608}}. "Similar triangles have similarly situated centers", p.&nbsp;164.</ref>
and the centroid is equivariant under [[affine transformation]]s.<ref>The centroid is the only affine equivariant center of a triangle, but more general convex bodies can have other affine equivariant centers; see e.g. {{citation
 | last = Neumann | first = B. H.
 | journal = Journal of the London Mathematical Society
 | mr = 0000978
 | pages = 262–272
 | series = Second Series
 | title = On some affine invariants of closed convex regions
 | volume = 14
 | year = 1939
 | issue = 4
 | doi = 10.1112/jlms/s1-14.4.262 }}.</ref>

The same function may be an invariant for one group of symmetries and equivariant for a different group of symmetries. For instance, under similarity transformations instead of congruences the area and perimeter are no longer invariant: scaling a triangle also changes its area and perimeter. However, these changes happen in a predictable way: if a triangle is scaled by a factor of {{mvar|s}}, the perimeter also scales by {{mvar|s}} and the area scales by {{math|''s''<sup>2</sup>}}. In this way, the function mapping each triangle to its area or perimeter can be seen as equivariant for a multiplicative group action of the scaling transformations on the positive real numbers.