An isometric transformation that maps each point to its mirror image across a specified line (in the plane) or plane (in space), fixing the reflecting subspace pointwise.

Principle

Reflection is an involutive Euclidean isometry whose fixed set is the reflecting subspace; algebraically it is an orthogonal linear map with determinant −1 (in an appropriate dimension) and satisfies R^2 = identity.

Demonstration

In the plane, reflection across the x-axis sends (x,y) to (x,−y). In R^3 reflecting across the plane z=0 sends (x,y,z) to (x,y,−z). As matrices, a reflection about a unit normal n has the form I−2nn^T.

Misapplication

Confusing reflection with a rotation or assuming it preserves orientation in all dimensions; applying the “mirror” formula with a non-unit normal or in a space without an inner product can produce incorrect or non-isometric maps.

Consequence

Reflection preserves distances, reverses orientation, and generates orthogonal groups together with other reflections; composing two reflections produces a rotation or a translation depending on the configuration of reflecting subspaces.

Reversal

A reflection is its own inverse (it is an involution): applying the same reflection twice returns the original configuration. Viewed oppositely, composing reflections can reverse the effect (two reflections about the same subspace cancel).

Boundary

Defined in Euclidean or inner-product spaces where orthogonality and distance are defined; lacks meaning in purely affine settings without a metric or in spaces where a notion of mirror symmetry is unavailable.

Semantic Tension

Sometimes conflated with inversion in a circle or with affine symmetries that fix a subspace but do not preserve distances; reflections are rigid and isometric, while inversions or glide reflections have different metric or orientation properties.

Synthesis

Reflection is the elementary mirror isometry: an involutive, distance-preserving map fixing a subspace pointwise and serving as a basic generator of orthogonal transformations by composition.