Problem

Define reflection \(r\ell~\) across a line \(\ell~\).

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A complete reflection definition must handle two kinds of points: those off the mirror and those on it. For an off-line point, characterize its image through the segment joining the pair and the mirror’s perpendicular-bisector role; then examine what that same rule implies when the point’s distance from the mirror is zero.

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for every P = translate by one common vector across ℓ; fixed points = points at the origin; distances equal

for every P = P and P′ are in opposite half-planes; fixed points = none; distances may differ

for P∉ℓ = PP′ is parallel to ℓ; for P∈ℓ = P moves along ℓ; distances equal

for P∉ℓ = ℓ is the perpendicular bisector of segment PP′, so P and P′ lie in opposite half-planes; for P∈ℓ = P′=P; distance consequence = d(P,ℓ)=d(P′,ℓ)

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Define reflection \(r\ell~\) across a line \(\ell~\).

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