Definition D1
Let ℓ_θ be the line through the origin making angle θ with the positive x-axis. Define two transformations in coordinates. σ_θ(x, y) = (x cos 2θ + y sin 2θ, x sin 2θ − y cos 2θ), R_φ(x, y) = (x cos φ − y sin φ, x sin φ + y cos φ)
Proof.
(1)The points of ℓ_θ are those of the form (t cos θ, t sin θ) for real t.A3, convention
(2)σ_θ and R_φ do not yet live up to their names. For now they are merely two formulas.convention
(3)That σ_θ is the reflection in ℓ_θ and R_φ the rotation by φ is verified in L1 and L2.(2), convention
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Lemma L1
σ_θ fixes every point of the line ℓ_θ and sends the direction perpendicular to ℓ_θ to its opposite.
Proof.
(1)Let P = (t cos θ, t sin θ). Then the x-coordinate of σ_θ(P) is t(cos 2θ cos θ + sin 2θ sin θ)D1, A8, A10
(2)cos 2θ cos θ + sin 2θ sin θ = cos(2θ − θ) = cos θA1
(3)The y-coordinate of σ_θ(P) is t(sin 2θ cos θ − cos 2θ sin θ) = t sin(2θ − θ) = t sin θD1, A8, A10, A1
(4)Hence σ_θ(P) = P, and every point of ℓ_θ is fixed.(1), (2), (3), E
(5)Q = (cos(θ+90°), sin(θ+90°)) is a point in the direction perpendicular to ℓ_θ, and by the addition theorems Q = (−sin θ, cos θ)A3, A1, A14, A12, convention
(6)The x-coordinate of σ_θ(Q) is (−sin θ)cos 2θ + cos θ sin 2θ = sin 2θ cos θ − cos 2θ sin θ = sin θD1, (5), A10, A12, A1
(7)The y-coordinate of σ_θ(Q) is (−sin θ)sin 2θ − cos θ cos 2θ = −(cos 2θ cos θ + sin 2θ sin θ) = −cos θD1, (5), A10, A12, A13, A1
(8)Hence σ_θ(Q) = (sin θ, −cos θ) = −Q, and the perpendicular direction is flipped.(5), (6), (7), A11, A12
(9)It fixes ℓ_θ and flips the direction perpendicular to it, so σ_θ is the reflection in ℓ_θ.(4), (8), convention
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Lemma L2
R_φ is the rotation by angle φ about the origin.
Proof.
(1)Write a point P other than the origin as P = (r cos α, r sin α).A3
(2)The x-coordinate of R_φ(P) is r cos α cos φ − r sin α sin φ = r(cos α cos φ − sin α sin φ) = r cos(α + φ)D1, (1), A8, A1
(3)The y-coordinate of R_φ(P) is r cos α sin φ + r sin α cos φ = r(sin α cos φ + cos α sin φ) = r sin(α + φ)D1, (1), A8, A10, A1
(4)The distance r from the origin is unchanged and only the angle grows by φ. Hence R_φ is the rotation by angle φ about the origin.(1), (2), (3), A3
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Theorem T1
When two lines meet at angle θ, composing the reflections in the two lines gives the rotation by angle 2θ about their intersection.
Proof.
(1)Taking the intersection as the origin, the two lines are written ℓ_α and ℓ_β; being distinct, they meet only at the origin. The angle between them is θ = β − αA5, A7, D1
(2)Let σ_α(x, y) = (u, v). Then u = x cos 2α + y sin 2α, v = x sin 2α − y cos 2αD1
(3)The x-coordinate of σ_β(u, v) is u cos 2β + v sin 2β = x(cos 2α cos 2β + sin 2α sin 2β) + y(sin 2α cos 2β − cos 2α sin 2β)D1, (2), A8, A9, A10, A12
(4)cos 2α cos 2β + sin 2α sin 2β = cos(2β − 2α) = cos 2θ(1), A1, A10, A8
(5)sin 2α cos 2β − cos 2α sin 2β = sin(2α − 2β) = −sin(2β − 2α) = −sin 2θ(1), A1, A2, A8, A11, A13
(6)Hence the x-coordinate is x cos 2θ − y sin 2θ(3), (4), (5), A12, E
(7)The y-coordinate of σ_β(u, v) is u sin 2β − v cos 2β = x(cos 2α sin 2β − sin 2α cos 2β) + y(sin 2α sin 2β + cos 2α cos 2β)D1, (2), A8, A9, A10, A12, A13
(8)cos 2α sin 2β − sin 2α cos 2β = sin(2β − 2α) = sin 2θ and sin 2α sin 2β + cos 2α cos 2β = cos(2β − 2α) = cos 2θ(1), A1, A10, A8
(9)Hence the y-coordinate is x sin 2θ + y cos 2θ(7), (8), E
(10)Comparing (6) and (9) with the R of D1 gives σ_β ∘ σ_α = R_{2θ}(6), (9), D1, E
(11)Hence the composition is the rotation by angle 2θ about the intersection of the two lines. The angle of rotation is twice the angle between them.(1), (10), L2
(12)If β = α, then 2θ = 0 and R₀ leaves every point unchanged. Reflecting twice in the same line returns everything to its place.(10), D1, A15
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Theorem T2
If the two lines are parallel, the composed reflections give not a rotation but a translation.
Proof.
(1)Taking the x-axis parallel to the two lines, they are written y = c₁, y = c₂ (c₁ ≠ c₂).A5
(2)Define the reflection in the line y = c by τ_c(x, y) = (x, 2c − y).convention
(3)τ_c(x, c) = (x, 2c − c) = (x, c), so every point of the line y = c is fixed.(2), A16, A9, A11
(4)(2c − y) − c = c − y = −(y − c), so the offset measured from the line has its sign flipped.(2), A16, A9, A11, A13
(5)Hence τ_c is the reflection in the line y = c.(3), (4), convention
(6)τ_{c₂}(τ_{c₁}(x, y)) = τ_{c₂}(x, 2c₁ − y) = (x, 2c₂ − (2c₁ − y)) = (x, y + 2(c₂ − c₁))(2), A8, A9, A10, A11, A13, E
(7)The x-coordinate is unchanged and the constant 2(c₂ − c₁) is added to the y-coordinate. This is a translation in the direction perpendicular to the two lines.(6), convention
(8)The distance between the two lines is |c₂ − c₁|, so the amount moved is twice that distance.(1), (7), A6, A16
(9)c₁ ≠ c₂, so 2(c₂ − c₁) ≠ 0 and no point is fixed. A rotation keeps its center; here nothing is kept.(1), (6), (7), A11, A17
(10)This block never cited A1. In the parallel case no trigonometric function appears in the computation.(6), (7), (8), (9)
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Corollary C
The result of two reflections preserves distance and angle.
Proof.
(1)Reflections preserve distance and angle.A4
(2)Composing transformations that preserve them leaves distance and angle unchanged.(1), convention
(3)For two parallel lines the composition is a translation, so translations preserve distance and angle.(2), T2
(4)For two intersecting lines the composition is a rotation, so rotations also preserve distance and angle.(2), T1
(5)There is no need to treat rotations and translations separately. Both are two reflections composed.(3), (4)
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