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Snapshots from Transformation Geometry Snapshots from Transformation Geometry Shailesh Shirali Community Mathematics Centre, Sahyadri School & Rishi Valley School (KFI) 29 November 2013, IMSc, Chennai SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 1 / 44 What is geometry? There are many different ways of defining ‘geometry’ but one of them is: Geometry is the study of shapes, and how their properties are affected by given groups of transformations: which properties are left unaltered, and which ones undergo a change. This view of geometry is due to the mathematician Felix Klein (1849–1925). SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 2 / 44 What is a ‘Geometric Transformation’? A transformation of the plane is a function defined on the plane, moving points around according to a definite law. Matters of interest: Is the function ‘well behaved’? Is it smooth? Does it preserve length? Angles? Orientation? Area? In today’s talk we shall see how the use of transformations can give rise to elegant proofs of some geometrical propositions. SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 3 / 44 Affine maps Let f be a bijection of the plane. We say that f is affine if it preserves the property of collinearity. Let the images of points A, B, C,... under f be A′, B′, C ′,.... Let the images of lines l, m under f be l ′, m′. Then: SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 4 / 44 Affine maps Let f be a bijection of the plane. We say that f is affine if it preserves the property of collinearity. Let the images of points A, B, C,... under f be A′, B′, C ′,.... Let the images of lines l, m under f be l ′, m′. Then: l m l m • ⇐⇒ ′ ′ SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 4 / 44 Affine maps Let f be a bijection of the plane. We say that f is affine if it preserves the property of collinearity. Let the images of points A, B, C,... under f be A′, B′, C ′,.... Let the images of lines l, m under f be l ′, m′. Then: l m l m • ⇐⇒ ′ ′ B is the midpoint of AC B is the midpoint of A C • ⇐⇒ ′ ′ ′ SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 4 / 44 Affine maps Let f be a bijection of the plane. We say that f is affine if it preserves the property of collinearity. Let the images of points A, B, C,... under f be A′, B′, C ′,.... Let the images of lines l, m under f be l ′, m′. Then: l m l m • ⇐⇒ ′ ′ B is the midpoint of AC B is the midpoint of A C • ⇐⇒ ′ ′ ′ A, B, C collinear = AB : BC = A B : B C • ⇒ ′ ′ ′ ′ SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 4 / 44 Affine maps Let f be a bijection of the plane. We say that f is affine if it preserves the property of collinearity. Let the images of points A, B, C,... under f be A′, B′, C ′,.... Let the images of lines l, m under f be l ′, m′. Then: l m l m • ⇐⇒ ′ ′ B is the midpoint of AC B is the midpoint of A C • ⇐⇒ ′ ′ ′ A, B, C collinear = AB : BC = A B : B C • ⇒ ′ ′ ′ ′ Interior of ABC is mapped to interior of A B C • △ △ ′ ′ ′ SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 4 / 44 Examples of affine maps 1 Isometries: mappings which preserve distance. Examples: SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 5 / 44 Examples of affine maps 1 Isometries: mappings which preserve distance. Examples: (a) Displacement (‘translation’) through a vector SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 5 / 44 Examples of affine maps 1 Isometries: mappings which preserve distance. Examples: (a) Displacement (‘translation’) through a vector (b) Mirror reflection in a line SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 5 / 44 Examples of affine maps 1 Isometries: mappings which preserve distance. Examples: (a) Displacement (‘translation’) through a vector (b) Mirror reflection in a line (c) Rotation about a point, through some angle SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 5 / 44 Examples of affine maps 1 Isometries: mappings which preserve distance. Examples: (a) Displacement (‘translation’) through a vector (b) Mirror reflection in a line (c) Rotation about a point, through some angle 2 Enlargement about a point, by some scale factor (‘homothety’) SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 5 / 44 Examples of affine maps 1 Isometries: mappings which preserve distance. Examples: (a) Displacement (‘translation’) through a vector (b) Mirror reflection in a line (c) Rotation about a point, through some angle 2 Enlargement about a point, by some scale factor (‘homothety’) 3 Shear SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 5 / 44 Examples of affine maps 1 Isometries: mappings which preserve distance. Examples: (a) Displacement (‘translation’) through a vector (b) Mirror reflection in a line (c) Rotation about a point, through some angle 2 Enlargement about a point, by some scale factor (‘homothety’) 3 Shear Note the progression: congruence geometry, similarity geometry, affine geometry. This is in keeping with Klein’s vision. SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 5 / 44 Notation Symbol Meaning TPQ Translation (‘displacement’) through vector PQ HP Half-turn centred at point P Mℓ Mirror reflection in line ℓ RP,θ Rotation centred at P, through angle θ EP,k Enlargement centred at P, with scale factor k 1 Note: (i) TPQ − = TQP (ii) HP and Mℓ are self-inverse (iii) inverse of RP,θ is RP, θ (iv) inverse of EP,k is EP,1/k (v) EP, 1 is the same as HP − − SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 6 / 44 Composition of two reflections: parallel mirrors If l m, then Ml followed by Mm is equivalent to a displacement. l m b b b ′ ′′ A A A Segments AA′′, BB′′, b b b CC ′′, DD′′ have equal ′ ′′ B B B length: each is twice the b b b ′′ ′ C C C distance between l & m. b b ′ ′′ D D D SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 7 / 44 Composition of two reflections: non-parallel mirrors If (l m), then Ml followed by Mm is equivalent to a rotation. ¬ m b A′′ ∠(l, O, m) y b y A′ b x x l O b A ∠AOA = 2 ∠(l, O, m) = twice the directed angle from l to m ′′ × SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 8 / 44 Composition of two rotations (With due apologies to Herr Klein) II B b Here we see a motif rotated first about A by 60◦, then RA,60◦ RB,30◦ about B by 30◦. From the III positions, it appears as though a single rotation could have b I I III A taken the motif from to . SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 9 / 44 Locating the centre of R R B,β ◦ A,α n Draw line AB; draw lines m, n b b l through A, B such that A 1 1 B α 2 β 2 ∠ 1 ∠ 1 (m, l) = 2 α, (l, n) = 2 β. 1 1 b 2 α + 2 β C Keep directions in mind! m ∠ 1 1 Let m, n meet at C. Then (m, n) = 2 α + 2 β. So: SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 10 / 44 Locating the centre of R R B,β ◦ A,α n Draw line AB; draw lines m, n b b l through A, B such that A 1 1 B α 2 β 2 ∠ 1 ∠ 1 (m, l) = 2 α, (l, n) = 2 β. 1 1 b 2 α + 2 β C Keep directions in mind! m ∠ 1 1 Let m, n meet at C. Then (m, n) = 2 α + 2 β. So: RA = Ml Mm, RB = Mn Ml , ,α ◦ ,β ◦ Locating the centre of R R B,β ◦ A,α n Draw line AB; draw lines m, n b b l through A, B such that A 1 1 B α 2 β 2 ∠ 1 ∠ 1 (m, l) = 2 α, (l, n) = 2 β. 1 1 b 2 α + 2 β C Keep directions in mind! m ∠ 1 1 Let m, n meet at C. Then (m, n) = 2 α + 2 β. So: RA = Ml Mm, RB = Mn Ml , ,α ◦ ,β ◦ ∴ RB RA = (Mn Ml ) (Ml Mm) . ,β ◦ ,α ◦ ◦ ◦ SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 10 / 44 So RB RA = Mn (Ml Ml ) Mm = Mn Mm and is therefore ,β ◦ ,α ◦ ◦ ◦ ◦ equivalent to the composite map Mn Mm. ◦ SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 11 / 44 So RB RA = Mn (Ml Ml ) Mm = Mn Mm and is therefore ,β ◦ ,α ◦ ◦ ◦ ◦ equivalent to the composite map Mn Mm. ◦ But Mn Mm is equivalent to a rotation about the point where m and n ◦ meet, through twice ∠(m, n). SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 11 / 44 So RB RA = Mn (Ml Ml ) Mm = Mn Mm and is therefore ,β ◦ ,α ◦ ◦ ◦ ◦ equivalent to the composite map Mn Mm. ◦ But Mn Mm is equivalent to a rotation about the point where m and n ◦ meet, through twice ∠(m, n). Therefore, RB RA is equivalent to the rotation RC . ,β ◦ ,α ,α+β SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 11 / 44 So RB RA = Mn (Ml Ml ) Mm = Mn Mm and is therefore ,β ◦ ,α ◦ ◦ ◦ ◦ equivalent to the composite map Mn Mm. ◦ But Mn Mm is equivalent to a rotation about the point where m and n ◦ meet, through twice ∠(m, n). Therefore, RB RA is equivalent to the rotation RC . ,β ◦ ,α ,α+β Could anything go wrong with this analysis? Yes: it could happen that m n, in which case the lines m, n do not meet at all! SAS (CoMaC) Snapshots from Transformation Geometry Nov 2013 11 / 44 This will happen if α + β is a multiple of 360◦.
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