CGAffineTransform はどう動いてるのか？〜Swift エンジニアのための線形代数〜 / How does CGAffineTransform work? ~A linearity lesson for Swift engineers~

Transcript

1. $("GpOF5SBOTGPSN
   ͸Ͳ͏ಇ͍ͯΔͷ͔ʁ
   ʙ4XJGU
   ΤϯδχΞͷͨΊͷઢܗ୅਺ʙ
   f o r
   J 0 4 % $
   +BQ B O
      
2. $("GpOF5SBOTGPSN
   ͸Ͳ͏ಇ͍ͯΔͷ͔ʁ
   ʙ4XJGU
   ΤϯδχΞͷͨΊͷઢܗ୅਺ʙ
   f o r
   J 0 4 % $
   +BQ B O
       嬭⓸
3. ઢܗ୅਺ ߦྻ ߦྻࣜ ݻ༗ۭؒ ઢܕ
   ํఔࣜ FUD ࠓ೔औΓ্͛Δൣғ
4. } var employedBy = "YUMEMI Inc." var job = "iOS Tech Lead" var favoriteLanguage = "Swift" var twitter = "@lovee" var qiita = "lovee" var github = "el-hoshino" var additionalInfo = """ ͸΍͘ίϩφऩଋͯ͠ग़ࣾۈ຿͍ͨ͠… """ final class Me: Developable, Talkable {
5. ͦ΋ͦ΋
   CGAffineTransform
   ͬͯʁ
6. Ҡಈ 5SBOTMBUJPO ֦ॖ 4DBMJOH ճస 3PUBUJPO
7. "GpOF
   5SBOTGPSN
   ΞϑΟϯม׵
8. O ΦϒδΣΫτݻ༗࠲ඪ ΞϑΟϯม׵ ΦϒδΣΫτදࣔ࠲ඪ O
9. ΞϑΟϯม׵͸࠲ඪม׵ͷҰछ
11. IUUQTTQFBLFSEFDLDPNMPWFFDHBGpOFUSBOTGPSNTIJKJBOSVNFO
12. IUUQTTQFBLFSEFDLDPNMPWFFDHBGpOFUSBOTGPSNTIJKJBOSVNFO
13. IUUQTTQFBLFSEFDLDPNMPWFFDHBGpOFUSBOTGPSNTIJKJBOSVNFO
14. CGAffineTransform
    ͸
    ΞϑΟϯม׵Λදݱ͢ΔͨΊʹ
    $PSF
    (SBQIJDT
    ্ͷσʔλߏ଄
16. ΞϑΟϯࣸ૾ ΞϑΟϯม׵ O O x ax + by + t x x′ y cx + dy + t y y′ P(x, y) P′ (x′ , y′ )
17. O O a = 1.5 c = 0.8 tx = 13 ty = 2 b = -0.1 d = 0.9 x = 10 y = 10 x' = 27 y' = 19 x ax + by + t x x′ y cx + dy + t y y′ P(10, 10) P′ (27, 19)
18. O O a = 1.5 c = 0.8 tx = 13 ty = 2 b = -0.1 d = 0.9 x = 10 y = 10 x' = 27 y' = 19 x = 0 y = 0 x' = 13 y' = 2 x ax + by + t x x′ y cx + dy + t y y′ O(0, 0) O′ (13, 2)
19. O O(0, 0) O′ (13, 2) O a = 1.5 c = 0.8 tx = 13 ty = 2 b = -0.1 d = 0.9 x = 10 y = 10 x' = 27 y' = 19 x = 0 y = 0 x' = 13 y' = 2 x ax + by + t x x′ y cx + dy + t y y′ CGAffineTransform
    ͷ
    a b c d tx ty
    Ͱ͢
20. ΞϑΟϯม׵͸ ࠲ඪม׵ͷϝιουͰ͋Γ ௚઀తʹҠಈ΍֦ॖΛѻΘͳ͍
21. Ҡಈ 5SBOTMBUJPO ֦ॖ 4DBMJOH ճస 3PUBUJPO
22. a = 1 c = 0 tx = 0 ty = 0 b = 0 d = 1 x = 10 y = 10 x' = 10 y' = 10 ม׵ޙɺݻ༗࠲ඪͱશ͘ಉ͡දࣔ࠲ඪ P(x, y) x ax + by + t x x′ y cx + dy + t y y′
23. Ҡಈ 5SBOTMBUJPO a = 1 c = 0 tx = tx ty = ty b = 0 d = 1 x = 10 y = 10 x' = 10 + tx y' = 10 + ty ม׵ޙɺݻ༗࠲ඪΑΓ
    Y
    ͕࣠
    UYɺZ
    ͕࣠
    UZ
    Ҡಈͨ͠࠲ඪ P(x, y) P′ (x′ , y′ ) x ax + by + t x x′ y cx + dy + t y y′
24. ֦ॖ 4DBMJOH a = a c = 0 tx = 0 ty = 0 b = 0 d = d x = 10 y = 10 x' = 10a y' = 10d ม׵ޙɺݻ༗࠲ඪΑΓ
    Y
    ͕࣠
    B
    ഒɺZ
    ͕࣠
    E
    ഒ֦େͨ͠࠲ඪ P(x, y) P′ (x′ , y′ ) x ax + by + t x x′ y cx + dy + t y y′
25. ճస 3PUBUJPO a = cos(θ) c = sin(θ) tx = 0 ty = 0 b = -sin(θ) d = cos(θ) x = 10 y = 10 x' = 10a + 10b y' = 10c + 10d ม׵ޙɺݻ༗࠲ඪΑΓ
    В˃ճసͨ͠࠲ඪ P(x, y) P′ (x′ , y′ ) x ax + by + t x x′ y cx + dy + t y y′
26. IUUQTTQFBLFSEFDLDPNMPWFFDHBGpOFUSBOTGPSNTIJKJBOSVNFO
27. ࿩͕͍ͩͿ௕͔ͬͨɻ ͱ͜ΖͰɺߦྻͬͯʁ
28. O O x ax + by + t x x′ y cx + dy + t y y′ P(x, y) P′ (x′ , y′ )
29. O O ߦྻ ( a b c d) × ( x y) + ( t x t y) = ( x′ y′ ) P(x, y) P′ (x′ , y′ )
30. ߦྻɿͦΕ͸ఆ·ͬͨߦͱྻʹΑΔ
    ɹɹɹ਺ࣈͷ૊Έ߹Θͤ ( 1 2 3 4) 458 488 718 911 595 124 ( 79.9 56 86 89 61 88) …
31. A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn N
    ߦ
    O
    ྻ w
    ߦ
    
    ྻͷߦྻΛ
    
    ܕߦྻͱݴ͏
    w
    ߦ໨
    
    ྻ໨ͷ਺
    
    Λ
    
    ͷ
    
    ੒෼ͱݴ͏ m n (m, n) i j a ij A (i, j)
32. ༷ʑͳߦྻʢʣɿྵߦྻ O = 0 0 … 0 0 0 … 0 ⋮ ⋮ ⋱ ⋮ 0 0 … 0 ྵߦྻ͸
    
    ͱ΋දه͞ΕΔ O શͯͷ੒෼͕
    
    ͷߦྻΛྵߦྻͱݺͿ 0
33. ༷ʑͳߦྻʢʣɿਖ਼ํߦྻ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a n1 a n2 … a nn
    ܕਖ਼ํߦྻ
    
    ࣍ਖ਼ํߦྻͱݺͿ (n, n) n ߦ਺ͱྻ਺͕౳͍͠ߦྻΛਖ਼ํߦྻͱݺͿ
34. ༷ʑͳߦྻʢʣɿ୯Ґߦྻ E n = 1 0 … 0 0 1 … 0 ⋮ ⋮ ⋱ ⋮ 0 0 … 1
    ࣍ਖ਼ํߦྻͰ΋ɺ
    ੒෼͕
    ɺͦΕҎ֎ͷ੒෼͕
    
    ͷ
    ߦྻΛ୯ҐߦྻͱݺͿ n a ii 1 0
    ࣍୯Ґߦྻ͸
    
    ͱ΋දه͞ΕΔ n E n
35. ߦྻͷܭࢉʢʣɿ૬౳ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn B = b 11 b 12 … b 1n b 21 b 22 … b 2n ⋮ ⋮ ⋱ ⋮ b m1 b m2 … b mn શͯͷ
    ɺ
    
    ʹର͠ɺ ͳ Βɺߦྻ
    
    ͱߦྻ
    
    ͕૬౳ʢٯ΋વΓʣ i j (1 ≤ i ≤ m, 1 ≤ j ≤ n) a ij = b ij A B ೋͭͷಉܕʢ
    ܕʣߦྻ
    ɺ
    ʹ͍ͭͯɿ (m, n) A B
36. ߦྻͷܭࢉʢʣɿ࿨ͱࠩ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn B = b 11 b 12 … b 1n b 21 b 22 … b 2n ⋮ ⋮ ⋱ ⋮ b m1 b m2 … b mn A ± B = a 11 ± b 11 a 12 ± b 12 … a 1n ± b 1n a 21 ± b 21 a 22 ± b 22 … a 2n ± b 2n ⋮ ⋮ ⋱ ⋮ a m1 ± b m1 a m2 ± b m2 … a mn ± b mn ೋͭͷಉܕʢ
    ܕʣߦྻ
    ɺ
    ʹ͍ͭͯɿ (m, n) A B
37. ߦྻͷܭࢉʢʣɿ࣮਺ഒ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn λ × A = λ × a 11 λ × a 12 … λ × a 1n λ × a 21 λ × a 22 … λ × a 2n ⋮ ⋮ ⋱ ⋮ λ × a m1 λ × a m2 … λ × a mn ࣮਺
    
    ͱߦྻ
    
    ʹ͍ͭͯɿ λ A
38. ߦྻͷܭࢉʢʣɿੵ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn B = b 11 b 12 … b 1l b 21 b 22 … b 2l ⋮ ⋮ ⋱ ⋮ b n1 b n2 … b nl
    ˞ A × B = ∑n k=1 a 1k × b k1 ∑n k=1 a 1k × b k2 … ∑n k=1 a 1k × b kl ∑n k=1 a 2k × b k1 ∑n k=1 a 2k × b k2 … ∑n k=1 a 2k × b kl ⋮ ⋮ ⋱ ⋮ ∑n k=1 a mk × b k1 ∑n k=1 a mk × b k2 … ∑n k=1 a mk × b kl n ∑ k=1 a 1k × b k1 = a 11 × b 11 + a 12 × b 21 + … + a 1n × b n1
    ܕߦྻ
    
    ͱɺ
    ܕߦྻ
    
    ʹ͍ͭͯɿ (m, n) A (n, l) B
39. ߦྻͷܭࢉʢʣɿੵ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn B = b 11 b 12 … b 1l b 21 b 22 … b 2l ⋮ ⋮ ⋱ ⋮ b n1 b n2 … b nl
    ˞ A × B = ∑n k=1 a 1k × b k1 ∑n k=1 a 1k × b k2 … ∑n k=1 a 1k × b kl ∑n k=1 a 2k × b k1 ∑n k=1 a 2k × b k2 … ∑n k=1 a 2k × b kl ⋮ ⋮ ⋱ ⋮ ∑n k=1 a mk × b k1 ∑n k=1 a mk × b k2 … ∑n k=1 a mk × b kl n ∑ k=1 a 1k × b k1 = a 11 × b 11 + a 12 × b 21 + … + a 1n × b n1
    ܕߦྻ
    
    ͱɺ
    ܕߦྻ
    
    ʹ͍ͭͯɿ (m, n) A (n, l) B
40. ߦྻͷܭࢉʢʣɿੵ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn B = b 11 b 12 … b 1l b 21 b 22 … b 2l ⋮ ⋮ ⋱ ⋮ b n1 b n2 … b nl
    ˞ A × B = ∑n k=1 a 1k × b k1 ∑n k=1 a 1k × b k2 … ∑n k=1 a 1k × b kl ∑n k=1 a 2k × b k1 ∑n k=1 a 2k × b k2 … ∑n k=1 a 2k × b kl ⋮ ⋮ ⋱ ⋮ ∑n k=1 a mk × b k1 ∑n k=1 a mk × b k2 … ∑n k=1 a mk × b kl n ∑ k=1 a 1k × b k1 = a 11 × b 11 + a 12 × b 21 + … + a 1n × b n1
    ܕߦྻ
    
    ͱɺ
    ܕߦྻ
    
    ʹ͍ͭͯɿ (m, n) A (n, l) B
41. ߦྻͷܭࢉʢʣɿੵ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn B = b 11 b 12 … b 1l b 21 b 22 … b 2l ⋮ ⋮ ⋱ ⋮ b n1 b n2 … b nl
    ˞ A × B = ∑n k=1 a 1k × b k1 ∑n k=1 a 1k × b k2 … ∑n k=1 a 1k × b kl ∑n k=1 a 2k × b k1 ∑n k=1 a 2k × b k2 … ∑n k=1 a 2k × b kl ⋮ ⋮ ⋱ ⋮ ∑n k=1 a mk × b k1 ∑n k=1 a mk × b k2 … ∑n k=1 a mk × b kl n ∑ k=1 a 1k × b k1 = a 11 × b 11 + a 12 × b 21 + … + a 1n × b n1
    ܕߦྻ
    
    ͱɺ
    ܕߦྻ
    
    ʹ͍ͭͯɿ (m, n) A (n, l) B
42. ߦྻͷܭࢉʢʣɿੵ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn B = b 11 b 12 … b 1l b 21 b 22 … b 2l ⋮ ⋮ ⋱ ⋮ b n1 b n2 … b nl
    ˞ A × B = ∑n k=1 a 1k × b k1 ∑n k=1 a 1k × b k2 … ∑n k=1 a 1k × b kl ∑n k=1 a 2k × b k1 ∑n k=1 a 2k × b k2 … ∑n k=1 a 2k × b kl ⋮ ⋮ ⋱ ⋮ ∑n k=1 a mk × b k1 ∑n k=1 a mk × b k2 … ∑n k=1 a mk × b kl n ∑ k=1 a 1k × b k1 = a 11 × b 11 + a 12 × b 21 + … + a 1n × b n1
    ܕߦྻ
    
    ͱɺ
    ܕߦྻ
    
    ʹ͍ͭͯɿ (m, n) A (n, l) B
43. ߦྻͷܭࢉʢʣɿੵ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn B = b 11 b 12 … b 1l b 21 b 22 … b 2l ⋮ ⋮ ⋱ ⋮ b n1 b n2 … b nl
    ˞ A × B = ∑n k=1 a 1k × b k1 ∑n k=1 a 1k × b k2 … ∑n k=1 a 1k × b kl ∑n k=1 a 2k × b k1 ∑n k=1 a 2k × b k2 … ∑n k=1 a 2k × b kl ⋮ ⋮ ⋱ ⋮ ∑n k=1 a mk × b k1 ∑n k=1 a mk × b k2 … ∑n k=1 a mk × b kl n ∑ k=1 a 1k × b k1 = a 11 × b 11 + a 12 × b 21 + … + a 1n × b n1
    ܕߦྻ
    
    ͱɺ
    ܕߦྻ
    
    ʹ͍ͭͯɿ (m, n) A (n, l) B
44. ߦྻͷܭࢉʢʣɿੵ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn B = b 11 b 12 … b 1l b 21 b 22 … b 2l ⋮ ⋮ ⋱ ⋮ b n1 b n2 … b nl
    ˞ A × B = ∑n k=1 a 1k × b k1 ∑n k=1 a 1k × b k2 … ∑n k=1 a 1k × b kl ∑n k=1 a 2k × b k1 ∑n k=1 a 2k × b k2 … ∑n k=1 a 2k × b kl ⋮ ⋮ ⋱ ⋮ ∑n k=1 a mk × b k1 ∑n k=1 a mk × b k2 … ∑n k=1 a mk × b kl n ∑ k=1 a 1k × b k1 = a 11 × b 11 + a 12 × b 21 + … + a 1n × b n1
    ܕߦྻ
    
    ͱɺ
    ܕߦྻ
    
    ʹ͍ͭͯɿ (m, n) A (n, l) B
45. ߦྻͷܭࢉʢʣɿੵ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn B = b 11 b 12 … b 1l b 21 b 22 … b 2l ⋮ ⋮ ⋱ ⋮ b n1 b n2 … b nl
    ˞ A × B = ∑n k=1 a 1k × b k1 ∑n k=1 a 1k × b k2 … ∑n k=1 a 1k × b kl ∑n k=1 a 2k × b k1 ∑n k=1 a 2k × b k2 … ∑n k=1 a 2k × b kl ⋮ ⋮ ⋱ ⋮ ∑n k=1 a mk × b k1 ∑n k=1 a mk × b k2 … ∑n k=1 a mk × b kl n ∑ k=1 a 1k × b k1 = a 11 × b 11 + a 12 × b 21 + … + a 1n × b n1
    ܕߦྻ
    
    ͱɺ
    ܕߦྻ
    
    ʹ͍ͭͯɿ (m, n) A (n, l) B
46. ߦྻͷܭࢉʢʣɿੵ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn B = b 11 b 12 … b 1l b 21 b 22 … b 2l ⋮ ⋮ ⋱ ⋮ b n1 b n2 … b nl
    ˞ A × B = ∑n k=1 a 1k × b k1 ∑n k=1 a 1k × b k2 … ∑n k=1 a 1k × b kl ∑n k=1 a 2k × b k1 ∑n k=1 a 2k × b k2 … ∑n k=1 a 2k × b kl ⋮ ⋮ ⋱ ⋮ ∑n k=1 a mk × b k1 ∑n k=1 a mk × b k2 … ∑n k=1 a mk × b kl n ∑ k=1 a 1k × b k1 = a 11 × b 11 + a 12 × b 21 + … + a 1n × b n1
    ܕߦྻ
    
    ͱɺ
    ܕߦྻ
    
    ʹ͍ͭͯɿ (m, n) A (n, l) B
47. ߦྻͷܭࢉʢʣɿੵ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn B = b 11 b 12 … b 1l b 21 b 22 … b 2l ⋮ ⋮ ⋱ ⋮ b n1 b n2 … b nl
    ˞ A × B = ∑n k=1 a 1k × b k1 ∑n k=1 a 1k × b k2 … ∑n k=1 a 1k × b kl ∑n k=1 a 2k × b k1 ∑n k=1 a 2k × b k2 … ∑n k=1 a 2k × b kl ⋮ ⋮ ⋱ ⋮ ∑n k=1 a mk × b k1 ∑n k=1 a mk × b k2 … ∑n k=1 a mk × b kl n ∑ k=1 a 1k × b k1 = a 11 × b 11 + a 12 × b 21 + … + a 1n × b n1
    ܕߦྻ
    
    ͱɺ
    ܕߦྻ
    
    ʹ͍ͭͯɿ (m, n) A (n, l) B
48. ߦྻͷܭࢉʢʣɿੵ A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a m1 a m2 … a mn B = b 11 b 12 … b 1l b 21 b 22 … b 2l ⋮ ⋮ ⋱ ⋮ b n1 b n2 … b nl
    ͷ݁Ռ͸
    
    ܕߦྻʹͳΔ A × B (m, l)
    ܕߦྻ
    
    ͱɺ
    ܕߦྻ
    
    ʹ͍ͭͯɿ (m, n) A (n, l) B
49. ߦྻͷܭࢉʢʣɿެࣜ w
    w
    w
    w
    w
    w A + B = B + A, A + O = O + A = A (A + B) + C = A + (B + C) A × E = E × A = A (A × B) × C = A × (B × C) (α + β)A = αA + βA, α(A + B) = αA + αBʢЋ Ќ͸࣮਺ʣ (A + B) × C = A × C + B × C
50. O O ( a b c d) × ( x y) + ( t x t y) = ( x′ y′ ) ( ax + by cx + dy) + ( t x t y) = ( x′ y′ ) ( ax + by + t x cx + dy + t y) = ( x′ y′ ) P(x, y) P′ (x′ , y′ )
51. O O a b t x c d t y 0 0 1 × ( x y 1) = x′ y′ 1 ax + by + t x cx + dy + t y 0x + 0y + 1 = x′ y′ 1 P(x, y) P′ (x′ , y′ )
52. ͳͥ
    
    ࣍ਖ਼ํߦྻͰΞϑΟϯม׵Λදݱ͢Δʁ w ෳ਺ճͷΞϑΟϯม׵Λѻ͏ͨΊʹਖ਼ํߦྻͷํָ͕
    
    
    
    ( x′ y′ ) = ( a b c d) × ( x y) + ( t x t y) , ( x′ ′ y′ ′ ) = ( a′ b′ c′ d′ ) × ( x′ y′ ) + ( t′ x t′ y) ( x′ ′ y′ ′ ) = ( a′ b′ c′ d′ ) × (( a b c d) × ( x y) + ( t x t y)) + ( t′ x t′ y) x′ ′ y′ ′ 1 = a′ b′ t′ x c′ d′ t′ y 0 0 1 × a b t x c d t y 0 0 1 × ( x y 1) x′ ′ y′ ′ 1 = a′ b′ t′ x c′ d′ t′ y 0 0 1 × a b t x c d t y 0 0 1 × ( x y 1) = a′ ′ b′ ′ t′ ′ x c′ ′ d′ ′ t′ ′ y 0 0 1 × ( x y 1)
53. ͳͥ
    
    ࣍ਖ਼ํߦྻͰΞϑΟϯม׵Λදݱ͢Δʁ w ෳ਺ճͷΞϑΟϯม׵Λѻ͏ͨΊʹਖ਼ํߦྻͷํָ͕
    
    
    
    ( x′ y′ ) = ( a b c d) × ( x y) + ( t x t y) , ( x′ ′ y′ ′ ) = ( a′ b′ c′ d′ ) × ( x′ y′ ) + ( t′ x t′ y) ( x′ ′ y′ ′ ) = ( a′ b′ c′ d′ ) × (( a b c d) × ( x y) + ( t x t y)) + ( t′ x t′ y) x′ ′ y′ ′ 1 = a′ b′ t′ x c′ d′ t′ y 0 0 1 × a b t x c d t y 0 0 1 × ( x y 1) x′ ′ y′ ′ 1 = a′ b′ t′ x c′ d′ t′ y 0 0 1 × a b t x c d t y 0 0 1 × ( x y 1) = a′ ′ b′ ′ t′ ′ x c′ ′ d′ ′ t′ ′ y 0 0 1 × ( x y 1) CGAffineTransform ެࣜɿ
    (A × B) × C = A × (B × C)
54. 
    (A × B) × C = A × (B × C) A × B = B × A Ұൠతʹɺߦྻͷֻ͚ࢉʹަ׵๏ଇ͸੒ཱ͠ͳ͍ɻ
55. A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a n1 a n2 … a nn B = b 11 b 12 … b 1n b 21 b 22 … b 2n ⋮ ⋮ ⋱ ⋮ b n1 b n2 … b nn
    ˞ A × B = ∑n k=1 a 1k × b k1 ∑n k=1 a 1k × b k2 … ∑n k=1 a 1k × b kn ∑n k=1 a 2k × b k1 ∑n k=1 a 2k × b k2 … ∑n k=1 a 2k × b kn ⋮ ⋮ ⋱ ⋮ ∑n k=1 a nk × b k1 ∑n k=1 a nk × b k2 … ∑n k=1 a nk × b kn n ∑ k=1 a 1k × b k1 = a 11 × b 11 + a 12 × b 21 + … + a 1n × b n1 ೋͭͷ
    
    ࣍ਖ਼ํߦྻ
    ɺ
    ʹ͍ͭͯɿ n A B
56. A = a 11 a 12 … a 1n a 21 a 22 … a 2n ⋮ ⋮ ⋱ ⋮ a n1 a n2 … a nn B = b 11 b 12 … b 1n b 21 b 22 … b 2n ⋮ ⋮ ⋱ ⋮ b n1 b n2 … b nn
    ˞ B × A = ∑n k=1 b 1k × a k1 ∑n k=1 b 1k × a k2 … ∑n k=1 b 1k × a kn ∑n k=1 b 2k × a k1 ∑n k=1 b 2k × a k2 … ∑n k=1 b 2k × a kn ⋮ ⋮ ⋱ ⋮ ∑n k=1 b nk × a k1 ∑n k=1 b nk × a k2 … ∑n k=1 b nk × a kn n ∑ k=1 b 1k × a k1 = b 11 × a 11 + b 12 × a 21 + … + b 1n × a n1 ೋͭͷ
    
    ࣍ߦྻ
    ɺ
    ʹ͍ͭͯɿ n A B
57. O
58. O 3PUBUJPO
59. O 3PUBUJPO4DBMF
60. O 3PUBUJPO4DBMF O
61. O 3PUBUJPO4DBMF O 4DBMF
62. O 3PUBUJPO4DBMF O 4DBMF3PUBUJPO w ߦྻͷֻ͚ࢉ͸ॱ൪ʹΑͬͯ݁Ռ͕ҧ͏
    w ΞϑΟϯม׵͸ॱ൪ʹΑͬͯ݁Ռ͕ҧ͏
63. ૢ࡞ ༧૝ ࣮ࡍ transform .identity transform .identity .scaled transform .identity .scaled .rotated !
64. ͳͥॻ͖ॱ௨Γͷม׵ʹͳ͍ͬͯͳ͍ͷʁ
    ( x′ y′ ) = ( a b c d) × ( x y) + ( t x t y) , ( x′ ′ y′ ′ ) = ( a′ b′ c′ d′ ) × ( x′ y′ ) + ( t′ x t′ y) x′ ′ y′ ′ 1 = a′ b′ t′ x c′ d′ t′ y 0 0 1 × a b t x c d t y 0 0 1 × ( x y 1) ֻ͚ࢉͷॱ൪͸ɺ߹੒ॱͷٯʹ
    ͳΔඞཁ͕͋Δ͕ɺ΋͔ͯ͠͠
    $PSF
    (SBQIJDT
    ͕͜͜Ͱ߹੒ॱ
    ௨ΓͰֻ͚ࢉͪ͠Όͬͨʂʁ
65. ͳͥॻ͖ॱ௨Γͷม׵ʹͳ͍ͬͯͳ͍ͷʁ ಉ͡
66. ͳͥॻ͖ॱ௨Γͷม׵ʹͳ͍ͬͯͳ͍ͷʁ ಉ͡ ֻ͚ࢉͷॱ൪͸
    ߹ͬͯΔͬΆ͍" ͭ·Γඳը࣌ͷܭࢉ͕ؒҧͬͯ
    b
    ͱ
    c
    ͷࢀর͕ٯͩͬͨʂʁ
68. IUUQTEFWFMPQFSBQQMFDPNEPDVNFOUBUJPODPSFHSBQIJDTDHBGpOFUSBOTGPSN
69. ͳͥॻ͖ॱ௨Γͷม׵ʹͳ͍ͬͯͳ͍ͷʁ ͦ΋ͦ΋͜Ε·Ͱͷલఏࣗମ͕ؒҧͬͯͨ#
    
    x′ ′ y′ ′ 1 = a′ b′ t′ x c′ d′ t′ y 0 0 1 × a b t x c d t y 0 0 1 × ( x y 1) (x′ ′ y′ ′ 1) = (x y 1) × a b 0 c d 0 t x t y 1 × a′ b′ 0 c′ d′ 0 t′ x t′ y 1 ී௨ͷΞϑΟϯม׵ղઆͰ
    Α͘࢖ΘΕΔܭࢉࣜ ࣮ࡍͷ
    $PSF
    (SBQIJDT
    Ͱ
    ࢖ΘΕͯΔܭࢉࣜ
70. ͳͥॻ͖ॱ௨Γͷม׵ʹͳ͍ͬͯͳ͍ͷʁ ͭ·Γ͜Ε͸࢓༷Ͱ͢ɻ
71. ͳͥॻ͖ॱ௨Γͷม׵ʹͳ͍ͬͯͳ͍ͷʁ ͭ·Γ͜Ε͸࢓༷Ͱ͢ɻ
    ԾઆͰ͕͢ɺӳޠͷදݱͱͯ͠ɺ"FE#FE
    ͷ઀ଓ͸ɺ ײ֮ͱͯ͠͸ʮઌʹ
    #
    ͞Ε͔ͯΒ
    "
    ͞Εͨʯ͔ͩΒɺ CGAffineTransform
    ͷ
    xxed
    ܥϝιου͸׶͑ͯ
    ͜Μͳ෩ʹ࡞ΒΕ͍ͯΔͷͰ͸ͳ͍͔ͱߟ͍͑ͯ·͢ɻ
72. ૢ࡞ ༧૝ ࣮ࡍ transform .identity transform .identity .scaled transform .identity .scaled .rotated ! ͦ΋ͦ΋ӳޠͰ͸͜Ε͸
    ઌʹ
    SPUBUFE
    ͞Ε͔ͯΒ
    TDBMFE
    ͞ΕΔײ͔֮ͩΒ
    ݁Ռ͸ݴޠײ֮ʹ߹க
73. ͡Ό͋ॻ͖ॱ௨Γʹม׵͔ͨͬͨ͠Βʁ DPODBUFOBUJOH
    ϝιουΛ࢖͑͹
    ͪΌΜͱ
    "ʷ#
    ͷॱ൪Ͱܭࢉͯ͠
    ͘ΕΔ$
74. ͡Ό͋ॻ͖ॱ௨Γʹม׵͔ͨͬͨ͠Βʁ %
75. ·ͱΊ w ΞϑΟϯม׵͸࠲ඪม׵ͷҰछ
    w ΞϑΟϯม׵ͷܭࢉ͸ߦྻͷԋࢉ
    w ߦྻͷԋࢉʢಛʹֻ͚ࢉʣ͸໘౗Ͱ͕͢೉͘͠͸ͳ͍
    w ߦྻͷֻ͚ࢉ͸ॱ൪ΛؾΛ͚ͭΔ΂͠
    w CGAffineTransform
    ͷ
    xxed
    "1*
    ͕ॻ͖ॱͱٯͷ ॱ൪Ͱඳը͞ΕΔͷ͸࢓༷
    w ॻ͖ॱ௨Γͷඳը͕ཉ͔ͬͨ͠Β
    concatenating
    Λ࢖͑͹͍͍
76. ॓୊ w CGAffineTransform.identity
    ͱ͸ͲΜͳߦྻ͔ɺ ߟ͑ͯΈΑ͏ɻ
    w ͦͯ͠ɺԿނ ͷܭࢉࣜͩͱɺࠓͷಈ͖ʢb
    ͱ
    c
    ͷ໾ׂ͕ٯͩͬͨΓɺ ߦྻͷֻ͚ࢉॱ൪͕߹੒ॱ൪ͱಉ͡ॱ൪ʹͳͬͨΓʣʹ ͳΔͷ͔ɺߟ͑ͯΈΑ͏ɻ (x′ ′ y′ ′ 1) = (x y 1) × a b 0 c d 0 t x t y 1 × a′ b′ 0 c′ d′ 0 t′ x t′ y 1
77. ࢀߟࢿྉ w ྫ୊ͱԋशͰϚελʔ͢Δઢܗ୅਺ɿ
    IUUQTXXXBNB[PODPKQEQ
    w $("⒏OF5SBOTGPSN
    ࣮ફೖ໳ൃදεϥΠυɿ
    IUUQTTQFBLFSEFDLDPNMPWFFDHB⒏OFUSBOTGPSNTIJKJBOSVNFO
    w ߴ౳ֶߍ਺ֶ$ߦྻɿ
    IUUQTKBXJLJCPPLTPSHXJLJߴ౳ֶߍ਺ֶ$ߦྻ
    
    w ׬શʹཧղ͢ΔΞϑΟϯม׵ɿ
    IUUQTRJJUBDPNLPTIJBOJUFNTDFFDCCG
    w $("⒏OF5SBOTGPSN
    ެࣜυΩϡϝϯτɿ
    IUUQTEFWFMPQFSBQQMFDPNEPDVNFOUBUJPODPSFHSBQIJDT DHB⒏OFUSBOTGPSN