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~

$("GpOF5SBOTGPSN͸Ͳ͏ಇ͍ͯΔͷ͔ʁ
ʙ4XJGUΤϯδχΞͷͨΊͷઢܗ୅਺ʙ
f o r J 0 4 % $ +BQ B O

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$("GpOF5SBOTGPSN͸Ͳ͏ಇ͍ͯΔͷ͔ʁ
ʙ4XJGUΤϯδχΞͷͨΊͷઢܗ୅਺ʙ
f o r J 0 4 % $ +BQ B O
嬭⓸

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ઢܗ୅਺
ߦྻ
ߦྻࣜ ݻ༗ۭؒ
ઢܕ
ํఔࣜ
FUD
ࠓ೔औΓ্͛Δൣғ

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}
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 {

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ͦ΋ͦ΋
CGAffineTransformͬͯʁ

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Ҡಈ
5SBOTMBUJPO
֦ॖ 4DBMJOH
ճస
3PUBUJPO

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"GpOF5SBOTGPSN
ΞϑΟϯม׵

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O
ΦϒδΣΫτݻ༗࠲ඪ ΞϑΟϯม׵ ΦϒδΣΫτදࣔ࠲ඪ
O

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ΞϑΟϯม׵͸࠲ඪม׵ͷҰछ

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IUUQTTQFBLFSEFDLDPNMPWFFDHBGpOFUSBOTGPSNTIJKJBOSVNFO

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CGAffineTransform͸
ΞϑΟϯม׵Λදݱ͢ΔͨΊʹ
$PSF(SBQIJDT্ͷσʔλߏ଄

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ΞϑΟϯࣸ૾
ΞϑΟϯม׵
O O
x ax + by + t
x
x′
y cx + dy + t
y
y′
P(x, y)
P′ (x′ , y′ )

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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)

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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)

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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′
CGAffineTransformͷ
a b c d tx
ty
Ͱ͢

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ΞϑΟϯม׵͸
࠲ඪม׵ͷϝιουͰ͋Γ
௚઀తʹҠಈ΍֦ॖΛѻΘͳ͍

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Ҡಈ
5SBOTMBUJPO
֦ॖ 4DBMJOH
ճస
3PUBUJPO

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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′

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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
Y͕࣠UYɺZ͕࣠UZҠಈͨ͠࠲ඪ
P(x, y) P′ (x′ , y′ )
x ax + by + t
x
x′
y cx + dy + t
y
y′

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֦ॖ
4DBMJOH
a = a
c = 0
tx = 0
ty = 0
b = 0
d = d
x = 10
y = 10
x' = 10a
y' = 10d
Y͕࣠BഒɺZ͕࣠Eഒ֦େͨ͠࠲ඪ
P(x, y)
P′ (x′ , y′ )
x ax + by + t
x
x′
y cx + dy + t
y
y′

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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
В˃ճసͨ͠࠲ඪ
P(x, y)
P′ (x′ , y′ )
x ax + by + t
x
x′
y cx + dy + t
y
y′

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IUUQTTQFBLFSEFDLDPNMPWFFDHBGpOFUSBOTGPSNTIJKJBOSVNFO

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࿩͕͍ͩͿ௕͔ͬͨɻ
ͱ͜ΖͰɺߦྻͬͯʁ

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O O
x ax + by + t
x
x′
y cx + dy + t
y
y′
P(x, y)
P′ (x′ , y′ )

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O O
ߦྻ
(
a b
c d)
×
(
x
y)
+
(
t
x
t
y)
=
(
x′
y′ )
P(x, y)
P′ (x′ , y′ )

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ߦྻɿͦΕ͸ఆ·ͬͨߦͱྻʹΑΔ
ɹɹɹ਺ࣈͷ૊Έ߹Θͤ
(
1 2
3 4)
458 488
718 911
595 124
(
79.9 56 86
89 61 88)
…

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A =
a
11
a
12
… a
1n
a
21
a
22
… a
2n
⋮ ⋮ ⋱ ⋮
a
m1
a
m2
… a
mn
NߦOྻ
w ߦ ྻͷߦྻΛ ܕߦྻͱݴ͏
w ߦ໨ ྻ໨ͷ਺ Λ ͷ ੒෼ͱݴ͏
m n (m, n)
i j a
ij
A (i, j)

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༷ʑͳߦྻʢʣɿྵߦྻ
O =
0 0 … 0
0 0 … 0
⋮ ⋮ ⋱ ⋮
0 0 … 0
ྵߦྻ͸ ͱ΋දه͞ΕΔ
O
શͯͷ੒෼͕ ͷߦྻΛྵߦྻͱݺͿ
0

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༷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

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༷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

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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

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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

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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

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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

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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

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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

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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′ )

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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′ )

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ͳͥ࣍ਖ਼ํߦྻͰΞϑΟϯม׵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)

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ͳͥ࣍ਖ਼ํߦྻͰΞϑΟϯม׵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)

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(A × B) × C = A × (B × C)
A × B = B × A
Ұൠతʹɺߦྻͷֻ͚ࢉʹަ׵๏ଇ͸੒ཱ͠ͳ͍ɻ

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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

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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

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O

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O
3PUBUJPO

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O
3PUBUJPO4DBMF

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O
3PUBUJPO4DBMF
O

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O
3PUBUJPO4DBMF
O
4DBMF

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O
3PUBUJPO4DBMF
O
4DBMF3PUBUJPO
w ߦྻͷֻ͚ࢉ͸ॱ൪ʹΑͬͯ݁Ռ͕ҧ͏
w ΞϑΟϯม׵͸ॱ൪ʹΑͬͯ݁Ռ͕ҧ͏

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ૢ࡞ ༧૝ ࣮ࡍ
transform
.identity
transform
.identity
.scaled
transform
.identity
.scaled
.rotated
!

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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
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ͦ΋ͦ΋͜Ε·Ͱͷલఏࣗମ͕ؒҧͬͯͨ#

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·ͱΊ
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॓୊
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ࢀߟࢿྉ
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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~

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~

Elvis Shi

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