winter 2021, lingqi yan, uc santa barbaralingqi/teaching/resources/cs...cs180 / 280, winter 2021...

Winter 2021, Lingqi Yan, UC Santa Barbara

TransformationLecture 3:

http://www.cs.ucsb.edu/~lingqi/teaching/cs180.html

CS180 / 280: Introduction to Computer Graphics

CS180 / 280, Winter 2021 Lingqi Yan, UC Santa Barbara

Announcements• TA-s’ office hours are all available on the course website

- When you join and see no one, don’t leave, just wait a while

• Assignment 1 will be released this Thursday (Jan 14)

- Weekly assignment, so it’s due in a week

• Policy update

- “Merit days”: you will have 3 days IN TOTAL to spend over the

quarter to make up for late assignments

2


Last Lecture• Vectors

- Basic operations: addition, multiplication

• Dot Product

- Forward / backward (dot product positive / negative)

• Cross Product

- Left / right (cross product outward / inward)

• Matrices

3


This Week

• Transformation!

• Today

- Why study transformation - 2D transformations: rotation, scale, shear - Homogeneous coordinates - Composing transforms

4


Today• Why study transformation

- Modeling

- Viewing

• 2D transformations

• Homogeneous coordinates

5


Why Transformation?

Modeling: translation6


Why Transformation?

Modeling: rotation7


Why Transformation?

Modeling: scaling8


Why Transformation?

9

Viewing: (3D to 2D) projection


Today• Why study transformation


- Representing transformations using matrices

- Rotation, scale, shear


10

CS180 / 280, Winter 2021 Lingqi Yan, UC Santa Barbara11Ren NgCS184/284A

Scale

S0.5


Scale Transform

S0.5

x0 = sx

y0 = sy


Scale Matrix

S0.5

x0

y0

�=

s 00 s

� xy

�


Scale (Non-Uniform)

S0.5,1.0

x0

y0

�=

sx 00 sy

� xy

�

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


Reflection Matrix

x0 =??

y0 =??

Horizontal reflection: x0

y0

�=

�1 00 1

� xy

�

AAACWHicbVHLSgMxFM2MrX34qnXpJli0biwzIuhGKLpxWcE+oFNKJnOnDc1khiQjLUN/UnChv+LG9IHY6oWQw7nnPnLiJ5wp7Tgflr2Ty+8WiqXy3v7B4VHluNpRcSoptGnMY9nziQLOBLQ10xx6iQQS+Ry6/uRxke++glQsFi96lsAgIiPBQkaJNtSwEns+jJjI/Ihoyabzad3z8KzugQh+uPstzZWLL7BjdI653Q3pdrdFsw3BsFJzGs4y8F/grkENraM1rLx5QUzTCISmnCjVd51EDzIiNaMc5mUvVZAQOiEj6BsoSARqkC2NmeNzwwQ4jKU5QuMl+7siI5FSs8g3SrPfWG3nFuR/uX6qw7tBxkSSahB0NShMOdYxXriMAyaBaj4zgFDJzK6YjokkVJu/KBsT3O0n/wWd64Zr8PNNrfmwtqOITtEZukQuukVN9IRaqI0oekdfVs7KW582sgt2aSW1rXXNCdoIu/oN3QK1LQ==AAACWHicbVHLSgMxFM2MrX34qnXpJli0biwzIuhGKLpxWcE+oFNKJnOnDc1khiQjLUN/UnChv+LG9IHY6oWQw7nnPnLiJ5wp7Tgflr2Ty+8WiqXy3v7B4VHluNpRcSoptGnMY9nziQLOBLQ10xx6iQQS+Ry6/uRxke++glQsFi96lsAgIiPBQkaJNtSwEns+jJjI/Ihoyabzad3z8KzugQh+uPstzZWLL7BjdI653Q3pdrdFsw3BsFJzGs4y8F/grkENraM1rLx5QUzTCISmnCjVd51EDzIiNaMc5mUvVZAQOiEj6BsoSARqkC2NmeNzwwQ4jKU5QuMl+7siI5FSs8g3SrPfWG3nFuR/uX6qw7tBxkSSahB0NShMOdYxXriMAyaBaj4zgFDJzK6YjokkVJu/KBsT3O0n/wWd64Zr8PNNrfmwtqOITtEZukQuukVN9IRaqI0oekdfVs7KW582sgt2aSW1rXXNCdoIu/oN3QK1LQ==AAACWHicbVHLSgMxFM2MrX34qnXpJli0biwzIuhGKLpxWcE+oFNKJnOnDc1khiQjLUN/UnChv+LG9IHY6oWQw7nnPnLiJ5wp7Tgflr2Ty+8WiqXy3v7B4VHluNpRcSoptGnMY9nziQLOBLQ10xx6iQQS+Ry6/uRxke++glQsFi96lsAgIiPBQkaJNtSwEns+jJjI/Ihoyabzad3z8KzugQh+uPstzZWLL7BjdI653Q3pdrdFsw3BsFJzGs4y8F/grkENraM1rLx5QUzTCISmnCjVd51EDzIiNaMc5mUvVZAQOiEj6BsoSARqkC2NmeNzwwQ4jKU5QuMl+7siI5FSs8g3SrPfWG3nFuR/uX6qw7tBxkSSahB0NShMOdYxXriMAyaBaj4zgFDJzK6YjokkVJu/KBsT3O0n/wWd64Zr8PNNrfmwtqOITtEZukQuukVN9IRaqI0oekdfVs7KW582sgt2aSW1rXXNCdoIu/oN3QK1LQ==AAACWHicbVHLSgMxFM2MrX34qnXpJli0biwzIuhGKLpxWcE+oFNKJnOnDc1khiQjLUN/UnChv+LG9IHY6oWQw7nnPnLiJ5wp7Tgflr2Ty+8WiqXy3v7B4VHluNpRcSoptGnMY9nziQLOBLQ10xx6iQQS+Ry6/uRxke++glQsFi96lsAgIiPBQkaJNtSwEns+jJjI/Ihoyabzad3z8KzugQh+uPstzZWLL7BjdI653Q3pdrdFsw3BsFJzGs4y8F/grkENraM1rLx5QUzTCISmnCjVd51EDzIiNaMc5mUvVZAQOiEj6BsoSARqkC2NmeNzwwQ4jKU5QuMl+7siI5FSs8g3SrPfWG3nFuR/uX6qw7tBxkSSahB0NShMOdYxXriMAyaBaj4zgFDJzK6YjokkVJu/KBsT3O0n/wWd64Zr8PNNrfmwtqOITtEZukQuukVN9IRaqI0oekdfVs7KW582sgt2aSW1rXXNCdoIu/oN3QK1LQ==

x0 = �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

y0 = yAAAB63icbZDLSgMxFIbP1Futt6pLN8EiuiozUtCNUHTjsoK9QDuUTJppQ5PMkGSEYegruHGhiFtfyJ1vY6adhbb+EPj4zznknD+IOdPGdb+d0tr6xuZWebuys7u3f1A9POroKFGEtknEI9ULsKacSdo2zHDaixXFIuC0G0zv8nr3iSrNIvlo0pj6Ao8lCxnBJrfS85t0WK25dXcutApeATUo1BpWvwajiCSCSkM41rrvubHxM6wMI5zOKoNE0xiTKR7TvkWJBdV+Nt91hs6sM0JhpOyTBs3d3xMZFlqnIrCdApuJXq7l5n+1fmLCaz9jMk4MlWTxUZhwZCKUH45GTFFieGoBE8XsrohMsMLE2HgqNgRv+eRV6FzWPcsPjVrztoijDCdwChfgwRU04R5a0AYCE3iGV3hzhPPivDsfi9aSU8wcwx85nz+neY34AAAB63icbZDLSgMxFIbP1Futt6pLN8EiuiozUtCNUHTjsoK9QDuUTJppQ5PMkGSEYegruHGhiFtfyJ1vY6adhbb+EPj4zznknD+IOdPGdb+d0tr6xuZWebuys7u3f1A9POroKFGEtknEI9ULsKacSdo2zHDaixXFIuC0G0zv8nr3iSrNIvlo0pj6Ao8lCxnBJrfS85t0WK25dXcutApeATUo1BpWvwajiCSCSkM41rrvubHxM6wMI5zOKoNE0xiTKR7TvkWJBdV+Nt91hs6sM0JhpOyTBs3d3xMZFlqnIrCdApuJXq7l5n+1fmLCaz9jMk4MlWTxUZhwZCKUH45GTFFieGoBE8XsrohMsMLE2HgqNgRv+eRV6FzWPcsPjVrztoijDCdwChfgwRU04R5a0AYCE3iGV3hzhPPivDsfi9aSU8wcwx85nz+neY34AAAB63icbZDLSgMxFIbP1Futt6pLN8EiuiozUtCNUHTjsoK9QDuUTJppQ5PMkGSEYegruHGhiFtfyJ1vY6adhbb+EPj4zznknD+IOdPGdb+d0tr6xuZWebuys7u3f1A9POroKFGEtknEI9ULsKacSdo2zHDaixXFIuC0G0zv8nr3iSrNIvlo0pj6Ao8lCxnBJrfS85t0WK25dXcutApeATUo1BpWvwajiCSCSkM41rrvubHxM6wMI5zOKoNE0xiTKR7TvkWJBdV+Nt91hs6sM0JhpOyTBs3d3xMZFlqnIrCdApuJXq7l5n+1fmLCaz9jMk4MlWTxUZhwZCKUH45GTFFieGoBE8XsrohMsMLE2HgqNgRv+eRV6FzWPcsPjVrztoijDCdwChfgwRU04R5a0AYCE3iGV3hzhPPivDsfi9aSU8wcwx85nz+neY34AAAB63icbZDLSgMxFIbP1Futt6pLN8EiuiozUtCNUHTjsoK9QDuUTJppQ5PMkGSEYegruHGhiFtfyJ1vY6adhbb+EPj4zznknD+IOdPGdb+d0tr6xuZWebuys7u3f1A9POroKFGEtknEI9ULsKacSdo2zHDaixXFIuC0G0zv8nr3iSrNIvlo0pj6Ao8lCxnBJrfS85t0WK25dXcutApeATUo1BpWvwajiCSCSkM41rrvubHxM6wMI5zOKoNE0xiTKR7TvkWJBdV+Nt91hs6sM0JhpOyTBs3d3xMZFlqnIrCdApuJXq7l5n+1fmLCaz9jMk4MlWTxUZhwZCKUH45GTFFieGoBE8XsrohMsMLE2HgqNgRv+eRV6FzWPcsPjVrztoijDCdwChfgwRU04R5a0AYCE3iGV3hzhPPivDsfi9aSU8wcwx85nz+neY34


Shear Matrix

x0 =??

y0 =??

x0

y0

�=

1 a0 1

� xy

�

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

Horizontal shift is 0 at y=0Horizontal shift is a at y=1

Hints:

a

1

Vertical shift is always 0


Rotate

R45

(about the origin (0, 0), CCW by default)


Rotation Matrix

Lecture 3 Math

x0 � x1 � x2 � x3 � y

x+ y � ✓

f(x)

f(x+ y) = f(x) + f(y)

f(ax) = af(x)

f(x) = af(x)

Scale:Sa(x) = ax

S2(x)� S2(x1)� S2(x2)� S2(x3 � S2(ax)� aS2(x)� S2(x)� S2(y)� S2(x+ y)

S2(x) = 2x

aS2(x) = 2ax

S2(ax) = 2ax

S2(ax) = aS2(x)

S2(x+ y) = 2(x+ y)

S2(x) + S2(y) = 2x+ 2y

S2(x+ y) = S2(x) + S2(y)

Rotations:

R✓(x)�R✓(x0)�R✓(x1)�R✓(x2)�R✓(x3)�R✓(ax)� aR✓(x)�R✓(y)�R✓(x+ y)

Translation:Ta,b(x0)� Ta,b(x1)� Ta,b(x2)� Ta,b(x3)

Lecture 3 Math

x0 � x1 � x2 � x3 � y

x+ y � ✓ � xx � xycos ✓ � sin ✓

f(x)

f(x+ y) = f(x) + f(y)

f(ax) = af(x)

f(x) = af(x)

Scale:Sa(x) = ax


S2(x) = 2x

aS2(x) = 2ax

S2(ax) = 2ax

S2(ax) = aS2(x)

S2(x+ y) = 2(x+ y)

S2(x) + S2(y) = 2x+ 2y

S2(x+ y) = S2(x) + S2(y)

Ss =

sx 00 sy

�

Ss =

0.5 00 2

�

s =⇥0.5 2

⇤T

Ssx0 � Ssx1 � Ssx2 � Ssx3

Lecture 3 Math

x0 � x1 � x2 � x3 � y

x+ y � ✓ � xx � xycos ✓ � sin ✓

f(x)

f(x+ y) = f(x) + f(y)

f(ax) = af(x)

f(x) = af(x)

Scale:Sa(x) = ax


S2(x) = 2x

aS2(x) = 2ax

S2(ax) = 2ax

S2(ax) = aS2(x)

S2(x+ y) = 2(x+ y)

S2(x) + S2(y) = 2x+ 2y

S2(x+ y) = S2(x) + S2(y)

Ss =

sx 00 sy

�

Ss =

0.5 00 2

�

s =⇥0.5 2

⇤T


Lecture 3 Math

x0 � x1 � x2 � x3 � y

x+ y � ✓ � xx � xycos ✓ � sin ✓

f(x)

f(x+ y) = f(x) + f(y)

f(ax) = af(x)

f(x) = af(x)

Scale:Sa(x) = ax


S2(x) = 2x

aS2(x) = 2ax

S2(ax) = 2ax

S2(ax) = aS2(x)

S2(x+ y) = 2(x+ y)

S2(x) + S2(y) = 2x+ 2y

S2(x+ y) = S2(x) + S2(y)

Ss =

sx 00 sy

�

Ss =

0.5 00 2

�

s =⇥0.5 2

⇤T


Lecture 3 Math

x0 � x1 � x2 � x3 � y

x+ y � ✓ � xx � xycos ✓ � sin ✓

f(x)

f(x+ y) = f(x) + f(y)

f(ax) = af(x)

f(x) = af(x)

Scale:Sa(x) = ax


S2(x) = 2x

aS2(x) = 2ax

S2(ax) = 2ax

S2(ax) = aS2(x)

S2(x+ y) = 2(x+ y)

S2(x) + S2(y) = 2x+ 2y

S2(x+ y) = S2(x) + S2(y)

Ss =

sx 00 sy

�

Ss =

0.5 00 2

�

s =⇥0.5 2

⇤T


Rotations:

R✓(x)�R✓(x0)�R✓(x1)�R✓(x2)�R✓(x3)�R✓(ax)� aR✓(x)�R✓(y)�R✓(x+ y)

R✓ =

cos ✓ � sin ✓sin ✓ cos ✓

�

Translation:Tb(x) = x+ b

Tb(x0)� Tb(x1)� Tb(x2)� Tb(x3)� Tb(x)� Tb(y)� Tb(x+ y)� Tb(x) + Tb(y)

Ta,b(x0)� Ta,b(x1)� Ta,b(x2)� Ta,b(x3)� Ta,b(x)� Ta,b(y)� Ta,b(x+ y)� Ta,b(x) + Ta,b(y)

Reflection:Rey(x0)�Rey(x1)�Rey(x2)�Rey(x3)

Rex(x0)�Rex(x1)�Rex(x2)�Rex(x3)

Shear:Hx(x0)�Hx(x1)�Hx(x2)�Hx(x3)

Misc:f(x0)� f(x1)� f(x2)� f(x3)

f(x) = T3,1(S0.5(x))��f(x) = S0.5(T3,1(x))

f(x) = g(x) + b

Euclidean:|f(x)� f(y)| = |x� y|

2


Deriving the Rotation Matrix

19

* Assuming we already know that a 2x2 matrix is able to perform rotations

• (1,0) → (cos θ, sin θ)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✓cos ✓sin ✓

◆=

✓A BC D

◆·✓10

◆

• , A = cos θ C = sin θ


Linear Transforms = Matrices

x0 = a x+ b y

y0 = c x+ d y

x0

y0

�=

a bc d

� xy

�

x0 = M x

(of the same dimension)


Today

• Why study transformation



- Why homogeneous coordinates - Affine transformation

21


Translate

T1,1Ttx,ty


Translation??

T1,1

x0 = x+ tx

y0 = y + ty

Ttx,ty


Why Homogeneous Coordinates

• Translation cannot be represented in matrix form

24

x0

y0

�=

a bc d

� xy

�+

txty

�

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

(So, translation is NOT linear transform!)

• But we don’t want translation to be a special case

• Is there a unified way to represent all transformations? (and what’s the cost?)


Solution: Homogenous Coordinates

Add a third coordinate (w-coordinate)

• 2D point = (x, y, 1)T • 2D vector = (x, y, 0)T

Matrix representation of translations�

⇤x�

y�

w�

⇥

⌅ =

�

⇤1 0 tx0 1 ty0 0 1

⇥

⌅ ·

�

⇤xy1

⇥

⌅ =

�

⇤x + txy + ty

1

⇥

⌅

What if you translate a vector?


Translating Points and Vectors• Translating a point (x, y,1)

26

• Translating a vector

- Recall: translational invariance of vectors

(x, y,0)

AAACnnicbVFRa9swEJa9dkvddsvWx7yIhpZCIdhjY30pdBuM0VLawtIG4hBk+ZKKyrKRziPG5Gftj/Rt/2ayY1jr9ODgu+/06T6dokwKg77/13FfbWy+ftPZ8rZ3dt++677/cGvSXHMY8lSmehQxA1IoGKJACaNMA0siCXfRw/eqf/cbtBGp+oVFBpOEzZWYCc7QUtPunzCCuVBlljDUYrH0AnpIfZs4XYShV6GgroqmqhkvBBX/14Q8TtFr31TpK1WIsMDaaakhXpbBsiU/XZcer8YXx6vB7YHTbt8f+HXQdRA0oE+auJ52H8M45XkCCrlkxowDP8NJyTQKLsEayg1kjD+wOYwtVCwBMylr00t6YJmYzlJtUyGt2aeKkiXGFElkT1qD96bdq8iXeuMcZyeTUqgsR1B8NWiWS4oprf6KxkIDR1lYwLgW1ivl90wzjvZHqyUE7Sevg9uPg+DzwL/51D/71qyjQ3pknxyRgHwhZ+QnuSZDwp2e89U5dy5c6v5wL92r1VHXaTR75Fm4o3+azci70

@1 0 tx0 1 ty0 0 1

1

A ·

0

@xy1

1

A =

0

@x+ txy + ty

1

1

A

AAAClnicdVFNT9wwEHVSaCHlY1sulbhYrIo4rRwEKhxaIVAFx0ViAWmzWjnO7GLhOJE9qXYV7U/qn+mNf4MTIgGBjjTSmzfz7OdxnCtpkbEHz/+wtPzx08pq8HltfWOz8+Xrtc0KI2AgMpWZ25hbUFLDACUquM0N8DRWcBPfn1X9mz9grMz0Fc5zGKV8quVECo6OGnf+RjFMpS7zlKORs0UQ0l3KXOJ4FkVBhcK6mjdVzQQR6ORZE4kkw6B9UqWvVBHCDGunpYFkUbJFS/7zv1LWmhx3uqzH6qBvQdiALmmiP+78i5JMFCloFIpbOwxZjqOSG5RCgTNSWMi5uOdTGDqoeQp2VNZmF/S7YxI6yYxLjbRmXypKnlo7T2M36Qze2XavIt/rDQucHI1KqfMCQYuniyaFopjR6o9oIg0IVHMHuDDSeaXijhsu0P1ktYSw/eS34Hq/Fx722OVB9+S0WccK2SY7ZI+E5Ac5IRekTwZEeFvesXfqnfnf/F/+b//8adT3Gs0WeRV+/xEPfMV80

@1 0 tx0 1 ty0 0 1

1

A ·

0

@xy0

1

A =

0

@xy0

1

A


Homogenous Coordinates

Valid operation if w-coordinate of result is 1 or 0

• vector + vector = vector • point – point = vector • point + vector = point • point + point = ??

In homogeneous coordinates, 0

@xyw

1

A is the 2D point

0

@x/wy/w1

1

A , w 6= 0

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


Affine Transformations

Affine map = linear map + translation

Using homogenous coordinates:�

⇤x�

y�

1

⇥

⌅ =

�

⇤a b txc d ty0 0 1

⇥

⌅ ·

�

⇤xy1

⇥

⌅

�x�

y�

⇥=

�a bc d

⇥·�

xy

⇥+

�txty

⇥


2D Transformations

Scale

Rotation

Translation

T(tx, ty) =

�

⇤1 0 tx0 1 ty0 0 1

⇥

⌅

S(sx, sy) =

�

⇤sx 0 00 sy 00 0 1

⇥

⌅

R(�) =

�

⇤cos � � sin � 0sin� cos � 0

0 0 1

⇥

⌅


Inverse Transform

   is the inverse of transform in both a matrix and geometric sense

M�1

M�1

M

M�1 M�1

CS180 / 280, Winter 2021 Lingqi Yan, UC Santa Barbara31

Composing Transforms


Composite Transform

?


Translate Then Rotate?

M = R45 · T(1,0) M = R45 · T(1,0)


Rotate Then Translate

M = T(1,0) ·R45 M = T(1,0) ·R45


Transform Ordering Matters!

M = R45 · T(1,0) M = R45 · T(1,0)

M = T(1,0) ·R45 M = T(1,0) ·R45

CS180 / 280, Winter 2021 Lingqi Yan, UC Santa Barbara36 Ren NgCS184/284A


Matrix multiplication is not commutative

M = R45 · T(1,0)M = T(1,0) ·R45≠

2

4cos 45� � sin 45� 0sin 45� cos 45� 0

0 0 1

3

5

2

41 0 10 1 00 0 1

3

5 6=

2

41 0 10 1 00 0 1

3

5

2

4cos 45� � sin 45� 0sin 45� cos 45� 0

0 0 1

3

5

Recall the matrix math represented by these symbols:

Note that matrices are applied right to left:

T(1,0) ·R45

2

4xy1

3

5 =

2

41 0 10 1 00 0 1

3

5

2

4cos 45� � sin 45� 0sin 45� cos 45� 0

0 0 1

3

5

2

4xy1

3

5

Ren NgCS184/284A


Matrix multiplication is not commutative

M = R45 · T(1,0)M = T(1,0) ·R45≠

2

4cos 45� � sin 45� 0sin 45� cos 45� 0

0 0 1

3

5

2

41 0 10 1 00 0 1

3

5 6=

2

41 0 10 1 00 0 1

3

5

2

4cos 45� � sin 45� 0sin 45� cos 45� 0

0 0 1

3

5

Recall the matrix math represented by these symbols:

Note that matrices are applied right to left:

T(1,0) ·R45

2

4xy1

3

5 =

2

41 0 10 1 00 0 1

3

5

2

4cos 45� � sin 45� 0sin 45� cos 45� 0

0 0 1

3

5

2

4xy1

3

5


An Implicit Order using Homogeneous Coordinates

• Linear transform first, then translate

37

Ren NgCS184/284A




⇤x�

y�

1

⇥

⌅ =

�


⇥

⌅ ·

�

⇤xy1

⇥

⌅

�x�

y�

⇥=

�a bc d

⇥·�

xy

⇥+

�txty

⇥Ren NgCS184/284A




⇤x�

y�

1

⇥

⌅ =

�


⇥

⌅ ·

�

⇤xy1

⇥

⌅

�x�

y�

⇥=

�a bc d

⇥·�

xy

⇥+

�txty

⇥


Composing Transforms

Sequence of affine transforms A1, A2, A3, ...

• Compose by matrix multiplication • Very important for performance!

An(. . . A2(A1(x))) = An · · · A2 · A1 ·

�

⇤xy1

⇥

⌅

Pre-multiply n matrices to obtain a  single matrix representing combined transform


Decomposing Complex Transforms

How to rotate around a given point c? 1. Translate center to origin 2. Rotate 3. Translate back

Matrix representation?

T(�c) T(c)R(�)

T(c) · R(�) · T(�c)

Thank you!(And thank Prof. Ravi Ramamoorthi and Prof. Ren Ng for many of the slides!)

winter 2021, lingqi yan, uc santa barbaralingqi/teaching/resources/cs...cs180 / 280, winter 2021...

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