Rotate: Difference between revisions

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{{Built-ins|Rotate|⌽|⊖}} is a [[dyadic]] [[primitive function]] which "rotates" the [[elements]] of the right [[argument]] around a specified [[axis]]. The name Rotate is typically used for the primitive <source lang=apl inline>⌽</source>, which rotates along the last axis, while <source lang=apl inline>⊖</source>, which rotates along the first axis, is called "Rotate First" or similar. APLs with [[function axis]] allow to choose a different axis from the default one. In the [[leading axis model]], specifying an axis is discouraged in favor of using <source lang=apl inline>⊖</source> with the [[Rank operator]].
{{Built-ins|Rotate|⌽|⊖}} is a [[dyadic]] [[primitive function]] which "rotates" the [[elements]] of the right [[argument]] around a specified [[axis]]. The name Rotate is typically used for the primitive <source lang=apl inline>⌽</syntaxhighlight>, which rotates along the last axis, while <source lang=apl inline>⊖</syntaxhighlight>, which rotates along the first axis, is called "Rotate First" or similar. APLs with [[function axis]] allow to choose a different axis from the default one. In the [[leading axis model]], specifying an axis is discouraged in favor of using <source lang=apl inline>⊖</syntaxhighlight> with the [[Rank operator]].


Rotate and Rotate First share the [[glyph|glyphs]] <source lang=apl inline>⌽</source> and <source lang=apl inline>⊖</source> with monadic functions [[Reverse]] and Reverse First, respectively.
Rotate and Rotate First share the [[glyph|glyphs]] <source lang=apl inline>⌽</syntaxhighlight> and <source lang=apl inline>⊖</syntaxhighlight> with monadic functions [[Reverse]] and Reverse First, respectively.


== Examples ==
== Examples ==
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       ¯4⌽1 2 3 4 5 6 7
       ¯4⌽1 2 3 4 5 6 7
4 5 6 7 1 2 3
4 5 6 7 1 2 3
</source>
</syntaxhighlight>


Rotate on a [[matrix]] rotates the elements around horizontally (to the left), while Rotate First does vertically (upwards).
Rotate on a [[matrix]] rotates the elements around horizontally (to the left), while Rotate First does vertically (upwards).
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IJKL
IJKL
ABCD
ABCD
</source>
</syntaxhighlight>


Higher-[[rank]] arrays can be rotated on an arbitrary axis if [[function axis]] or [[Rank operator]] is supported:
Higher-[[rank]] arrays can be rotated on an arbitrary axis if [[function axis]] or [[Rank operator]] is supported:
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│IJKL│MNOP│VWXU│
│IJKL│MNOP│VWXU│
└────┴────┴────┘
└────┴────┴────┘
</source>
</syntaxhighlight>


Usage of non-scalar X differs between implementations. [[Dyalog APL]] allows to specify different amount of rotation for different "columns" or 1-dimensional [[subarray|subarrays]] which Rotate is applied to. [[J]] allows to specify rotations for multiple leading axes at once.
Usage of non-scalar X differs between implementations. [[Dyalog APL]] allows to specify different amount of rotation for different "columns" or 1-dimensional [[subarray|subarrays]] which Rotate is applied to. [[J]] allows to specify rotations for multiple leading axes at once.
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4 1 2 3
4 1 2 3
3 4 1 2
3 4 1 2
</source>{{Works in|[[Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}


<source lang=j>
<source lang=j>
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10 11 8 9
10 11 8 9
  2  3 0 1
  2  3 0 1
</source>{{Works in|[[J]]}}
</syntaxhighlight>{{Works in|[[J]]}}


== Description ==
== Description ==
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In languages with [[function axis]], exactly one argument axis may be specified.
In languages with [[function axis]], exactly one argument axis may be specified.


Rotating a [[scalar]] always yields that scalar unchanged. Otherwise, Rotate operates on a particular axis of its right argument. This axis is the specified axis if one is given, and otherwise the last axis for <source lang=apl inline>⌽</source>, or the first axis for <source lang=apl inline>⊖</source>.
Rotating a [[scalar]] always yields that scalar unchanged. Otherwise, Rotate operates on a particular axis of its right argument. This axis is the specified axis if one is given, and otherwise the last axis for <source lang=apl inline>⌽</syntaxhighlight>, or the first axis for <source lang=apl inline>⊖</syntaxhighlight>.


The result array has the same [[shape]] and [[elements]] as the right argument array, but the elements cyclically move around the rotation axis. Consequently if the length of this axis is 0 or 1 then rotation has no effect.
The result array has the same [[shape]] and [[elements]] as the right argument array, but the elements cyclically move around the rotation axis. Consequently if the length of this axis is 0 or 1 then rotation has no effect.
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=== APL model ===
=== APL model ===


The rotation of a vector <source lang=apl inline>Y</source> by <source lang=apl inline>X</source> units may be written in any APL, assuming <source lang=apl inline>⎕IO←0</source>, as <source lang=apl inline>Y[(⍴Y)|X+⍳⍴Y]</source>. To rotate an arbitrary array [[Squad indexing]] with axis (or [[Rank operator|rank]]) is helpful.
The rotation of a vector <source lang=apl inline>Y</syntaxhighlight> by <source lang=apl inline>X</syntaxhighlight> units may be written in any APL, assuming <source lang=apl inline>⎕IO←0</syntaxhighlight>, as <source lang=apl inline>Y[(⍴Y)|X+⍳⍴Y]</syntaxhighlight>. To rotate an arbitrary array [[Squad indexing]] with axis (or [[Rank operator|rank]]) is helpful.
<source lang=apl>
<source lang=apl>
RotateAxis ← {                  ⍝ R ← X (K RotateAxis) Y
RotateAxis ← {                  ⍝ R ← X (K RotateAxis) Y
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     (⊂l|⍺+⍳l) ⌷[⍺⍺] ⍵          ⍝ Rotate with indexing
     (⊂l|⍺+⍳l) ⌷[⍺⍺] ⍵          ⍝ Rotate with indexing
}
}
</source>
</syntaxhighlight>


== External links ==
== External links ==

Revision as of 21:27, 10 September 2022

⌽ ⊖

Rotate (⌽, ⊖) is a dyadic primitive function which "rotates" the elements of the right argument around a specified axis. The name Rotate is typically used for the primitive <source lang=apl inline>⌽</syntaxhighlight>, which rotates along the last axis, while <source lang=apl inline>⊖</syntaxhighlight>, which rotates along the first axis, is called "Rotate First" or similar. APLs with function axis allow to choose a different axis from the default one. In the leading axis model, specifying an axis is discouraged in favor of using <source lang=apl inline>⊖</syntaxhighlight> with the Rank operator.

Rotate and Rotate First share the glyphs <source lang=apl inline>⌽</syntaxhighlight> and <source lang=apl inline>⊖</syntaxhighlight> with monadic functions Reverse and Reverse First, respectively.

Examples

The left argument is usually restricted to a scalar. Rotate on a vector right argument rotates the elements to the left, wrapping around as necessary. The left argument can be large or negative.

<source lang=apl> 

     3⌽1 2 3 4 5 6 7

4 5 6 7 1 2 3 

     8⌽1 2 3 4 5 6 7

2 3 4 5 6 7 1 

     ¯4⌽1 2 3 4 5 6 7

4 5 6 7 1 2 3 </syntaxhighlight>

Rotate on a matrix rotates the elements around horizontally (to the left), while Rotate First does vertically (upwards).

<source lang=apl> 

     ⎕←M←3 4⍴⎕A

ABCD EFGH IJKL 

     2⌽M  ⍝ Horizontally rotate twice

CDAB GHEF KLIJ 

     1⊖M  ⍝ Vertically rotate once

EFGH IJKL ABCD </syntaxhighlight>

Higher-rank arrays can be rotated on an arbitrary axis if function axis or Rank operator is supported:

<source lang=apl> 

     ⎕←M←2 3 4⍴⎕A

ABCD EFGH IJKL

MNOP QRST UVWX 

     (1⌽[1]M)(1⌽[2]M)(1⌽[3]M)  ⍝ Rotate once over 1st, 2nd, or 3rd axis, using function axis

┌────┬────┬────┐ │MNOP│EFGH│BCDA│ │QRST│IJKL│FGHE│ │UVWX│ABCD│JKLI│ │ │ │ │ │ABCD│QRST│NOPM│ │EFGH│UVWX│RSTQ│ │IJKL│MNOP│VWXU│ └────┴────┴────┘ 

     (1⊖⍤3⊢M)(1⊖⍤2⊢M)(1⊖⍤1⊢M)  ⍝ Same as above, using Rotate First with Rank

┌────┬────┬────┐ │MNOP│EFGH│BCDA│ │QRST│IJKL│FGHE│ │UVWX│ABCD│JKLI│ │ │ │ │ │ABCD│QRST│NOPM│ │EFGH│UVWX│RSTQ│ │IJKL│MNOP│VWXU│ └────┴────┴────┘ </syntaxhighlight>

Usage of non-scalar X differs between implementations. Dyalog APL allows to specify different amount of rotation for different "columns" or 1-dimensional subarrays which Rotate is applied to. J allows to specify rotations for multiple leading axes at once.

<source lang=apl> 

     ⎕←M←3 4⍴⍳4

1 2 3 4 1 2 3 4 1 2 3 4 

     1 3 2⌽M  ⍝ Rotate 1st row once, 2nd row thrice, and 3rd row twice

2 3 4 1 4 1 2 3 3 4 1 2

</syntaxhighlight>

Works in: Dyalog APL

<source lang=j> 

     i.3 4

0 1 2 3 4 5 6 7 8 9 10 11 

     1 2|.i.3 4  NB. Rotate vertically once, horizontally twice
6  7 4 5

10 11 8 9 

2  3 0 1

</syntaxhighlight>

Works in: J

Description

In languages with function axis, exactly one argument axis may be specified.

Rotating a scalar always yields that scalar unchanged. Otherwise, Rotate operates on a particular axis of its right argument. This axis is the specified axis if one is given, and otherwise the last axis for <source lang=apl inline>⌽</syntaxhighlight>, or the first axis for <source lang=apl inline>⊖</syntaxhighlight>.

The result array has the same shape and elements as the right argument array, but the elements cyclically move around the rotation axis. Consequently if the length of this axis is 0 or 1 then rotation has no effect.

APL model

The rotation of a vector <source lang=apl inline>Y</syntaxhighlight> by <source lang=apl inline>X</syntaxhighlight> units may be written in any APL, assuming <source lang=apl inline>⎕IO←0</syntaxhighlight>, as <source lang=apl inline>Y[(⍴Y)|X+⍳⍴Y]</syntaxhighlight>. To rotate an arbitrary array Squad indexing with axis (or rank) is helpful. <source lang=apl> RotateAxis ← { ⍝ R ← X (K RotateAxis) Y 

   ⎕IO←0
   0=≢⍴⍵: ⍵                    ⍝ Return a scalar unchanged
   l ← ⍺⍺ ⌷ ⍴⍵                 ⍝ Length of rotation axis
   (⊂l|⍺+⍳l) ⌷[⍺⍺] ⍵           ⍝ Rotate with indexing

} </syntaxhighlight>

External links

Lessons

Documentation


APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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