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{{Built-in|Circular|○}}, or '''Circle Function''', is a [[dyadic]] [[scalar function]], which is essentially a collection of related monadic functions. Circular encompasses [[wikipedia:trigonometric functions|trigonometric functions]], [[wikipedia:hyperbolic functions|hyperbolic functions]], and a few relationships between them. On some implementations with [[complex number|complex numbers]], Circular also includes functions to decompose and assemble a complex number. It shares the [[glyph]] < | {{Built-in|Circular|○}}, or '''Circle Function''', is a [[dyadic]] [[scalar function]], which is essentially a collection of related monadic functions. Circular encompasses [[wikipedia:trigonometric functions|trigonometric functions]], [[wikipedia:hyperbolic functions|hyperbolic functions]], and a few relationships between them. On some implementations with [[complex number|complex numbers]], Circular also includes functions to decompose and assemble a complex number. It shares the [[glyph]] <syntaxhighlight lang=apl inline>○</syntaxhighlight> with the [[monadic]] arithmetic function [[Pi Times]]. | ||
== Design == | == Design == | ||
The Circular function, as found in today's APL implementations, was designed by [[Gene McDonnell]], based on three things:<ref>[[Gene McDonnell|McDonnell, Gene]]. [https://www.jsoftware.com/papers/eem/storyofo.htm The Story of < | The Circular function, as found in today's APL implementations, was designed by [[Gene McDonnell]], based on three things:<ref>[[Gene McDonnell|McDonnell, Gene]]. [https://www.jsoftware.com/papers/eem/storyofo.htm The Story of <syntaxhighlight lang=apl inline>○</syntaxhighlight>]. Recreational APL. [[APL Quote-Quad]], Volume 8, Number 2, 1977-12.</ref> | ||
* [[Ken Iverson]] stating that [[Semantic density|a concise programming language]] must group together related monadic functions as a dyadic function that takes a "controlling parameter" as left argument to select specific function. | * [[Ken Iverson]] stating that [[Semantic density|a concise programming language]] must group together related monadic functions as a dyadic function that takes a "controlling parameter" as left argument to select specific function. | ||
* The sine, tangent, hyperbolic sine, and hyperbolic tangent are [[wikipedia:odd function|odd functions]]. | * The sine, tangent, hyperbolic sine, and hyperbolic tangent are [[wikipedia:odd function|odd functions]]. | ||
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{| class="wikitable c" style="margin: 1em auto 1em auto" | {| class="wikitable c" style="margin: 1em auto 1em auto" | ||
! style="width:33%" | < | ! style="width:33%" | <syntaxhighlight lang=apl inline>(-X)○Y</syntaxhighlight> !! style="width:33%" | <syntaxhighlight lang=apl inline>X</syntaxhighlight> !! <syntaxhighlight lang=apl inline>X○Y</syntaxhighlight> | ||
|- | |- | ||
| → || 0 || <math>\sqrt{1-Y^2}</math> | | → || 0 || <math>\sqrt{1-Y^2}</math> | ||
| Line 34: | Line 34: | ||
The following shows the values of sine, cosine, and tangent at <math>0, \frac{\pi}{2}, \pi</math>. Note that zeros are not exact and <math>\tan{\frac{\pi}{2}}</math> does not signal [[DOMAIN ERROR]] because of limited precision of <math>\pi</math>. | The following shows the values of sine, cosine, and tangent at <math>0, \frac{\pi}{2}, \pi</math>. Note that zeros are not exact and <math>\tan{\frac{\pi}{2}}</math> does not signal [[DOMAIN ERROR]] because of limited precision of <math>\pi</math>. | ||
< | <syntaxhighlight lang=apl> | ||
⎕PP←5 | ⎕PP←5 | ||
'sin' 'cos' 'tan',1 2 3∘.○ ○0 0.5 1 | 'sin' 'cos' 'tan',1 2 3∘.○ ○0 0.5 1 | ||
| Line 40: | Line 40: | ||
cos 1 6.1232E¯17 ¯1 | cos 1 6.1232E¯17 ¯1 | ||
tan 0 1.6331E16 ¯1.2246E¯16 | tan 0 1.6331E16 ¯1.2246E¯16 | ||
</ | </syntaxhighlight>{{Works in|[[Dyalog APL]]}} | ||
< | <syntaxhighlight lang=apl inline>0○Y</syntaxhighlight> and <syntaxhighlight lang=apl inline>4○Y</syntaxhighlight>/<syntaxhighlight lang=apl inline>¯4○Y</syntaxhighlight> reflect the identities <math>\sin^2{Y} + \cos^2{Y} = 1</math> and <math>\cosh^2{Y} - \sinh^2{Y} = 1</math> respectively. In APL code, <syntaxhighlight lang=apl inline>0○Y</syntaxhighlight> is equivalent to <syntaxhighlight lang=apl inline>1○¯2○Y</syntaxhighlight> and <syntaxhighlight lang=apl inline>2○¯1○Y</syntaxhighlight>, <syntaxhighlight lang=apl inline>4○Y</syntaxhighlight> is equivalent to <syntaxhighlight lang=apl inline>6○¯5○Y</syntaxhighlight>, and <syntaxhighlight lang=apl inline>¯4○Y</syntaxhighlight> is equivalent to <syntaxhighlight lang=apl inline>5○¯6○Y</syntaxhighlight>. | ||
< | <syntaxhighlight lang=apl> | ||
0○¯0.9 ¯0.3 0 0.3 0.9 | 0○¯0.9 ¯0.3 0 0.3 0.9 | ||
0.43589 0.95394 1 0.95394 0.43589 | 0.43589 0.95394 1 0.95394 0.43589 | ||
| Line 61: | Line 61: | ||
5○¯6○1 3 10 | 5○¯6○1 3 10 | ||
0 2.8284 9.9499 | 0 2.8284 9.9499 | ||
</ | </syntaxhighlight> | ||
On implementations with complex number support, < | On implementations with complex number support, <syntaxhighlight lang=apl inline>¯4○Y</syntaxhighlight> is modified to <math>(Y+1)\sqrt{\frac{Y-1}{Y+1}}</math> so that it agrees with <syntaxhighlight lang=apl inline>5○¯6○Y</syntaxhighlight> for complex numbers. | ||
As a [[mnemonics|mnemonic]], the functions for even X are [[wikipedia:even function|even functions]], and those for odd X are [[wikipedia:odd function|odd functions]]. Also, < | As a [[mnemonics|mnemonic]], the functions for even X are [[wikipedia:even function|even functions]], and those for odd X are [[wikipedia:odd function|odd functions]]. Also, <syntaxhighlight lang=apl inline>X○Y</syntaxhighlight> and <syntaxhighlight lang=apl inline>(-X)○Y</syntaxhighlight> are inverses of each other. | ||
=== Functions added with complex number support === | === Functions added with complex number support === | ||
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{| class="wikitable c" style="margin: 1em auto 1em auto" | {| class="wikitable c" style="margin: 1em auto 1em auto" | ||
! style="width:33%" | < | ! style="width:33%" | <syntaxhighlight lang=apl inline>(-X)○Y</syntaxhighlight> !! style="width:33%" | <syntaxhighlight lang=apl inline>X</syntaxhighlight> !! <syntaxhighlight lang=apl inline>X○Y</syntaxhighlight> | ||
|- | |- | ||
| <math>-\sqrt{-1-Y^2}</math> || 8 || <math>\sqrt{-1-Y^2}</math> | | <math>-\sqrt{-1-Y^2}</math> || 8 || <math>\sqrt{-1-Y^2}</math> | ||
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The left arguments 9 and 11 extract the real and imaginary parts (the [[wikipedia:cartesian coordinate system|Cartesian coordinates]] on the complex plane), while 10 and 12 extract the magnitude and angle (the [[wikipedia:polar coordinate system|polar coordinates]]). The negative counterparts can be used to [[complex (function)|assemble a single complex number]] from the Cartesian or polar coordinates. | The left arguments 9 and 11 extract the real and imaginary parts (the [[wikipedia:cartesian coordinate system|Cartesian coordinates]] on the complex plane), while 10 and 12 extract the magnitude and angle (the [[wikipedia:polar coordinate system|polar coordinates]]). The negative counterparts can be used to [[complex (function)|assemble a single complex number]] from the Cartesian or polar coordinates. | ||
< | <syntaxhighlight lang=apl> | ||
⎕←mat←9 11∘.○1J2 3J¯4 ¯5J¯6 ⍝ first row is real, second row is imag | ⎕←mat←9 11∘.○1J2 3J¯4 ¯5J¯6 ⍝ first row is real, second row is imag | ||
1 3 ¯5 | 1 3 ¯5 | ||
| Line 99: | Line 99: | ||
¯10 ¯12×.○mat | ¯10 ¯12×.○mat | ||
1J2 3J¯4 ¯5J¯6 | 1J2 3J¯4 ¯5J¯6 | ||
</ | </syntaxhighlight>{{Works in|[[Dyalog APL]]}} | ||
[[J]] has separate built-in functions for these operations: [https://code.jsoftware.com/wiki/Vocabulary/plusdot '''Real/Imag'''] < | [[J]] has separate built-in functions for these operations: [https://code.jsoftware.com/wiki/Vocabulary/plusdot '''Real/Imag'''] <syntaxhighlight lang=j inline>+.</syntaxhighlight> (Cartesian decomposition), [https://code.jsoftware.com/wiki/Vocabulary/stardot '''Length/Angle'''] <syntaxhighlight lang=j inline>*.</syntaxhighlight> (polar decomposition), [https://code.jsoftware.com/wiki/Vocabulary/jdot#dyadic '''Complex'''] <syntaxhighlight lang=j inline>j.</syntaxhighlight> (Cartesian assembly), and [https://code.jsoftware.com/wiki/Vocabulary/rdot#dyadic '''Polar'''] <syntaxhighlight lang=j inline>r.</syntaxhighlight> (polar assembly). Note that, unlike the APL version above, <syntaxhighlight lang=j inline>+.</syntaxhighlight> and <syntaxhighlight lang=j inline>*.</syntaxhighlight> create a trailing axis of length 2. | ||
< | <syntaxhighlight lang=j> | ||
]mat =: +. 1j2 3j_4 _5j_6 | ]mat =: +. 1j2 3j_4 _5j_6 | ||
1 2 | 1 2 | ||
| Line 117: | Line 117: | ||
r./"1 mat | r./"1 mat | ||
1j2 3j_4 _5j_6 | 1j2 3j_4 _5j_6 | ||
</ | </syntaxhighlight>{{Works in|[[J]]}} | ||
== References == | == References == | ||