[Python-ideas] Trigonometry in degrees

Tue Jun 12 08:27:36 EDT 2018

Sym: SymPy, SymEngine, PySym, SymCXX, Diofant

(re: \pi, symbolic computation and trigonometry
instead of surprisingly useful piecewise optimizations)

On Fri, Jun 8, 2018 at 10:09 PM Wes Turner <wes.turner at gmail.com> wrote:

> # Python, NumPy, SymPy, mpmath, sage trigonometric functions
> https://en.wikipedia.org/wiki/Trigonometric_functions
>
> ## Python math module
> https://docs.python.org/3/library/math.html#trigonometric-functions
> - degrees(radians): Float degrees
> - radians(degrees): Float degrees
>
> ## NumPy
>
> https://docs.scipy.org/doc/numpy/reference/routines.math.html#trigonometric-functions
> - degrees(radians) : List[float] degrees
> - rad2deg(radians): List[float] degrees
> - radians(degrees) : List[float] radians
> - deg2rad(degrees): List[float] radians
>
> https://docs.scipy.org/doc/numpy/reference/generated/numpy.sin.html
>
>
# Symbolic computation

>
> ## SymPy
>
> http://docs.sympy.org/latest/modules/functions/elementary.html#sympy-functions-elementary-trigonometric
>
> http://docs.sympy.org/latest/modules/functions/elementary.html#trionometric-functions
>
> - sympy.mpmath.degrees(radians): Float degrees
> - sympy.mpmath.radians(degrees): Float radians
>
> - https://stackoverflow.com/questions/31072815/cosd-and-sind-with-sympy
>   - cosd, sind
>   -
> https://stackoverflow.com/questions/31072815/cosd-and-sind-with-sympy#comment50176770_31072815
>
>
    > Let x, theta, phi, etc. be Symbols representing quantities in
> radians. Keep a list of these symbols: angles = [x, theta, phi]. Then, at
> the very end, use y.subs([(angle, angle*pi/180) for angle in angles]) to
> change the meaning of the symbols to degrees"
>

http://docs.sympy.org/latest/tutorial/simplification.html#trigonometric-simplification

https://github.com/sympy/sympy/blob/master/sympy/functions/elementary/trigonometric.py
https://github.com/sympy/sympy/blob/master/sympy/functions/elementary/tests/test_trigonometric.py#
https://github.com/sympy/sympy/blob/master/sympy/simplify/trigsimp.py
https://github.com/sympy/sympy/blob/master/sympy/simplify/tests/test_trigsimp.py
https://github.com/sympy/sympy/blob/master/sympy/integrals/trigonometry.py
https://github.com/sympy/sympy/blob/master/sympy/integrals/tests/test_trigonometry.py

https://github.com/sympy/sympy/blob/master/sympy/utilities/tests/test_wester.py
https://github.com/sympy/sympy/blob/master/sympy/utilities/tests/test_wester.py#L593
(I. Trigonometry)

## Sym
Src: https://github.com/bjodah/sym
PyPI: https://pypi.org/project/sym/
> sym provides a unified wrapper to some symbolic manipulation libraries in
Python. It

## SymEngine
- Src: https://github.com/symengine/symengine
- Src: https://github.com/symengine/symengine.py
- Docs: https://github.com/symengine/symengine/blob/master/doc/design.md
- SymEngine / SymPy compatibility tests:

https://github.com/symengine/symengine.py/blob/master/symengine/tests/test_sympy_compat.py

## Diofant
Src: https://github.com/diofant/diofant
https://diofant.readthedocs.io/en/latest/tutorial/intro.html
https://diofant.readthedocs.io/en/latest/tutorial/basics.html#substitution
https://diofant.readthedocs.io/en/latest/tutorial/simplification.html#trigonometric-functions

from diofant import symbols, pi
x,y,z,_pi = symbols('x y z _pi')
expr =  pi**x # TODO: see diofant/tests/test_wester.py#L511
expr.subs(x, 1e11)
print(operator.sub(
      expr.subs(pi, 3.14),
      expr.subs(pi, 3.14159265)))
assert expr.subs(pi, 3.14) != expr.subs(pi, 3.14159265)
print(expr.subs(pi, 3.14159).evalf(70))

- CAS capability tests:

https://github.com/diofant/diofant/blob/master/diofant/tests/test_wester.py

> """ Tests from Michael Wester's 1999 paper "Review of CAS mathematical
> capabilities".
> http://www.math.unm.edu/~wester/cas/book/Wester.pdf
> See also http://math.unm.edu/~wester/cas_review.html for detailed output
of
> each tested system.
"""

https://github.com/diofant/diofant/blob/79ae584e949a08/diofant/tests/test_wester.py#L511

# I. Trigonometry
> @pytest.mark.xfail
> def test_I1():
>     assert tan(7*pi/10) == -sqrt(1 + 2/sqrt(5))
> @pytest.mark.xfail
> def test_I2():
>     assert sqrt((1 + cos(6))/2) == -cos(3)
> def test_I3():
>     assert cos(n*pi) + sin((4*n - 1)*pi/2) == (-1)**n - 1
> def test_I4():
>     assert cos(pi*cos(n*pi)) + sin(pi/2*cos(n*pi)) == (-1)**n - 1
> @pytest.mark.xfail
> def test_I5():
>     assert sin((n**5/5 + n**4/2 + n**3/3 - n/30) * pi) == 0

diofant.sin.eval() has a number of interesting conditionals in there:
https://github.com/diofant/diofant/blob/master/diofant/functions/elementary/trigonometric.py#L200

The tests for diofant.functions.elementary.trigonometric likely have a
number of helpful tests for implementing methods dealing with pi and
trigonometric identities:
https://github.com/diofant/diofant/blob/master/diofant/functions/elementary/tests/test_trigonometric.py

https://github.com/diofant/diofant/blob/master/diofant/simplify/trigsimp.py
https://github.com/diofant/diofant/blob/master/diofant/simplify/tests/test_trigsimp.py
https://github.com/diofant/diofant/blob/master/diofant/integrals/tests/test_trigonometry.py
https://github.com/diofant/diofant/blob/master/diofant/functions/elementary/tests/test_trigonometric.py

## mpmath
> http://mpmath.org/doc/current/functions/trigonometric.html
> - sympy.mpmath.degrees(radians): Float degrees
> - sympy.mpmath.radians(degrees): Float radians
>
>
> ## Sage
>
> https://doc.sagemath.org/html/en/reference/functions/sage/functions/trig.html
>
>
>
> On Friday, June 8, 2018, Robert Vanden Eynde <
> robertvandeneynde at hotmail.com> wrote:
>
>> - Thanks for pointing out a language (Julia) that already had a name
>> convention. Interestingly they don't have a atan2d function. Choosing the
>> same convention as another language is a big plus.
>>
>> - Adding trig function using floats between 0 and 1 is nice, currently
>> one needs to do sin(tau * t) which is not so bad (from math import tau, tau
>> sounds like turn).
>>
>> - Julia has sinpi for sin(pi*x), one could have sintau(x) for sin(tau*x)
>> or sinturn(x).
>>
>> Grads are in the idea of turns but with more problems, as you guys said,
>> grads are used by noone, but turns are more useful. sin(tau * t) For The
>> Win.
>>
>> - Even though people mentionned 1/6 not being exact, so that advantage
>> over radians isn't that obvious ?
>>
>> from math import sin, tau
>> from fractions import Fraction
>> sin(Fraction(1,6) * tau)
>> sindeg(Fraction(1,6) * 360)
>>
>> These already work today by the way.
>>
>> - As you guys pointed out, using radians implies knowing a little bit
>> about floating point arithmetic and its limitations. Integer are more
>> simple and less error prone. Of course it's useful to know about floats but
>> in many case it's not necessary to learn about it right away, young
>> students just want their player in the game move in a straight line when
>> angle = 90.
>>
>> - sin(pi/2) == 1 but cos(pi/2) != 0 and sin(3*pi/2) != 1 so sin(pi/2) is
>> kind of an exception.
>>
>>
>>
>>
>> Le ven. 8 juin 2018 à 09:11, Steven D'Aprano <steve at pearwood.info> a
>> écrit :
>>
>>> On Fri, Jun 08, 2018 at 03:55:34PM +1000, Chris Angelico wrote:
>>> > On Fri, Jun 8, 2018 at 3:45 PM, Steven D'Aprano <steve at pearwood.info>
>>> wrote:
>>> > > Although personally I prefer the look of d as a prefix:
>>> > >
>>> > > dsin, dcos, dtan
>>> > >
>>> > > That's more obviously pronounced "d(egrees) sin" etc rather than
>>> "sined"
>>> > > "tanned" etc.
>>> >
>>> > Having it as a suffix does have one advantage. The math module would
>>> > need a hyperbolic sine function which accepts an argument in; and
>>> > then, like Charles Napier [1], Python would finally be able to say "I
>>> > have sindh".
>>>
>>> Ha ha, nice pun, but no, the hyperbolic trig functions never take
>>> arguments in degrees. Or radians for that matter. They are "hyperbolic
>>> angles", which some electrical engineering text books refer to as
>>> "hyperbolic radians", but all the maths text books I've seen don't call
>>> them anything other than a real number. (Or sometimes a complex number.)
>>>
>>> But for what it's worth, there is a correspondence of a sort between the
>>> hyperbolic angle and circular angles. The circular angle going between 0
>>> to 45° corresponds to the hyperbolic angle going from 0 to infinity.
>>>
>>> https://en.wikipedia.org/wiki/Hyperbolic_angle
>>>
>>> https://en.wikipedia.org/wiki/Hyperbolic_function
>>>
>>>
>>> > [1] Apocryphally, alas.
>>>
>>> Don't ruin a good story with facts ;-)
>>>
>>>
>>>
>>> --
>>> Steve
>>> _______________________________________________
>>> Python-ideas mailing list
>>> Python-ideas at python.org
>>> https://mail.python.org/mailman/listinfo/python-ideas
>>> Code of Conduct: http://python.org/psf/codeofconduct/
>>>
>>
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