I’m not sure if this fully addresses your question, but I wanted to share
an idea about working with Sympy’s assumptions system when you need
something like a (0,1] range.
Sympy’s default assumptions (e.g., positive, real) can feel inflexible, and
there isn’t a direct way to specify an interval like (0,1]. However, one
possible workaround is to use a known function that naturally stays in the
interval (0,1). A classic choice is the logistic sigmoid function:
```python
x = Symbol('x', real=True)
sigmoid = 1 / (1 + exp(-x))
```
The sigmoid has a range of (0,1) for all real x. If you need something that
stays in (0,1) symbolically, you can substitute the sigmoid in place of
expressions that you want to treat as (0,1)-valued.
For instance, the derivative of the sigmoid function,
```python
dsigmoid = sigmoid.diff(x)
```
has a range of (0, 1/4]. If you want it to go up to 1 instead of 1/4, you
can normalize it:
```python
dsigmoid_normalized = dsigmoid / dsigmoid.subs(x, 0)
```
That way, `dsigmoid_normalized` will lie in (0,1].
Suppose you have:
```python
A, INC = symbols('A INC')
zxh = Matrix([[-sin(A)*sin(INC)], [cos(A)*sin(INC)], [0]])
```
If you only need a symbolic expression with a value in (0,1] rather than
specifically sin(INC), you can do a direct substitution with the normalized
sigmoid derivative:
```python
zxh_sub = zxh.subs(sin(INC), dsigmoid_normalized)
```
Now, if you also want to normalize the resulting vector or matrix:
```python
zxh_normalized = zxh_sub.normalized()
```
and then simplify:
```python
result = simplify(zxh_normalized)
print(result)
```
You might see absolute values in the denominator, but that often simplifies
if you add further assumptions (you may be missing the variable A can be
real)
Another area of interest is whether Sympy can automatically prove
statements like dsigmoid_normalized <= 1 by simplification.
If it is possible, then even with restricted supply of assumptions like
real, positive, we can possibly extend the system to handle more broader
set of assumptions like interval assumption.
On Wednesday, January 22, 2025 at 5:30:08 PM UTC+1 [email protected]
wrote:
> Dear Conrad,
>
>
>
> my email is [email protected]
>
>
>
> Take care!
>
>
>
> Peter
>
>
>
> *From:* 'Conrad Schiff' via sympy <[email protected]>
> *Sent:* Wednesday, January 22, 2025 11:28 AM
> *To:* [email protected]
> *Subject:* Re: [sympy] trigsimp, assumptions, and vector magnitudes
>
>
>
> Thank you, Peter, for both your interest and for pointing me to another
> place to ask the question. I certainly would like to see what you are
> doing with the mechanics module so perhaps you can also point me to your
> work.
>
>
>
> Conrad
> ------------------------------
>
> *From:* [email protected] <[email protected]> on behalf of
> [email protected] <[email protected]>
> *Sent:* Wednesday, January 22, 2025 2:22 AM
> *To:* [email protected] <[email protected]>
> *Subject:* RE: [sympy] trigsimp, assumptions, and vector magnitudes
>
>
>
> Dear Conrad,
>
>
>
> I have never seen this range restriction – but then again I use mostly a
> sub library of sympy, called sympy.physics.mechanics (to set up symbolic
> equations of motion of multibody systems), so I am no sympy expert at all!
>
>
>
> Maybe you could ask your question here, too:
>
> https://github.com/sympy/sympy
>
>
>
> Peter
>
>
>
>
>
> *From:* 'Conrad Schiff' via sympy <[email protected]>
> *Sent:* Wednesday, January 22, 2025 2:28 AM
> *To:* [email protected]
> *Subject:* Re: [sympy] trigsimp, assumptions, and vector magnitudes
>
>
>
> Thank you, Peter but there is not quite what I was looking for.
>
>
>
> Technically your result simply proves that the square root of the square
> of a real number x is the absolute value of x (\sqrt{x^2} = |x| for x \in
> reals). Or in your terms:
>
>
>
> test = |sin(INC)|/\sqrt{sin(INC)^2} for INC real
>
>
>
> is always, identically, equal to 1.
>
>
>
> For my case, sin(INC)/|sin(INC)| is what I am looking at (i.e., the
> numerator has no absolute value), and it is equal to 1 for my assumptions
> but how can sympy know that without knowing that I've restricted INC to the
> range [0,\pi]? If, for example, INC were in the range [\pi,2\pi] then
> sin(INC)/|sin(INC)| = -1.
>
>
>
> If there is a way to say something like:
>
>
>
> INC = sm.symbols(‘INC’, real=True,range=[0,pi])
>
>
>
> then I say that the result is proved. But I can't anything like this in
> the documentation (perhaps I simply don't know where to look).
>
>
>
> Conrad
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
> ------------------------------
>
> *From:* [email protected] <[email protected]> on behalf of
> [email protected] <[email protected]>
> *Sent:* Tuesday, January 21, 2025 9:39 AM
> *To:* [email protected] <[email protected]>
> *Subject:* RE: [sympy] trigsimp, assumptions, and vector magnitudes
>
>
>
> I guess sympy does not know that INC is real and sin(INC) > 0.
>
> I just tried:
>
> INC = sm.symbols(‘INC’, real=True)
>
> test = sympy.Abs(sympy.sin(INC)) / sympy.sqrt(sympy.sin**2(INC))
>
>
>
> test = 1 was the result, as expected
>
>
>
> *From:* 'Conrad Schiff' via sympy <[email protected]>
> *Sent:* Tuesday, January 21, 2025 2:57 PM
> *To:* [email protected]
> *Subject:* [sympy] trigsimp, assumptions, and vector magnitudes
>
>
>
> I am writing a textbook on orbital mechanics and I would like to use sympy
> for certain calculations (e.g., deriving expressions for certain vectors in
> terms of Keplerian orbital elements) so that the readers realize the power
> of sympy. I have a specific problem I am hoping this group could help me
> with as I am also fairly new to sympy. Here is a minimum working example:
>
>
>
> ********************************
>
> from sympy import *
>
>
>
> A, INC = symbols('A INC')
>
> zxh = Matrix([[-sin(A)*sin(INC)], [cos(A)*sin(INC)], [0]])
>
> ********************************
>
>
>
> At this point, I want to unitize 'zxh' so that the sin(INC) divides out.
> This is a legitimate step since the range of INC is restricted to 0 to \pi
> and the sin(INC) is always non-negative (I am ignoring the singularities on
> the boundaries of 0 and \pi). I tried the following but I get the errors
> below. Some help in coaxing sympy to do what I know can be done would be
> appreciated.
>
>
>
> ********************************
>
> mag2_zxh = trigsimp(zxh.dot(zxh))
>
> -> gives sin**2(INC)
>
>
>
> mag_zxh = sqrt(mag2_zxh)
>
> -> gives sqrt(sin(INC)**2)
>
>
>
> zxh/mag_zxh
>
> -> gives Matrix([[-sin(A)*sin(INC)/sqrt(sin(INC)**2)],
> [sin(INC)*cos(A)/sqrt(sin(INC)**2)], [0]])
>
>
>
> ********************************
>
> How can I coax sympy into recognizing that "sin(INC)/sqrt(sin(INC)**2) = 1"?
>
>
>
> Thanks for any help,
>
> Conrad Schiff, PhD
>
> Professor of Physics
>
> Capitol Technology University
>
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