Do you have a write up to go with the picture so I can understand
exactly what are the inputs to the problem and what you are trying to
solve for (I am less clear on the input parameters than what you are
solving for). There may be a way of using 2D conformal geometric
algebra to solve the problem.
On 8/6/22 12:14 PM, Oscar Benjamin wrote:
The equations you are attempting to solve lead to a very complicated
Groebner basis that is slow to compute (I'm not sure how long it
takes) and probably gives quite a complicated expression for the
solution.
It might be better if you can derive the equations without introducing
radicals i.e. to work with the quadrances rather than square rooting
them to get lengths. I'm not sure how you derived this but I imagine
the intention of rational geometry is to avoid things like square
roots.
On Sat, 6 Aug 2022 at 16:22, Kevin Pauba <klpa...@gmail.com> wrote:
That might be another approach but this represents an applied geometry problem
(mechanics). The segment DP (along with s) is an input measure that is easy to
determine/specify.
On Saturday, August 6, 2022 at 4:45:13 AM UTC-7 smi...@gmail.com wrote:
Kevin, I took a look at some of the rational geometry links -- it makes me
wonder if you might approach the relationship between Q1 and Q2 and the area of
the triangle extending past the line instead of simply the length DP.
/c
On Friday, August 5, 2022 at 9:11:39 PM UTC-5 klp...@gmail.com wrote:
Hi Oscar,
That very well may be (as you've noticed, I'm not well versed in math as I once
was). This is an applied geometry problem (see attached diagram) where the
triangle rotates about point A. I am solving for the relationships between q1
and q2, the perpendicular distance from P to the base of the triangle (segment
DP) and the angular measure s using rational trigonometry (see
https://stijnoomes.com/laws-of-rational-trigonometry/). I have a solution
using sin, cos but I'm interested in this alternative method.
equ1 in my working example solves for the segment DP given q1, q2 and s (and
angular measure). The problem presented here is to determine q1 and q2 given
two known segments and angular measures (q_dp1, s1) and (q_dp2, s2).
I hope this makes sense and that the information might help you help me.
Thanks for taking the time looking into it!
On Friday, August 5, 2022 at 4:50:09 PM UTC-7 Oscar wrote:
I just had a quick look and I think that maybe this has a positive
dimensional solution set.
On Fri, 5 Aug 2022 at 16:08, Kevin Pauba <klp...@gmail.com> wrote:
Here's the minimal working example (except for it hanging on solve()):
import sympy as sym
from sympy import sqrt
q1, q2, s, s1, s2, q_dp1, q_dp2 = sym.symbols('Q_1, Q_2, s, s_1, s_2, Q_dp_1,
Q_dp_2')
equ1 = q1*s - q2*s + 2*q2 - 2*sqrt(q2*(q1*s - q2*s + q2 + 2*sqrt(q1*q2*s*(1 -
s)))) + 2*sqrt(q1*q2*s*(1 - s))
eq1 = equ1.subs({ s: s1 }) - q_dp1
print(f"{eq1} = 0")
eq2 = equ1.subs({ s: s2 }) - q_dp2
print(f"{eq2} = 0")
soln = sym.solve([eq1, eq2], (q1, q2), simplify=False)
print(f"soln = {soln}")
On Friday, August 5, 2022 at 7:47:07 AM UTC-7 Oscar wrote:
You should be able to obtain a parametric Groebner basis to represent
the solutions of this system. Whether that leads to an explicit
solution in radicals is hard to say without trying.
I would demonstrate how to do this but the code for putting together
the equations is incomplete.
On Fri, 5 Aug 2022 at 14:01, Chris Smith <smi...@gmail.com> wrote:
If you remove the radicals (`sympy.solvers.solvers.unrad(eq1)`) and replace
`Q1` and `Q2` with `x` and `y` and `Q_dp_1` with a and `s1` with `b` you will
get an expression that is of degree 4 in every variable and can be split into a
term with `a` and `b` and a term with only `b` -- both with `x` and `y`.
u1 = a*(a**3 - 4*a**2*y*(2 - b) - 2*a*y**2*(-3*b**2 + 8*b - 8) - 4*b**3*x**3 +
2*b**2*(-x**2*(-3*a + 2*y*(b - 2)) + 2*y**3*(b - 2)) - 4*b*x*(a**2 + a*y*(b -
2) + y**2*(-b**2 - 8*b + 8))) + \
b**2*(b*(x**2 + y**2) + 2*x*y*(b - 2))**2
Replace a,b with c,d (for `q_dp_2` and `s2`) to get `u2`. I can't imagine that
solving a pair of quartics is going to give a nice solution. But solving this
system with known values for `a` and `b` would be straightforward with `nsolve`.
/c
On Thursday, August 4, 2022 at 11:54:24 PM UTC-5 klp...@gmail.com wrote:
And eq1=Q_1*s_1 - Q_2*s_1 + 2*Q_2 - Q_dp_1 - 2*sqrt(Q_2*(Q_1*s_1 - Q_2*s_1 +
Q_2 + 2*sqrt(Q_1*Q_2*s_1*(1 - s_1)))) + 2*sqrt(Q_1*Q_2*s_1*(1 - s_1)) ...
sorry, long day!
On Thursday, August 4, 2022 at 9:39:53 PM UTC-7 Kevin Pauba wrote:
Sorry, Jeremy. Good suggestion!
s1, s2, q_dp1, q_dp2 = sym.symbols('s_1, s_2, Q_dp_1, Q_dp_2')
eq1 = equ1.subs({ s: s1 }) - q_dp1
md( "$" + sym.latex(eq1) + " = 0$\n" )
eq2 = equ1.subs({ s: s2 }) - q_dp2
md( "$" + sym.latex(eq2) + " = 0$\n" )
soln = sym.solve([eq1, eq2], q1, q2)
print(f"soln = {soln}")
I'll set simplify to False and see how it goes ...
On Thursday, August 4, 2022 at 8:18:00 PM UTC-7 Kevin Pauba wrote:
I've attached a portion of a jupyter notebook. I'm attempting to solve a
simultaneous equation using sympy. The sym.solve() in the green input box
doesn't return (well, I waited over night on my macbook pro). Might the
solution be intractable? Is there another way to get a solution? Any help is
greatly appreciated.
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