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Solve error: not a valid variable

J E
Posted 1 year ago

I have not used Mathematica in probably 10 years, so need some guidance. Specifically solving combinations of linear equations and circles. (Mohr circle analysis for engineers).

Anyway, hitting lots of roadblocks in syntax.

Here is an example: Solved for intercept of line y=mw+b with circle centered at xo and radius R. Coordinates are (x,w). You get two solutions:

x   =   -((b - (b + m xo)/(1 + m^2) +/- 
  Sqrt[(-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo -   m^4 xo^2)/(1 + m^2)^2])/m)

for

w   =  (b + m xo)/(1 + m^2) +/-
 Sqrt[(-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo -   m^4 xo^2)/(1 + m^2)^2]

I am not trying to solve for a special case of when I have a tangent. and the solution for this case as Sin[w], related to Mohr circles.

So for determinant:

(-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo -  m^4 xo^2) = 0

There one two many variables here, should be three: but I have b, m, R and xo R and xo should set the circle, and m the line slope which is tangent. So I am trying to solve for one of these, so I can plug back into eqs for (x,w). In particular, I need to get Sin[phi], where Tan[phi=m.

When I enter:

ClearAll
y = Sin[phiw]
xo = (sig1 + J)/(1 + y) 
R = y*xo
m = Sqrt[y/(1 - y^2)]
b = J*m 
Solve[-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo - m^4 xo^2 == 
0, b, Reals, 
Assumptions -> R > 0 && b > 0 && J > 0 && m > 0 && y > 0 && sig1 > 0]

I am getting an error for b=J*m as

Solve::ivar: J Sqrt[Sin[phiw]/(1-Sin[<<1>>]^2)] is not a valid variable.

in which case Solve just spits the entry back out at me.

I am sure this is some sort of beginner syntax issue.

Also, what is the command to clear everything. Clear All leaves variable definitions of something like that. I should have some beginning commands to always reset everything. Now I close and reopen Mathematica to ensure this.
POSTED BY: J E
15 Replies
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J E
J E
Posted 1 year ago

Still working this problem of intersection of line and circle. Trying to simplify. Here is initial calculation:

Code 1:

ClearAll["Global`*"]
ClearAll[x, w, b, m, R, xo, phiw, y, J, solns1]
{x, w, b, m, R, xo, phiw} \[Element] Reals

Solve[{x, w} \[Element] 
   InfiniteLine[{{0, b}, {-b/m, 0}}] && {x, w} \[Element] 
   Circle[{xo, 0}, R], {x, w}]
FullSimplify[%, 
 0 < phiw < Pi/2 && R >= 0 && m >= 0 && b >= 0 && xo >= 0]

% /. m -> Tan[phiw]
solns1 = 
 FullSimplify[%, Reals, 
  Assumptions -> 0 < phiw < Pi/2 && b >= 0 && J >= 0 && xo >= 0 ]
x /. solns1[[2]]

This gives two solution for x and w. The algebraic is easier to follow, but the trig gives a simpler solution as last step. After some manipulation, I can clean these up. For example, for one solution, if I do by hand, I get:

Code 2:

m = Tan[phiw]

w = (b + m (xo + Sqrt[(1 + m^2) R^2 - (b + m xo)^2]))/(
  1 + m^2) = (b + m xo + m Sqrt[(1 + m^2) R^2 - (b + m xo)^2])/(
   1 + m^2) = (b + m xo)/(Sqrt[(1 + m^2)] Sqrt[(1 + m^2)]) + (
    m Sqrt[(1 + m^2) R^2 - (b + m xo)^2])/(
    Sqrt[(1 + m^2)] Sqrt[(1 + m^2)])
= (b + m xo)/(Sqrt[(1 + m^2)] Sqrt[(1 + m^2)]) + (
   m Sqrt[(1 + m^2) R^2 - (b + m xo)^2])/(
   Sqrt[(1 + m^2)] Sqrt[(1 + m^2)]) = (b + m xo)/(
    Sqrt[(1 + m^2)] Sqrt[(1 + m^2)]) + (
    m Sqrt[R^2 - (b + m xo)^2/(1 + m^2)])/Sqrt[(1 + m^2)] = 
   Ro/Sqrt[(1 + m^2)] + (m Sqrt[R^2 - Ro^2])/Sqrt[(1 + m^2)]

w = Ro Cos[phiw] + Sqrt[R^2 - Ro^2] Sin[phiw]

(b + m xo)^2/(1 + m^2) = Ro^2

x = (-b m + xo + Sqrt[(1 + m^2) R^2 - (b + m xo)^2])/(
  1 + m^2) = (-b m + xo )/(1 + m^2) + Sqrt[
    R^2 - (b + m xo)^2/(1 + m^2)]/Sqrt[(1 + m^2)] = (-b m + xo )/(
    1 + m^2) + Sqrt[R^2 - Ro^2]/Sqrt[(1 + m^2)]

x = -b Cos[phiw] Sin[phiw]  +  xo Cos[phiw]^2  + 
  Sqrt[R^2 - Ro^2] Sin[phiw]

Note m is the slope of the line. And you get a little simpler after using Tan[phi]. Ro is the specific circle radius that gives one solution, or the tangent line.

First question, how do I do these manipulations in Mathematica. Doing by defeats the purpose.

Second, when I try and equate the initial solution set with my manipulated set, they are clearly equal. But I can't get == or === or in combination with Expand or Reduce, can't get a True. This was the code for this comparison next. What is the secret to showing these are equal?

Workbook is attached.

Code 3:

ClearAll["Global`*"]
ClearAll[x, w, b, m, R, xo, phiw, y, J, solns1, x1, x2, s1, s2]
{x, w, b, m, R, xo, phiw} \[Element] Reals

m = Tan[phiw]
x1 = (-b m + xo + Sqrt[(1 + m^2) R^2 - (b + m xo)^2])/(1 + m^2)
x2 = -b Cos[phiw] Sin[phiw]  +  xo Cos[phiw]^2  + 
  Sqrt[R^2 - (b + m xo)^2/(1 + m^2)] Sin[phiw]
x1 == x2
s1 = FullSimplify[x1, Reals, 
  Assumptions -> 0 < phiw < Pi/2 && b >= 0 && J >= 0 && xo >= 0 ]
s2 = FullSimplify[x2, Reals, 
  Assumptions -> 0 < phiw < Pi/2 && b >= 0 && J >= 0 && xo >= 0 ]

s1 == s2
Attachments:
Wall_Friction_Ca...nb
POSTED BY: J E
J E
J E
Posted 1 year ago

I have figured out how to pick solutions... sort of. Often, I will need to pick a positive solution, or more important, pick the maximum or minimum of x, for variable {x,w}.

Here is another question below. The intersection of the line and circle gives two solution, but when I impose the condition of the determinant (expression under square root) is zero, this becomes the tangent of the line and circle, with only one solution.

Up until the very last step, I have only one solution in {x,w}. If you look at the two solutions, there are actually identical. Why does Mathematic retain the redundant solution. How do I get rid of.

In[468]:= ClearAll["Global`*"]
ClearAll[x, w, b, m, R, xo, phiw, y, solns1]
{x, w, b, m, R, xo, phiw} \[Element] Reals

Solve[{x, w} \[Element] 
   InfiniteLine[{{0, b}, {-b/m, 0}}] && {x, w} \[Element] 
   Circle[{xo, 0}, R], {x, w}]
FullSimplify[%, 
 0 < phiw < Pi/2 && R >= 0 && m >= 0 && b >= 0 && xo >= 0]

% /. R -> (b + m xo)/Sqrt[1 + m^2]
% /. m -> Tan[phiw]

FullSimplify[%, Reals, 
 Assumptions -> 
  0 < phiw < Pi/2 && b >= 0 && xo >= 0 && 0 < Cos[phiw] < 1 && 
   0 < Sin[phiw] < 1]


Out[470]= (x | w | b | m | R | xo | phiw) \[Element] Reals

Out[471]= {{x -> 
   ConditionalExpression[-((
     b - (b + m xo)/(1 + m^2) - 
      Sqrt[(-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo - 
       m^4 xo^2)/(1 + m^2)^2])/m), Or[
And[R > (b^2 + xo^2)^Rational[1, 2], 
Element[
Alternatives[b, m], Reals], xo < 0], 
And[R > (b^2 + xo^2)^Rational[1, 2], xo > 0, 
Element[
Alternatives[b, m], Reals]], 
And[
Element[
Alternatives[b, m], Reals], xo < 0, 
Inequality[0, Less, R, Less, -xo], 
Inequality[-b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2], Less, m, 
       Less, -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2]]], 
And[
Element[
Alternatives[b, m], Reals], xo < 0, 
Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
      m < -b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2]], 
And[m > -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2], 
Element[
Alternatives[b, m], Reals], xo < 0, 
Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
And[m > -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2], xo > 0, 
Element[
Alternatives[b, m], Reals], 
Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
And[xo > 0, 
Element[
Alternatives[b, m], Reals], 
Inequality[-b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2], Less, m, 
       Less, -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2]], 
Inequality[0, Less, R, Less, xo]], 
And[xo > 0, 
Element[
Alternatives[b, m], Reals], 
      m < -b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2], 
Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]]]], 
  w -> ConditionalExpression[(b + m xo)/(1 + m^2) + 
     Sqrt[(-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo - 
      m^4 xo^2)/(1 + m^2)^2], Or[
And[R > (b^2 + xo^2)^Rational[1, 2], 
Element[
Alternatives[b, m], Reals], xo < 0], 
And[R > (b^2 + xo^2)^Rational[1, 2], xo > 0, 
Element[
Alternatives[b, m], Reals]], 
And[
Element[
Alternatives[b, m], Reals], xo < 0, 
Inequality[0, Less, R, Less, -xo], 
Inequality[-b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2], Less, m, 
       Less, -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2]]], 
And[
Element[
Alternatives[b, m], Reals], xo < 0, 
Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
      m < -b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2]], 
And[m > -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2], 
Element[
Alternatives[b, m], Reals], xo < 0, 
Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
And[m > -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2], xo > 0, 
Element[
Alternatives[b, m], Reals], 
Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
And[xo > 0, 
Element[
Alternatives[b, m], Reals], 
Inequality[-b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2], Less, m, 
       Less, -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2]], 
Inequality[0, Less, R, Less, xo]], 
And[xo > 0, 
Element[
Alternatives[b, m], Reals], 
      m < -b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2], 
Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]]]]}, {x -> 
   ConditionalExpression[-((
     b - (b + m xo)/(1 + m^2) + 
      Sqrt[(-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo - 
       m^4 xo^2)/(1 + m^2)^2])/m), Or[
And[R > (b^2 + xo^2)^Rational[1, 2], 
Element[
Alternatives[b, m], Reals], xo < 0], 
And[R > (b^2 + xo^2)^Rational[1, 2], xo > 0, 
Element[
Alternatives[b, m], Reals]], 
And[
Element[
Alternatives[b, m], Reals], xo < 0, 
Inequality[0, Less, R, Less, -xo], 
Inequality[-b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2], Less, m, 
       Less, -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2]]], 
And[
Element[
Alternatives[b, m], Reals], xo < 0, 
Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
      m < -b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2]], 
And[m > -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2], 
Element[
Alternatives[b, m], Reals], xo < 0, 
Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
And[m > -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2], xo > 0, 
Element[
Alternatives[b, m], Reals], 
Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
And[xo > 0, 
Element[
Alternatives[b, m], Reals], 
Inequality[-b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2], Less, m, 
       Less, -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2]], 
Inequality[0, Less, R, Less, xo]], 
And[xo > 0, 
Element[
Alternatives[b, m], Reals], 
      m < -b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2], 
Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]]]], 
  w -> ConditionalExpression[(b + m xo)/(1 + m^2) - 
     Sqrt[(-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo - 
      m^4 xo^2)/(1 + m^2)^2], Or[
And[R > (b^2 + xo^2)^Rational[1, 2], 
Element[
Alternatives[b, m], Reals], xo < 0], 
And[R > (b^2 + xo^2)^Rational[1, 2], xo > 0, 
Element[
Alternatives[b, m], Reals]], 
And[
Element[
Alternatives[b, m], Reals], xo < 0, 
Inequality[0, Less, R, Less, -xo], 
Inequality[-b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2], Less, m, 
       Less, -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2]]], 
And[
Element[
Alternatives[b, m], Reals], xo < 0, 
Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
      m < -b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2]], 
And[m > -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2], 
Element[
Alternatives[b, m], Reals], xo < 0, 
Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
And[m > -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2], xo > 0, 
Element[
Alternatives[b, m], Reals], 
Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
And[xo > 0, 
Element[
Alternatives[b, m], Reals], 
Inequality[-b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2], Less, m, 
       Less, -b xo/(-R^2 + 
        xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
        Rational[1, 2]], 
Inequality[0, Less, R, Less, xo]], 
And[xo > 0, 
Element[
Alternatives[b, m], Reals], 
      m < -b xo/(-R^2 + 
        xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
       Rational[1, 2], 
Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]]]]}}

Out[472]= {{x -> 
   ConditionalExpression[(-b m + xo + 
     Sqrt[(1 + m^2) R^2 - (b + m xo)^2])/(1 + m^2), And[xo > 0, 
Or[
And[
Inequality[0, Less, R, Less, xo], 
       m < (R^2 - xo^2)^(-1) (b xo - 
         R (b^2 - R^2 + xo^2)^Rational[1, 2]), (R - xo) (R + 
         xo) (-m R^2 + xo (b + m xo) + 
         R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0], 
And[
Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
Or[(R - xo) (R + xo) (m R^2 - xo (b + m xo) + 
          R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0, 
        m > (R^2 - xo^2)^(-1) (b xo + 
          R (b^2 - R^2 + xo^2)^Rational[1, 2])]], 
      R > (b^2 + xo^2)^Rational[1, 2]]]], 
  w -> ConditionalExpression[(
    b + m (xo + Sqrt[(1 + m^2) R^2 - (b + m xo)^2]))/(1 + m^2), And[
    xo > 0, 
Or[
And[
Inequality[0, Less, R, Less, xo], 
       m < (R^2 - xo^2)^(-1) (b xo - 
         R (b^2 - R^2 + xo^2)^Rational[1, 2]), (R - xo) (R + 
         xo) (-m R^2 + xo (b + m xo) + 
         R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0], 
And[
Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
Or[(R - xo) (R + xo) (m R^2 - xo (b + m xo) + 
          R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0, 
        m > (R^2 - xo^2)^(-1) (b xo + 
          R (b^2 - R^2 + xo^2)^Rational[1, 2])]], 
      R > (b^2 + xo^2)^Rational[1, 2]]]]}, {x -> 
   ConditionalExpression[(-b m + xo - 
     Sqrt[(1 + m^2) R^2 - (b + m xo)^2])/(1 + m^2), And[xo > 0, 
Or[
And[
Inequality[0, Less, R, Less, xo], 
       m < (R^2 - xo^2)^(-1) (b xo - 
         R (b^2 - R^2 + xo^2)^Rational[1, 2]), (R - xo) (R + 
         xo) (-m R^2 + xo (b + m xo) + 
         R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0], 
And[
Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
Or[(R - xo) (R + xo) (m R^2 - xo (b + m xo) + 
          R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0, 
        m > (R^2 - xo^2)^(-1) (b xo + 
          R (b^2 - R^2 + xo^2)^Rational[1, 2])]], 
      R > (b^2 + xo^2)^Rational[1, 2]]]], 
  w -> ConditionalExpression[(
    b + m (xo - Sqrt[(1 + m^2) R^2 - (b + m xo)^2]))/(1 + m^2), And[
    xo > 0, 
Or[
And[
Inequality[0, Less, R, Less, xo], 
       m < (R^2 - xo^2)^(-1) (b xo - 
         R (b^2 - R^2 + xo^2)^Rational[1, 2]), (R - xo) (R + 
         xo) (-m R^2 + xo (b + m xo) + 
         R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0], 
And[
Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
Or[(R - xo) (R + xo) (m R^2 - xo (b + m xo) + 
          R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0, 
        m > (R^2 - xo^2)^(-1) (b xo + 
          R (b^2 - R^2 + xo^2)^Rational[1, 2])]], 
      R > (b^2 + xo^2)^Rational[1, 2]]]]}}

Out[473]= {{x -> 
   ConditionalExpression[(-b m + xo)/(1 + m^2), And[xo > 0, 
Or[
And[
Inequality[0, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), Less, xo], 
       m < (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
         b xo - (1 + m^2)^Rational[-1, 2] (b + 
          m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
          Rational[1, 2]), (-
          xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
         xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (xo (b + m xo) - 
         m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (b + 
           m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
           Rational[1, 2]) < 0], 
And[
Inequality[
       xo, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), 
        Less, (b^2 + xo^2)^Rational[1, 2]], 
Or[(-xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
          xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (-xo (b + m xo) + 
          m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (
            b + m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
            Rational[1, 2]) < 0, 
        m > (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
          b xo + (1 + m^2)^Rational[-1, 2] (b + 
            m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
            Rational[1, 2])]], (1 + m^2)^Rational[-1, 2] (b + 
        m xo) > (b^2 + xo^2)^Rational[1, 2]]]], 
  w -> ConditionalExpression[(b + m xo)/(1 + m^2), And[xo > 0, 
Or[
And[
Inequality[0, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), Less, xo], 
       m < (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
         b xo - (1 + m^2)^Rational[-1, 2] (b + 
          m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
          Rational[1, 2]), (-
          xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
         xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (xo (b + m xo) - 
         m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (b + 
           m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
           Rational[1, 2]) < 0], 
And[
Inequality[
       xo, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), 
        Less, (b^2 + xo^2)^Rational[1, 2]], 
Or[(-xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
          xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (-xo (b + m xo) + 
          m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (
            b + m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
            Rational[1, 2]) < 0, 
        m > (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
          b xo + (1 + m^2)^Rational[-1, 2] (b + 
            m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
            Rational[1, 2])]], (1 + m^2)^Rational[-1, 2] (b + 
        m xo) > (b^2 + xo^2)^Rational[1, 2]]]]}, {x -> 
   ConditionalExpression[(-b m + xo)/(1 + m^2), And[xo > 0, 
Or[
And[
Inequality[0, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), Less, xo], 
       m < (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
         b xo - (1 + m^2)^Rational[-1, 2] (b + 
          m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
          Rational[1, 2]), (-
          xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
         xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (xo (b + m xo) - 
         m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (b + 
           m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
           Rational[1, 2]) < 0], 
And[
Inequality[
       xo, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), 
        Less, (b^2 + xo^2)^Rational[1, 2]], 
Or[(-xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
          xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (-xo (b + m xo) + 
          m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (
            b + m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
            Rational[1, 2]) < 0, 
        m > (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
          b xo + (1 + m^2)^Rational[-1, 2] (b + 
            m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
            Rational[1, 2])]], (1 + m^2)^Rational[-1, 2] (b + 
        m xo) > (b^2 + xo^2)^Rational[1, 2]]]], 
  w -> ConditionalExpression[(b + m xo)/(1 + m^2), And[xo > 0, 
Or[
And[
Inequality[0, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), Less, xo], 
       m < (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
         b xo - (1 + m^2)^Rational[-1, 2] (b + 
          m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
          Rational[1, 2]), (-
          xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
         xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (xo (b + m xo) - 
         m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (b + 
           m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
           Rational[1, 2]) < 0], 
And[
Inequality[
       xo, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), 
        Less, (b^2 + xo^2)^Rational[1, 2]], 
Or[(-xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
          xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (-xo (b + m xo) + 
          m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (
            b + m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
            Rational[1, 2]) < 0, 
        m > (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
          b xo + (1 + m^2)^Rational[-1, 2] (b + 
            m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
            Rational[1, 2])]], (1 + m^2)^Rational[-1, 2] (b + 
        m xo) > (b^2 + xo^2)^Rational[1, 2]]]]}}

Out[474]= {{x -> 
   ConditionalExpression[(xo - b Tan[phiw])/(1 + Tan[phiw]^2), And[
    xo > 0, 
Or[
And[
Inequality[
       0, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2], 
        Less, xo], 
       Tan[phiw] < (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
           Tan[phiw]^2))^(-1) (
         b xo - (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
          Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
           Tan[phiw]^2))^Rational[1, 2]), (-
          xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
         xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
         xo (b + xo Tan[phiw]) - Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
         Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
           Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
            Tan[phiw]^2))^Rational[1, 2]) < 0], 
And[
Inequality[
       xo, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2],
         Less, (b^2 + xo^2)^Rational[1, 2]], 
Or[(-xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
          xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
            Rational[-1, 2]) (-xo (b + xo Tan[phiw]) + 
          Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
           Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
            Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
             Tan[phiw]^2))^Rational[1, 2]) < 0, 
        Tan[phiw] > (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
            Tan[phiw]^2))^(-1) (
          b xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
            Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
             Tan[phiw]^2))^Rational[1, 2])]], (b + 
        xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2] > (b^2 + 
        xo^2)^Rational[1, 2]]]], 
  w -> ConditionalExpression[(b + xo Tan[phiw])/(1 + Tan[phiw]^2), 
    And[xo > 0, 
Or[
And[
Inequality[
       0, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2], 
        Less, xo], 
       Tan[phiw] < (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
           Tan[phiw]^2))^(-1) (
         b xo - (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
          Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
           Tan[phiw]^2))^Rational[1, 2]), (-
          xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
         xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
         xo (b + xo Tan[phiw]) - Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
         Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
           Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
            Tan[phiw]^2))^Rational[1, 2]) < 0], 
And[
Inequality[
       xo, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2],
         Less, (b^2 + xo^2)^Rational[1, 2]], 
Or[(-xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
          xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
            Rational[-1, 2]) (-xo (b + xo Tan[phiw]) + 
          Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
           Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
            Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
             Tan[phiw]^2))^Rational[1, 2]) < 0, 
        Tan[phiw] > (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
            Tan[phiw]^2))^(-1) (
          b xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
            Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
             Tan[phiw]^2))^Rational[1, 2])]], (b + 
        xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2] > (b^2 + 
        xo^2)^Rational[1, 2]]]]}, {x -> 
   ConditionalExpression[(xo - b Tan[phiw])/(1 + Tan[phiw]^2), And[
    xo > 0, 
Or[
And[
Inequality[
       0, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2], 
        Less, xo], 
       Tan[phiw] < (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
           Tan[phiw]^2))^(-1) (
         b xo - (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
          Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
           Tan[phiw]^2))^Rational[1, 2]), (-
          xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
         xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
         xo (b + xo Tan[phiw]) - Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
         Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
           Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
            Tan[phiw]^2))^Rational[1, 2]) < 0], 
And[
Inequality[
       xo, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2],
         Less, (b^2 + xo^2)^Rational[1, 2]], 
Or[(-xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
          xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
            Rational[-1, 2]) (-xo (b + xo Tan[phiw]) + 
          Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
           Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
            Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
             Tan[phiw]^2))^Rational[1, 2]) < 0, 
        Tan[phiw] > (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
            Tan[phiw]^2))^(-1) (
          b xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
            Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
             Tan[phiw]^2))^Rational[1, 2])]], (b + 
        xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2] > (b^2 + 
        xo^2)^Rational[1, 2]]]], 
  w -> ConditionalExpression[(b + xo Tan[phiw])/(1 + Tan[phiw]^2), 
    And[xo > 0, 
Or[
And[
Inequality[
       0, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2], 
        Less, xo], 
       Tan[phiw] < (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
           Tan[phiw]^2))^(-1) (
         b xo - (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
          Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
           Tan[phiw]^2))^Rational[1, 2]), (-
          xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
         xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
         xo (b + xo Tan[phiw]) - Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
         Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
           Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
            Tan[phiw]^2))^Rational[1, 2]) < 0], 
And[
Inequality[
       xo, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2],
         Less, (b^2 + xo^2)^Rational[1, 2]], 
Or[(-xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
          xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
            Rational[-1, 2]) (-xo (b + xo Tan[phiw]) + 
          Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
           Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
            Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
             Tan[phiw]^2))^Rational[1, 2]) < 0, 
        Tan[phiw] > (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
            Tan[phiw]^2))^(-1) (
          b xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
            Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
             Tan[phiw]^2))^Rational[1, 2])]], (b + 
        xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2] > (b^2 + 
        xo^2)^Rational[1, 2]]]]}}

Out[475]= {{x -> 
   ConditionalExpression[Cos[phiw] (xo Cos[phiw] - b Sin[phiw]), And[
    xo > 0, 
Or[
And[
Inequality[
       xo, Less, b Cos[phiw] + xo Sin[phiw], Less, (b^2 + xo^2)^
        Rational[1, 2]], 
Or[xo Cos[phiw] >= 
        b Sin[phiw], (b Cos[phiw] + xo (-1 + Sin[phiw])) (
          xo Cos[phiw] - b Sin[phiw]) (xo + b Cos[phiw] + 
          xo Sin[phiw]) (b + xo Tan[phiw]) > 0], 
Or[xo Cos[phiw] < 
        b Sin[phiw], (-xo + b Tan[phiw]) (b + xo Tan[phiw]) (b^2 - 
          xo^2 + 2 b xo Tan[phiw]) > 0]], 
      b Cos[phiw] + xo Sin[phiw] > (b^2 + xo^2)^Rational[1, 2]]]], 
  w -> ConditionalExpression[Cos[phiw] (b Cos[phiw] + xo Sin[phiw]), 
    And[xo > 0, 
Or[
And[
Inequality[
       xo, Less, b Cos[phiw] + xo Sin[phiw], Less, (b^2 + xo^2)^
        Rational[1, 2]], 
Or[xo Cos[phiw] >= 
        b Sin[phiw], (b Cos[phiw] + xo (-1 + Sin[phiw])) (
          xo Cos[phiw] - b Sin[phiw]) (xo + b Cos[phiw] + 
          xo Sin[phiw]) (b + xo Tan[phiw]) > 0], 
Or[xo Cos[phiw] < 
        b Sin[phiw], (-xo + b Tan[phiw]) (b + xo Tan[phiw]) (b^2 - 
          xo^2 + 2 b xo Tan[phiw]) > 0]], 
      b Cos[phiw] + xo Sin[phiw] > (b^2 + xo^2)^
       Rational[1, 2]]]]}, {x -> 
   ConditionalExpression[Cos[phiw] (xo Cos[phiw] - b Sin[phiw]), And[
    xo > 0, 
Or[
And[
Inequality[
       xo, Less, b Cos[phiw] + xo Sin[phiw], Less, (b^2 + xo^2)^
        Rational[1, 2]], 
Or[xo Cos[phiw] >= 
        b Sin[phiw], (b Cos[phiw] + xo (-1 + Sin[phiw])) (
          xo Cos[phiw] - b Sin[phiw]) (xo + b Cos[phiw] + 
          xo Sin[phiw]) (b + xo Tan[phiw]) > 0], 
Or[xo Cos[phiw] < 
        b Sin[phiw], (-xo + b Tan[phiw]) (b + xo Tan[phiw]) (b^2 - 
          xo^2 + 2 b xo Tan[phiw]) > 0]], 
      b Cos[phiw] + xo Sin[phiw] > (b^2 + xo^2)^Rational[1, 2]]]], 
  w -> ConditionalExpression[Cos[phiw] (b Cos[phiw] + xo Sin[phiw]), 
    And[xo > 0, 
Or[
And[
Inequality[
       xo, Less, b Cos[phiw] + xo Sin[phiw], Less, (b^2 + xo^2)^
        Rational[1, 2]], 
Or[xo Cos[phiw] >= 
        b Sin[phiw], (b Cos[phiw] + xo (-1 + Sin[phiw])) (
          xo Cos[phiw] - b Sin[phiw]) (xo + b Cos[phiw] + 
          xo Sin[phiw]) (b + xo Tan[phiw]) > 0], 
Or[xo Cos[phiw] < 
        b Sin[phiw], (-xo + b Tan[phiw]) (b + xo Tan[phiw]) (b^2 - 
          xo^2 + 2 b xo Tan[phiw]) > 0]], 
      b Cos[phiw] + xo Sin[phiw] > (b^2 + xo^2)^Rational[1, 2]]]]}}In[468]:= ClearAll["Global`*"]
      ClearAll[x, w, b, m, R, xo, phiw, y, solns1]
      {x, w, b, m, R, xo, phiw} \[Element] Reals

      Solve[{x, w} \[Element] 
         InfiniteLine[{{0, b}, {-b/m, 0}}] && {x, w} \[Element] 
         Circle[{xo, 0}, R], {x, w}]
      FullSimplify[%, 
       0 < phiw < Pi/2 && R >= 0 && m >= 0 && b >= 0 && xo >= 0]

      % /. R -> (b + m xo)/Sqrt[1 + m^2]
      % /. m -> Tan[phiw]

      FullSimplify[%, Reals, 
       Assumptions -> 
        0 < phiw < Pi/2 && b >= 0 && xo >= 0 && 0 < Cos[phiw] < 1 && 
         0 < Sin[phiw] < 1]


      Out[470]= (x | w | b | m | R | xo | phiw) \[Element] Reals

      Out[471]= {{x -> 
         ConditionalExpression[-((
           b - (b + m xo)/(1 + m^2) - 
            Sqrt[(-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo - 
             m^4 xo^2)/(1 + m^2)^2])/m), Or[
      And[R > (b^2 + xo^2)^Rational[1, 2], 
      Element[
      Alternatives[b, m], Reals], xo < 0], 
      And[R > (b^2 + xo^2)^Rational[1, 2], xo > 0, 
      Element[
      Alternatives[b, m], Reals]], 
      And[
      Element[
      Alternatives[b, m], Reals], xo < 0, 
      Inequality[0, Less, R, Less, -xo], 
      Inequality[-b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2], Less, m, 
             Less, -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2]]], 
      And[
      Element[
      Alternatives[b, m], Reals], xo < 0, 
      Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
            m < -b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2]], 
      And[m > -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2], 
      Element[
      Alternatives[b, m], Reals], xo < 0, 
      Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
      And[m > -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2], xo > 0, 
      Element[
      Alternatives[b, m], Reals], 
      Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
      And[xo > 0, 
      Element[
      Alternatives[b, m], Reals], 
      Inequality[-b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2], Less, m, 
             Less, -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2]], 
      Inequality[0, Less, R, Less, xo]], 
      And[xo > 0, 
      Element[
      Alternatives[b, m], Reals], 
            m < -b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2], 
      Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]]]], 
        w -> ConditionalExpression[(b + m xo)/(1 + m^2) + 
           Sqrt[(-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo - 
            m^4 xo^2)/(1 + m^2)^2], Or[
      And[R > (b^2 + xo^2)^Rational[1, 2], 
      Element[
      Alternatives[b, m], Reals], xo < 0], 
      And[R > (b^2 + xo^2)^Rational[1, 2], xo > 0, 
      Element[
      Alternatives[b, m], Reals]], 
      And[
      Element[
      Alternatives[b, m], Reals], xo < 0, 
      Inequality[0, Less, R, Less, -xo], 
      Inequality[-b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2], Less, m, 
             Less, -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2]]], 
      And[
      Element[
      Alternatives[b, m], Reals], xo < 0, 
      Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
            m < -b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2]], 
      And[m > -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2], 
      Element[
      Alternatives[b, m], Reals], xo < 0, 
      Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
      And[m > -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2], xo > 0, 
      Element[
      Alternatives[b, m], Reals], 
      Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
      And[xo > 0, 
      Element[
      Alternatives[b, m], Reals], 
      Inequality[-b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2], Less, m, 
             Less, -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2]], 
      Inequality[0, Less, R, Less, xo]], 
      And[xo > 0, 
      Element[
      Alternatives[b, m], Reals], 
            m < -b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2], 
      Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]]]]}, {x -> 
         ConditionalExpression[-((
           b - (b + m xo)/(1 + m^2) + 
            Sqrt[(-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo - 
             m^4 xo^2)/(1 + m^2)^2])/m), Or[
      And[R > (b^2 + xo^2)^Rational[1, 2], 
      Element[
      Alternatives[b, m], Reals], xo < 0], 
      And[R > (b^2 + xo^2)^Rational[1, 2], xo > 0, 
      Element[
      Alternatives[b, m], Reals]], 
      And[
      Element[
      Alternatives[b, m], Reals], xo < 0, 
      Inequality[0, Less, R, Less, -xo], 
      Inequality[-b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2], Less, m, 
             Less, -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2]]], 
      And[
      Element[
      Alternatives[b, m], Reals], xo < 0, 
      Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
            m < -b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2]], 
      And[m > -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2], 
      Element[
      Alternatives[b, m], Reals], xo < 0, 
      Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
      And[m > -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2], xo > 0, 
      Element[
      Alternatives[b, m], Reals], 
      Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
      And[xo > 0, 
      Element[
      Alternatives[b, m], Reals], 
      Inequality[-b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2], Less, m, 
             Less, -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2]], 
      Inequality[0, Less, R, Less, xo]], 
      And[xo > 0, 
      Element[
      Alternatives[b, m], Reals], 
            m < -b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2], 
      Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]]]], 
        w -> ConditionalExpression[(b + m xo)/(1 + m^2) - 
           Sqrt[(-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo - 
            m^4 xo^2)/(1 + m^2)^2], Or[
      And[R > (b^2 + xo^2)^Rational[1, 2], 
      Element[
      Alternatives[b, m], Reals], xo < 0], 
      And[R > (b^2 + xo^2)^Rational[1, 2], xo > 0, 
      Element[
      Alternatives[b, m], Reals]], 
      And[
      Element[
      Alternatives[b, m], Reals], xo < 0, 
      Inequality[0, Less, R, Less, -xo], 
      Inequality[-b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2], Less, m, 
             Less, -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2]]], 
      And[
      Element[
      Alternatives[b, m], Reals], xo < 0, 
      Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
            m < -b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2]], 
      And[m > -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2], 
      Element[
      Alternatives[b, m], Reals], xo < 0, 
      Inequality[-xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
      And[m > -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2], xo > 0, 
      Element[
      Alternatives[b, m], Reals], 
      Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]], 
      And[xo > 0, 
      Element[
      Alternatives[b, m], Reals], 
      Inequality[-b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2], Less, m, 
             Less, -b xo/(-R^2 + 
              xo^2) + ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
              Rational[1, 2]], 
      Inequality[0, Less, R, Less, xo]], 
      And[xo > 0, 
      Element[
      Alternatives[b, m], Reals], 
            m < -b xo/(-R^2 + 
              xo^2) - ((R^2 - xo^2)^(-2) (b^2 R^2 - R^4 + R^2 xo^2))^
             Rational[1, 2], 
      Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]]]]]}}

      Out[472]= {{x -> 
         ConditionalExpression[(-b m + xo + 
           Sqrt[(1 + m^2) R^2 - (b + m xo)^2])/(1 + m^2), And[xo > 0, 
      Or[
      And[
      Inequality[0, Less, R, Less, xo], 
             m < (R^2 - xo^2)^(-1) (b xo - 
               R (b^2 - R^2 + xo^2)^Rational[1, 2]), (R - xo) (R + 
               xo) (-m R^2 + xo (b + m xo) + 
               R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0], 
      And[
      Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
      Or[(R - xo) (R + xo) (m R^2 - xo (b + m xo) + 
                R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0, 
              m > (R^2 - xo^2)^(-1) (b xo + 
                R (b^2 - R^2 + xo^2)^Rational[1, 2])]], 
            R > (b^2 + xo^2)^Rational[1, 2]]]], 
        w -> ConditionalExpression[(
          b + m (xo + Sqrt[(1 + m^2) R^2 - (b + m xo)^2]))/(1 + m^2), And[
          xo > 0, 
      Or[
      And[
      Inequality[0, Less, R, Less, xo], 
             m < (R^2 - xo^2)^(-1) (b xo - 
               R (b^2 - R^2 + xo^2)^Rational[1, 2]), (R - xo) (R + 
               xo) (-m R^2 + xo (b + m xo) + 
               R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0], 
      And[
      Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
      Or[(R - xo) (R + xo) (m R^2 - xo (b + m xo) + 
                R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0, 
              m > (R^2 - xo^2)^(-1) (b xo + 
                R (b^2 - R^2 + xo^2)^Rational[1, 2])]], 
            R > (b^2 + xo^2)^Rational[1, 2]]]]}, {x -> 
         ConditionalExpression[(-b m + xo - 
           Sqrt[(1 + m^2) R^2 - (b + m xo)^2])/(1 + m^2), And[xo > 0, 
      Or[
      And[
      Inequality[0, Less, R, Less, xo], 
             m < (R^2 - xo^2)^(-1) (b xo - 
               R (b^2 - R^2 + xo^2)^Rational[1, 2]), (R - xo) (R + 
               xo) (-m R^2 + xo (b + m xo) + 
               R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0], 
      And[
      Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
      Or[(R - xo) (R + xo) (m R^2 - xo (b + m xo) + 
                R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0, 
              m > (R^2 - xo^2)^(-1) (b xo + 
                R (b^2 - R^2 + xo^2)^Rational[1, 2])]], 
            R > (b^2 + xo^2)^Rational[1, 2]]]], 
        w -> ConditionalExpression[(
          b + m (xo - Sqrt[(1 + m^2) R^2 - (b + m xo)^2]))/(1 + m^2), And[
          xo > 0, 
      Or[
      And[
      Inequality[0, Less, R, Less, xo], 
             m < (R^2 - xo^2)^(-1) (b xo - 
               R (b^2 - R^2 + xo^2)^Rational[1, 2]), (R - xo) (R + 
               xo) (-m R^2 + xo (b + m xo) + 
               R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0], 
      And[
      Inequality[xo, Less, R, Less, (b^2 + xo^2)^Rational[1, 2]], 
      Or[(R - xo) (R + xo) (m R^2 - xo (b + m xo) + 
                R (b^2 - R^2 + xo^2)^Rational[1, 2]) < 0, 
              m > (R^2 - xo^2)^(-1) (b xo + 
                R (b^2 - R^2 + xo^2)^Rational[1, 2])]], 
            R > (b^2 + xo^2)^Rational[1, 2]]]]}}

      Out[473]= {{x -> 
         ConditionalExpression[(-b m + xo)/(1 + m^2), And[xo > 0, 
      Or[
      And[
      Inequality[0, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), Less, xo], 
             m < (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
               b xo - (1 + m^2)^Rational[-1, 2] (b + 
                m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                Rational[1, 2]), (-
                xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
               xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (xo (b + m xo) - 
               m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (b + 
                 m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                 Rational[1, 2]) < 0], 
      And[
      Inequality[
             xo, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), 
              Less, (b^2 + xo^2)^Rational[1, 2]], 
      Or[(-xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
                xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (-xo (b + m xo) + 
                m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (
                  b + m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                  Rational[1, 2]) < 0, 
              m > (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
                b xo + (1 + m^2)^Rational[-1, 2] (b + 
                  m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                  Rational[1, 2])]], (1 + m^2)^Rational[-1, 2] (b + 
              m xo) > (b^2 + xo^2)^Rational[1, 2]]]], 
        w -> ConditionalExpression[(b + m xo)/(1 + m^2), And[xo > 0, 
      Or[
      And[
      Inequality[0, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), Less, xo], 
             m < (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
               b xo - (1 + m^2)^Rational[-1, 2] (b + 
                m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                Rational[1, 2]), (-
                xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
               xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (xo (b + m xo) - 
               m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (b + 
                 m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                 Rational[1, 2]) < 0], 
      And[
      Inequality[
             xo, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), 
              Less, (b^2 + xo^2)^Rational[1, 2]], 
      Or[(-xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
                xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (-xo (b + m xo) + 
                m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (
                  b + m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                  Rational[1, 2]) < 0, 
              m > (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
                b xo + (1 + m^2)^Rational[-1, 2] (b + 
                  m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                  Rational[1, 2])]], (1 + m^2)^Rational[-1, 2] (b + 
              m xo) > (b^2 + xo^2)^Rational[1, 2]]]]}, {x -> 
         ConditionalExpression[(-b m + xo)/(1 + m^2), And[xo > 0, 
      Or[
      And[
      Inequality[0, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), Less, xo], 
             m < (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
               b xo - (1 + m^2)^Rational[-1, 2] (b + 
                m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                Rational[1, 2]), (-
                xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
               xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (xo (b + m xo) - 
               m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (b + 
                 m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                 Rational[1, 2]) < 0], 
      And[
      Inequality[
             xo, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), 
              Less, (b^2 + xo^2)^Rational[1, 2]], 
      Or[(-xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
                xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (-xo (b + m xo) + 
                m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (
                  b + m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                  Rational[1, 2]) < 0, 
              m > (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
                b xo + (1 + m^2)^Rational[-1, 2] (b + 
                  m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                  Rational[1, 2])]], (1 + m^2)^Rational[-1, 2] (b + 
              m xo) > (b^2 + xo^2)^Rational[1, 2]]]], 
        w -> ConditionalExpression[(b + m xo)/(1 + m^2), And[xo > 0, 
      Or[
      And[
      Inequality[0, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), Less, xo], 
             m < (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
               b xo - (1 + m^2)^Rational[-1, 2] (b + 
                m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                Rational[1, 2]), (-
                xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
               xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (xo (b + m xo) - 
               m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (b + 
                 m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                 Rational[1, 2]) < 0], 
      And[
      Inequality[
             xo, Less, (1 + m^2)^Rational[-1, 2] (b + m xo), 
              Less, (b^2 + xo^2)^Rational[1, 2]], 
      Or[(-xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (
                xo + (1 + m^2)^Rational[-1, 2] (b + m xo)) (-xo (b + m xo) + 
                m (1 + m^2)^(-1) (b + m xo)^2 + (1 + m^2)^Rational[-1, 2] (
                  b + m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                  Rational[1, 2]) < 0, 
              m > (-xo^2 + (1 + m^2)^(-1) (b + m xo)^2)^(-1) (
                b xo + (1 + m^2)^Rational[-1, 2] (b + 
                  m xo) (b^2 + xo^2 - (1 + m^2)^(-1) (b + m xo)^2)^
                  Rational[1, 2])]], (1 + m^2)^Rational[-1, 2] (b + 
              m xo) > (b^2 + xo^2)^Rational[1, 2]]]]}}

      Out[474]= {{x -> 
         ConditionalExpression[(xo - b Tan[phiw])/(1 + Tan[phiw]^2), And[
          xo > 0, 
      Or[
      And[
      Inequality[
             0, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2], 
              Less, xo], 
             Tan[phiw] < (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
                 Tan[phiw]^2))^(-1) (
               b xo - (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                 Tan[phiw]^2))^Rational[1, 2]), (-
                xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
               xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
               xo (b + xo Tan[phiw]) - Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
               Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                 Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                  Tan[phiw]^2))^Rational[1, 2]) < 0], 
      And[
      Inequality[
             xo, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2],
               Less, (b^2 + xo^2)^Rational[1, 2]], 
      Or[(-xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
                xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                  Rational[-1, 2]) (-xo (b + xo Tan[phiw]) + 
                Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
                 Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                  Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                   Tan[phiw]^2))^Rational[1, 2]) < 0, 
              Tan[phiw] > (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
                  Tan[phiw]^2))^(-1) (
                b xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                  Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                   Tan[phiw]^2))^Rational[1, 2])]], (b + 
              xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2] > (b^2 + 
              xo^2)^Rational[1, 2]]]], 
        w -> ConditionalExpression[(b + xo Tan[phiw])/(1 + Tan[phiw]^2), 
          And[xo > 0, 
      Or[
      And[
      Inequality[
             0, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2], 
              Less, xo], 
             Tan[phiw] < (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
                 Tan[phiw]^2))^(-1) (
               b xo - (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                 Tan[phiw]^2))^Rational[1, 2]), (-
                xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
               xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
               xo (b + xo Tan[phiw]) - Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
               Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                 Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                  Tan[phiw]^2))^Rational[1, 2]) < 0], 
      And[
      Inequality[
             xo, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2],
               Less, (b^2 + xo^2)^Rational[1, 2]], 
      Or[(-xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
                xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                  Rational[-1, 2]) (-xo (b + xo Tan[phiw]) + 
                Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
                 Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                  Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                   Tan[phiw]^2))^Rational[1, 2]) < 0, 
              Tan[phiw] > (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
                  Tan[phiw]^2))^(-1) (
                b xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                  Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                   Tan[phiw]^2))^Rational[1, 2])]], (b + 
              xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2] > (b^2 + 
              xo^2)^Rational[1, 2]]]]}, {x -> 
         ConditionalExpression[(xo - b Tan[phiw])/(1 + Tan[phiw]^2), And[
          xo > 0, 
      Or[
      And[
      Inequality[
             0, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2], 
              Less, xo], 
             Tan[phiw] < (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
                 Tan[phiw]^2))^(-1) (
               b xo - (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                 Tan[phiw]^2))^Rational[1, 2]), (-
                xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
               xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
               xo (b + xo Tan[phiw]) - Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
               Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                 Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                  Tan[phiw]^2))^Rational[1, 2]) < 0], 
      And[
      Inequality[
             xo, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2],
               Less, (b^2 + xo^2)^Rational[1, 2]], 
      Or[(-xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
                xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                  Rational[-1, 2]) (-xo (b + xo Tan[phiw]) + 
                Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
                 Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                  Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                   Tan[phiw]^2))^Rational[1, 2]) < 0, 
              Tan[phiw] > (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
                  Tan[phiw]^2))^(-1) (
                b xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                  Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                   Tan[phiw]^2))^Rational[1, 2])]], (b + 
              xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2] > (b^2 + 
              xo^2)^Rational[1, 2]]]], 
        w -> ConditionalExpression[(b + xo Tan[phiw])/(1 + Tan[phiw]^2), 
          And[xo > 0, 
      Or[
      And[
      Inequality[
             0, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2], 
              Less, xo], 
             Tan[phiw] < (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
                 Tan[phiw]^2))^(-1) (
               b xo - (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                 Tan[phiw]^2))^Rational[1, 2]), (-
                xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
               xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
               xo (b + xo Tan[phiw]) - Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
               Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                 Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                  Tan[phiw]^2))^Rational[1, 2]) < 0], 
      And[
      Inequality[
             xo, Less, (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2],
               Less, (b^2 + xo^2)^Rational[1, 2]], 
      Or[(-xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2]) (
                xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                  Rational[-1, 2]) (-xo (b + xo Tan[phiw]) + 
                Tan[phiw] (b + xo Tan[phiw])^2/(1 + 
                 Tan[phiw]^2) + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                  Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                   Tan[phiw]^2))^Rational[1, 2]) < 0, 
              Tan[phiw] > (-xo^2 + (b + xo Tan[phiw])^2/(1 + 
                  Tan[phiw]^2))^(-1) (
                b xo + (b + xo Tan[phiw]) (1 + Tan[phiw]^2)^
                  Rational[-1, 2] (b^2 + xo^2 - (b + xo Tan[phiw])^2/(1 + 
                   Tan[phiw]^2))^Rational[1, 2])]], (b + 
              xo Tan[phiw]) (1 + Tan[phiw]^2)^Rational[-1, 2] > (b^2 + 
              xo^2)^Rational[1, 2]]]]}}

      Out[475]= {{x -> 
         ConditionalExpression[Cos[phiw] (xo Cos[phiw] - b Sin[phiw]), And[
          xo > 0, 
      Or[
      And[
      Inequality[
             xo, Less, b Cos[phiw] + xo Sin[phiw], Less, (b^2 + xo^2)^
              Rational[1, 2]], 
      Or[xo Cos[phiw] >= 
              b Sin[phiw], (b Cos[phiw] + xo (-1 + Sin[phiw])) (
                xo Cos[phiw] - b Sin[phiw]) (xo + b Cos[phiw] + 
                xo Sin[phiw]) (b + xo Tan[phiw]) > 0], 
      Or[xo Cos[phiw] < 
              b Sin[phiw], (-xo + b Tan[phiw]) (b + xo Tan[phiw]) (b^2 - 
                xo^2 + 2 b xo Tan[phiw]) > 0]], 
            b Cos[phiw] + xo Sin[phiw] > (b^2 + xo^2)^Rational[1, 2]]]], 
        w -> ConditionalExpression[Cos[phiw] (b Cos[phiw] + xo Sin[phiw]), 
          And[xo > 0, 
      Or[
      And[
      Inequality[
             xo, Less, b Cos[phiw] + xo Sin[phiw], Less, (b^2 + xo^2)^
              Rational[1, 2]], 
      Or[xo Cos[phiw] >= 
              b Sin[phiw], (b Cos[phiw] + xo (-1 + Sin[phiw])) (
                xo Cos[phiw] - b Sin[phiw]) (xo + b Cos[phiw] + 
                xo Sin[phiw]) (b + xo Tan[phiw]) > 0], 
      Or[xo Cos[phiw] < 
              b Sin[phiw], (-xo + b Tan[phiw]) (b + xo Tan[phiw]) (b^2 - 
                xo^2 + 2 b xo Tan[phiw]) > 0]], 
            b Cos[phiw] + xo Sin[phiw] > (b^2 + xo^2)^
             Rational[1, 2]]]]}, {x -> 
         ConditionalExpression[Cos[phiw] (xo Cos[phiw] - b Sin[phiw]), And[
          xo > 0, 
      Or[
      And[
      Inequality[
             xo, Less, b Cos[phiw] + xo Sin[phiw], Less, (b^2 + xo^2)^
              Rational[1, 2]], 
      Or[xo Cos[phiw] >= 
              b Sin[phiw], (b Cos[phiw] + xo (-1 + Sin[phiw])) (
                xo Cos[phiw] - b Sin[phiw]) (xo + b Cos[phiw] + 
                xo Sin[phiw]) (b + xo Tan[phiw]) > 0], 
      Or[xo Cos[phiw] < 
              b Sin[phiw], (-xo + b Tan[phiw]) (b + xo Tan[phiw]) (b^2 - 
                xo^2 + 2 b xo Tan[phiw]) > 0]], 
            b Cos[phiw] + xo Sin[phiw] > (b^2 + xo^2)^Rational[1, 2]]]], 
        w -> ConditionalExpression[Cos[phiw] (b Cos[phiw] + xo Sin[phiw]), 
          And[xo > 0, 
      Or[
      And[
      Inequality[
             xo, Less, b Cos[phiw] + xo Sin[phiw], Less, (b^2 + xo^2)^
              Rational[1, 2]], 
      Or[xo Cos[phiw] >= 
              b Sin[phiw], (b Cos[phiw] + xo (-1 + Sin[phiw])) (
                xo Cos[phiw] - b Sin[phiw]) (xo + b Cos[phiw] + 
                xo Sin[phiw]) (b + xo Tan[phiw]) > 0], 
      Or[xo Cos[phiw] < 
              b Sin[phiw], (-xo + b Tan[phiw]) (b + xo Tan[phiw]) (b^2 - 
                xo^2 + 2 b xo Tan[phiw]) > 0]], 
            b Cos[phiw] + xo Sin[phiw] > (b^2 + xo^2)^Rational[1, 2]]]]}}
POSTED BY: J E
J E
J E
Posted 1 year ago

I have not seen a tutorial, in written or video format, that takes me through all the obvious steps in setting up simultaneous equations and simplification and manipulations of results. Is there one??? I little disappointed in Mathematica here. Lots of and lots of tutorials, but seems to be a whole lot less on this, which seems weird. Most are either over simplistic or focused on a random set of topics, and not core math and engineering calculations.
POSTED BY: J E

Have you seen http://reference.wolfram.com/language/tutorial/ManipulatingEquationsAndInequalities.html? If so, what's missing?

(I found it by going to the documentation page for Solve[] and clicking on one of the linked tutorials. About half the links are linked to some subsection of this tutorial.)
POSTED BY: Michael Rogers

You can use SolveValues:

SolveValues[y == m*x + b, y]
y = First[%]

or the original Solve, with a more complicated syntax:

Solve[y == m*x + b, y]
y = y /. First[%]
POSTED BY: Gianluca Gorni
J E
J E
Posted 1 year ago

thanks, that worked on full simply and can have && for multiple assumptions, yes?

How can I restrict all calculations and variables to real domain, or eg. positive reals and zero. Is that possible for whole workbook? and does it get passed down to future expressions.

Or at least the results of trig functions.

also trying to assign the result of a solution back to one of the variables, see last post
POSTED BY: J E

There is no mechanism to work automatically on real variables only in a whole notebook. You must explicitly declare which variables are real, for example with $Assumptions:

$Assumptions = Element[x | y, Reals] && z >= 0
POSTED BY: Gianluca Gorni
J E
J E
Posted 1 year ago

Here is another, clearly something simple:

Solve[y == mx + b, y]
y = %

How do I assign a solution to the original variable after a solution
POSTED BY: J E

You should check the syntax of FullSimplify:

Solve[-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo - m^4 xo^2 == 0, R]
% /. m -> Tan[phiw]
FullSimplify[%, 0 <= phiw < Pi/2]
POSTED BY: Gianluca Gorni
J E
J E
Posted 1 year ago

Making some small progress. Is there a tutorial that deals with detailed solving of simultaneous equations and with significant simplification. These involved combinations of squares and other powers, roots, and trig. I can clearly see patterns but can't get Mathematica to do its part. All is in real domain. For example:

ClearAll["Global`*"]
ClearAll[x, w, b, m, R, xo, phiw]
{x, w, b, m, R, xo, phiw} \[Element] Reals
$Assumptions = R >= 0 && m >= 0 && b >= 0 && xo >= 0
Solve[-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo - m^4 xo^2 == 0, R]
FullSimplify[%, Reals, 
 Assumptions -> R >= 0 && m >= 0 && b >= 0 && xo >= 0]
% /. m -> Tan[phiw]
FullSimplify[%, Reals, 
 Assumptions -> 
  phiw >= 0 && phiw <= Pi/2 && R >= 0 && m >= 0 && b >= 0 && xo >= 0]

I get {{R -> Cos[phiw] Sign[Sec[phiw]] (b + xo Tan[phiw])}}

Clearly Sign of the sec should be positive, as Sec will be between 1 and + Inf, when phiw runs from 0 to Pi/2.

Reduce, fractor etc... expand?? what functions do I need.`
POSTED BY: J E
Eric Rimbey
Eric Rimbey
Posted 1 year ago

ClearAll does not clear all... for example, if x=sin(b), it will remember that.

In your code snippets above, you have just the raw head ClearAll. You need to pass it an argument(s). So, in your x example:

ClearAll[x]
(* or *) 
ClearAll["x"]

If you want to clear everything in the Global namespace:

ClearAll["Global`*"]
POSTED BY: Eric Rimbey
J E
J E
Posted 1 year ago

Here are some general questions:

ClearAll does not clear all... for example, if x=sin(b), it will remember that. What expression do I use to clear everything so variables don't creep into other calculations down the line? Right now, I have to close and reopen Mathematica.

Second, I am dealing with Mohr's stress circles, so a great deal of trig pops up, and it gets messy quick. Is it better to use variables like x instead of Sin(theta) for x. Put another way, when solving simultaneous eqs. with trig involved, do I put the trig in last with substitutions, or last.

Is there a good resource for manipulating trig equations to get them reduced, esp with simultaneous equations? I am missing something here in terms of best approach.

For example, if I try to solve for the intersection of a line and circle, esp if the line and circle have some trig definitions for slope, and center and radius for circle, I get an absolute mess. But if I use:

Solve[{x, w} \[Element] 
InfiniteLine[{{0, b}, {-b/m, 0}}] && {x, w} \[Element] 
Circle[{xo, 0}, R], {x, w}]

I get the expected two solutions. But the above will have lots of hidden trig. For example, for y=mx+b, m being m=tan(theta), you will see 1+m^2 terms a lot, which is really just sec(theta).

Anywhere, I am missing best practice of how to approach this.

I will try a bit more today and post a notebook.

A bit more explanation.

These are the basic relationships: tau=m*sig + c and (sig - sigavg)^2 + (tau)^2 = sigdev^2 (Mohr circle)

In turn sigavg=1/(1+sin(phi)) * sig1 ---> average stress or center of circle sigdev=sin(phi)/(1+sin(phi)) * sig1 ---> deviatoric stress or radius of circle. m=Tan(phi-w), wall friction

There are two solution, none or one. And I imagine there are much simpler expressions, but have not been able to simplify.

The one solution or tangent is what I am trying to check. For the tangent, the wall friction line should coincide with the powder friction line under the right conditions.

I should get e.g. Sin(phi-w) = Sin(phi), if determinant is zero and other conditions. More later.
POSTED BY: J E

First, while Solve does accept complicated expressions as variables, it seems to rule out ones with certain heads, like Plus, Times, and Power. I surmise the reason is that these have internal simplification rules that are applied automatically. It seems difficult to disentangle b when b^2 m^2 becomes J^2 m^4 (or worse, since m has a complicated value). You could try solving for J instead of J*m and then recover b from J * m /. solution.

Second, the desired answer indicated in one of posts suggests you want to simplify or reduce, rather than solve for b. However, doing so indicates a problem in your setup. Maybe there's an error in one of the formulas or assumptions?

Simplify[-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo - m^4 xo^2 == 0, 
 Assumptions -> R > 0 && b > 0 && J > 0 && m > 0 && y > 0 && sig1 > 0]
(*  False  *)

Reduce[-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo - m^4 xo^2 == 0 && 
  R > 0 && b > 0 && J > 0 && m > 0 && y > 0 && sig1 > 0, Reals]
(*  False  *)

For reference, here are the definitions I copied:

y = Sin[phiw];
xo = (sig1 + J)/(1 + y);
R = y*xo;
m = Sqrt[y/(1 - y^2)];
b = J*m;
POSTED BY: Michael Rogers

To get Sin[phi], where Tan[phi]=m, I would do like this:

Sin[phi] /. Solve[Tan[phi] == m && -Pi/2 < phi < Pi/2, phi]

or, shortly,

Sin[ArcTan[m]]
POSTED BY: Gianluca Gorni
J E
J E
Posted 1 year ago

Above text not clear, not sure why post runs it all together. Repeated againenter code here:

   ClearAll
y = Sin[phiw]
xo = (sig1 + J)/(1 + y)
R = y*xo
m = Sqrt[y/(1 - y^2)]
b = J*m

Solve[-b^2 m^2 + m^2 R^2 + m^4 R^2 - 2 b m^3 xo - m^4 xo^2 == 
  0, b, Reals,
 Assumptions -> R > 0 && b > 0 && J > 0 && m > 0 && y > 0 && sig1 > 0]

The final answer should look something like :

   Sin[w/x] = -(((J + sig1) Sin[phiw])/(-sig1 + J Sin[phiw]))

btw m = tan[phiw], or tangent to the circle given by xo and R . Which are related as given above, with y = sin[phiw] .
POSTED BY: J E
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