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Solve error: not a valid variable

J E
Posted 1 year ago
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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