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Bug introduced in 13.0 or earlier and fixed in 13.1.0

Let me look at a circle case:

sol = SolveValues[{x1, y1} ∈ Circle[] && {x2, y2} ∈ 
     Circle[{4, 5}, 2] && 
    GeometricTest[{Circle[], 
      InfiniteLine[{{x1, y1}, {x2, y2}}]}, {"Tangent", {x1, y1}}] && 
    GeometricTest[{Circle[{4, 5}, 2], 
      InfiniteLine[{{x1, y1}, {x2, y2}}]}, {"Tangent", {x2, 
       y2}}], {x1, y1, x2, y2}];
Graphics[{Circle[], Circle[{4, 5}, 2], Red, 
  Line /@ ArrayReshape[sol, {4, 2, 2}]}]

enter image description here

I can find the common tangent lines perfect. Then let change the circle into a ellipse:

sol = SolveValues[{x1, y1} ∈ 
     Circle[{0, 0}, {1, 2}] && {x2, y2} ∈ 
     Circle[{4, 5}, {2, 3}] && 
    GeometricTest[{Circle[{0, 0}, {1, 2}], 
      InfiniteLine[{{x1, y1}, {x2, y2}}]}, {"Tangent", {x1, y1}}] && 
    GeometricTest[{Circle[{4, 5}, {2, 3}], 
      InfiniteLine[{{x1, y1}, {x2, y2}}]}, {"Tangent", {x2, 
       y2}}], {x1, y1, x2, y2}];
Graphics[{Circle[{0, 0}, {1, 2}], Circle[{4, 5}, {2, 3}], Red, 
  InfiniteLine /@ ArrayReshape[sol, {4, 2, 2}]}]

enter image description here

Obviously what I found is no longer a tangent. Is this a bug about function GeometricTest? Please help to check..

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2 Answers 2

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19
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Yep, it's a simple mistake by WRI, that is, a bug.

GeometricTest[{Circle[{x0, y0}, {a, b}], 
  InfiniteLine[{{x1, y1}, {x2, y2}}]}, {"Tangent", {x1, y1}}]
(*
Exists[{C[1]},
     x0 + a*Cos[C[1]] == x1 && 
     y0 + b*Sin[C[1]] == y1 && 
     b (x0 - x1) (x1 - x2) + a (y0 - y1) (y1 - y2) == 0 && 
     a > 0 && b > 0 && 0 <= C[1] <= 2*Pi]
*)

The fourth line

b (x0 - x1) (x1 - x2) + a (y0 - y1) (y1 - y2) == 0

should be (note a^2 and b^2)

b^2 (x0 - x1) (x1 - x2) + a^2 (y0 - y1) (y1 - y2) == 0

There are many ways to fix it, probably. Here's one:

fix = Function[Null, 
   Hold[#] /. 
    Hold[
      GeometricTest[{Circle[p0_, {a_, b_}], 
        InfiniteLine[{p1_, p2_}]}, {"Tangent", 
        p_}]] :> (GeometricTest[{Circle[p0, {a, b}^2], 
         InfiniteLine[{p1, p2}]}, {"Tangent", p}] /. {c_Cos :> c/a, 
        s_Sin :> s/b}),
   HoldAll];
sol = SolveValues[{x1, y1} \[Element] 
     Circle[{0, 0}, {1, 2}] && {x2, y2} \[Element] 
     Circle[{4, 5}, {2, 3}] && 
    fix@GeometricTest[{Circle[{0, 0}, {1, 2}], 
       InfiniteLine[{{x1, y1}, {x2, y2}}]}, {"Tangent", {x1, y1}}] && 
    fix@GeometricTest[{Circle[{4, 5}, {2, 3}], 
       InfiniteLine[{{x1, y1}, {x2, y2}}]}, {"Tangent", {x2, y2}}],
   {x1, y1, x2, y2}];
Graphics[{Circle[{0, 0}, {1, 2}], Circle[{4, 5}, {2, 3}], Red, 
  InfiniteLine /@ ArrayReshape[sol, {4, 2, 2}],
  Blue, Point /@ ArrayReshape[sol, {4, 2, 2}]}]
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2
  • $\begingroup$ And GeometricTest doesn't work with ImplicitRegion at all. Do you think this is also a bug? $\endgroup$
    – yode
    Commented Jan 4, 2022 at 9:17
  • 3
    $\begingroup$ @yode Since it doesn’t evaluate at all in my single test, I would guess it is unimplemented on implicit regions. The Basic Geometric Objects in the guide Synthetic Geometry do not include any general regions. It’s really a question for WRI whether synthetic geometry will include analytic geometry at some point. Given the direction indicated by the geometric scenes, assertions, and conjectures, I doubt it is easy to make everything work well with arbitrary regions. If you need it, you should ask WRI. $\endgroup$
    – Michael E2
    Commented Jan 4, 2022 at 17:56
1
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In the 13.1:

sol = SolveValues[{GeometricTest[{Circle[{0, 0}, {1, 2}], 
      InfiniteLine[{{x1, y1}, {x2, y2}}]}, {"Tangent", {x1, y1}}], 
    GeometricTest[{Circle[{4, 5}, {3, 2}], 
      InfiniteLine[{{x1, y1}, {x2, y2}}]}, {"Tangent", {x2, y2}}]}, {x1, y1, x2, y2}, Reals];
Graphics[{Circle[{0, 0}, {1, 2}], Circle[{4, 5}, {3, 2}], Red, 
  InfiniteLine /@ ArrayReshape[sol, {4, 2, 2}]}]

enter image description here

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4
  • 1
    $\begingroup$ That looks bad. Here is my screenshot from 13.1 wolfram cloud. i.imgur.com/TXneLRp.png $\endgroup$
    – Валерий Заподовников
    Commented Oct 13, 2022 at 19:33
  • $\begingroup$ @ВалерийЗаподовников That's...magic..... $\endgroup$
    – yode
    Commented Oct 14, 2022 at 12:53
  • $\begingroup$ Nope. Just bad sibpixel rendering in the windows app or whatever OS you have. $\endgroup$
    – Валерий Заподовников
    Commented Oct 15, 2022 at 13:08
  • $\begingroup$ @ВалерийЗаподовников I have just tested several platforms and confirmed that this is indeed a specific bug in Windows $\endgroup$
    – yode
    Commented Oct 16, 2022 at 4:21

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