Solve this equation for three points determining a circular arc

Question

Given three points A, B and C on a circular arc, I want to get a parametric equation for them, where A is the starting point, C the end point and B the intermediate point on the Arc.

I started with a simple circular arc in the XZ plane, parametric in t:

arcXZ = r {Cos[ω1 + t (ω2 - ω1)], 0, Sin[ω1 + t (ω2 - ω1)]};

Next I rotate and translate it into an arbitrary position

arc={cx, cy, cz} + RotationMatrix[ϕ, {0, 0, 1}].RotationMatrix[θ, {1, 0, 0}].arcXZ;

This gives me three equations, for A, B, and C respectively.

eq1 = arc /. t -> 0;
eq2 = arc /. t-> 1/2;
eq3 = arc /. t-> 1;

I'm trying to solve them using

Solve[
 eq1 == {a1, a2, a3} &&
  eq2 == {b1, b2, b3} &&
  eq3 == {c1, c2, c3},
 {cx, cy, cz, r, ϕ, θ, ω1, ω2}
 ]

But this runs now for over an hour without any result. How can I solve this?

I also tried adding the inequality r>0 and restricting the domain to Reals in trial runs with simpler equations but that seemed to increase the runtime by a lot.

Responding to Carl Woll's comment. For

cx = 0,
cy = 0,
cz = 0,
ϕ = π/4,
θ = π/4,
ω1 = π/3,
ω2 = 3 π/4,
r = 10,

and t={0,1/2,1}

I get Circumsphere does not exist:

Circumsphere[{{10 (1/(2 Sqrt[2]) + Sqrt[3]/4), 
   10 (1/(2 Sqrt[2]) - Sqrt[3]/4), 
   5 Sqrt[3/2]}, {10 (1/2 Cos[\[Pi]/24] - Sin[\[Pi]/24]/Sqrt[2]), 
   10 (-(1/2) Cos[\[Pi]/24] - Sin[\[Pi]/24]/Sqrt[2]), 
   5 Sqrt[2] Cos[\[Pi]/24]}, {10 (-(1/2) + 1/(2 Sqrt[2])), 
   10 (-(1/2) - 1/(2 Sqrt[2])), 5}}]

Answer 1

• Method-1

Clear["Global`*"];
{a, b, c} = RandomReal[{-5, 5}, {3, 3}];

normal = Cross[c - a, c - b] // Simplify;
center = {c1, c2, c3};
sol = NSolve[{(center - a) . (center - a) == (center - b) . (center - 
        b) == (center - c) . (center - c), (center - a) . normal == 
     0}, center];
center = center /. sol[[1]];
r = Sqrt[(center - a) . (center - a)] // Simplify;

{e1, e2} = Orthogonalize[{a - center, b - center}];
circle[t_] = center + r {Cos[t], Sin[t]} . {e1, e2};
{t1, t2, t3} = 
  Sort[NArgMin[{(circle[t] - #) . (circle[t] - #), 0 <= t <= 2 π},
       t] & /@ {a, b, c}];
Show[{ParametricPlot3D[circle[t], {t, t1, t2}, PlotStyle -> Red], 
  ParametricPlot3D[circle[t], {t, t2, t3}, PlotStyle -> Green], 
  ParametricPlot3D[circle[t], {t, t1, t3}, 
   PlotStyle -> Directive[AbsoluteThickness[8], Dashed]]}, 
 PlotRange -> All, ViewPoint -> normal, 
 ViewProjection -> "Orthographic"]

• Method-2

Clear["Global`*"];
{a, b, c} = RandomReal[{-5, 5}, {3, 3}];
plane[u_, v_] = a + u*(c - a) + v*(b - a);
sol = Block[{center = plane[u, v]}, 
    NSolve[{(center - a) . (center - a) == (center - b) . (center - 
          b) == (center - c) . (center - c)}, {u, v}, Reals]] // First;
center = plane[u, v] /. sol;
{e1, e2} = Orthogonalize[{a - center, b - center}];
x1 = Dot[a - center, e1];
y1 = Dot[a - center, e2];
x2 = Dot[b - center, e1];
y2 = Dot[b - center, e2];
x3 = Dot[c - center, e1];
y3 = Dot[c - center, e2];

{θ1, θ2, θ3} = {ArcTan[x1, y1], ArcTan[x2, y2], 
  ArcTan[x3, y3]}

r = Norm[a - center];
{ParametricPlot3D[
  center + r*{Cos[t], Sin[t]} . {e1, e2}, {t, θ1, θ2}, 
  PlotStyle -> Green], 
 ParametricPlot3D[
  center + r*{Cos[t], Sin[t]} . {e1, e2}, {t, θ2, θ3}, 
  PlotStyle -> Blue], 
 ParametricPlot3D[
  center + r*{Cos[t], Sin[t]} . {e1, e2}, {t, θ3, θ1}, 
  PlotStyle -> Red]}

Answer 2

EDIT: Here's a 3D implicit fit, parametric fit is left as an exercise to the reader:

With[{pts =
   {{1, 2, 3},
    {3, 4, 5},
    {5, 6, 8}}},
 RegionIntersection[
    Sphere[{x, y, z}, r],
    Hyperplane[{u, v, w}, {x, y, z} . {u, v, w}]] /.
   First[Solve[
     Append[
      Thread@Element[# - {x, y, z} & /@ pts,
        RegionIntersection[
         Sphere[{0, 0, 0}, r],
         Hyperplane[{u, v, w}, 0]]],
      Element[{u, v, w}, Sphere[]]],
     {x, y, z, r, u, v, w}, Reals]] //
  Show[
    Region[Style[Echo[#], Black]],
    Graphics3D[{Red, PointSize[Large], Point@pts}]] &]

(* BooleanRegion[#1&&#2&,{Sphere[{-(23/4),-(19/4),39/2},(3 Sqrt[323/2])/2],
    Hyperplane[{-(1/Sqrt[2]),1/Sqrt[2],0},1/Sqrt[2]]}] *)

This is an explicit instead of a parametric fit, but I'd guess it's a start:

With[{pts =
   {{1, 2},
    {3, 4},
    {4, 6}}},
 Graphics[
  {Echo[Circle[{x, y}, r] /.
     First[Solve[Thread[Element[pts, Circle[{x, y}, r]]], {x, y, r}]]],
   Red, PointSize[Large], Point[pts]}]]

(* Circle[{-(7/2),17/2},5 Sqrt[5/2]] *)

Here's a parametric solution on basis of the above:

With[{pts =
   {{1, 2},
    {3, 4},
    {4, 6}}},
 r {Sin[t], Cos[t]} + {x, y} /.
  First[Solve[Thread[Element[pts, Circle[{x, y}, r]]], {x, y, r}]]]

(* {-(7/2) + 5 Sqrt[5/2] Sin[t], 17/2 + 5 Sqrt[5/2] Cos[t]} *)

Answer 3

Here is an example how to get a circular arc in 3D, parametrized by the angle. We first choose some arbitrary points, assuming p1 is the start, p2 somewhere in between and p3 the end:

SeedRandom[1];
{p1, p3, p2} = RandomReal[{-1, 1}, {3, 3}];

Then we get the midpoints of the first and second segment and name the, not yet known, center of the circle: mid:

m1 = (p1 + p2)/2;
m2 = (p2 + p3)/2;
mid = {xm, ym, zm};

With this we can write the equations for the center and calculate the center:

cen = mid /. 
   ToRules[Reduce[{(m1 - mid) . (p1 - p2) == 
       0, (m2 - mid) . (p2 - p3) == 0, 
      mid == p1 + t1 (p1 - p2) + t2 (p1 - p3)}, {xm, y, zm, , t1, 
      t2}]];

We also need the angle between start and end. The angle can be positive or negative. :

ang = VectorAngle[(p1 - cen), (p3 - cen)];
If[(p2 - p1) . (p3 - p2) < 0, ang = ang - 2 Pi];

With this info we can define the parametric arc and draw it:

arc[phi_] := 
  RotationTransform[phi, Cross[p1 - cen, p3 - cen], cen][p1];

d = 3;
Show[{
  ParametricPlot3D[arc[phi], {phi, 0, ang}, PlotRange -> {-d, d}]
  , Graphics3D[{PointSize[0.02], Point[{p1, p2, p3}], Red, Point[cen],
     Green, Line[{{cen, p1}, {cen, p3}}]}]}
 ]