用相切性定义正交性

hbghlyj

Two intersecting circles $H$ and $K$ are orthogonal if one of them (say $H$) belongs to a set of three mutually tangent circles that touch each other at three distinct points of $K$.
The following Manipulate shows the central circle is orthogonal to the black one by showing the other two circles (blue).

1. squareDistance[h_, k_] := Chop[(h - k) . (h - k)];
2. Manipulate[
3. Module[{rc, p, q, r, s, x, y, d, e, re},
4.   If[c . c < 1, c = 1.001 Normalize[c]];
5.   rc = Sqrt[c . c - 1];
6.   {p, q} = {x, y} /.
7.     Simplify[
8.      Quiet@NSolve[{x^2 + y^2 == 1,
9.         squareDistance[c, {x, y}] == rc^2}, {x, y}]];
10.   d = p + rd Normalize[p - c];
11.   {r, s} = {x, y} /.
12.     Simplify[
13.      Quiet@NSolve[{x^2 + y^2 == 1,
14.         squareDistance[d, {x, y}] == rd^2}, {x, y}]];
15.   {r} = Complement[{r, s}, {p, q},
16.     SameTest -> (Norm[#1 - #2] < 0.001 &)];
17.   re = Norm[q - r]/Norm[Normalize[r - d] - Normalize[q - c]];
18.   e = q + re Normalize[q - c];
19.   Graphics[{Circle[{0, 0}, 1], Circle[c, rc], Blue, Circle[d, rd],
20.     Circle[e, re], Red, Disk[#, 0.06] & /@ {p, q, r}},
21.    PlotRange -> {{-4, 4}, {-4, 4}}]],
22. {{c, {-2., 1}}, Locator},
23. {{rd, 1.5, "radius of blue circles"}, 0.01, 5,
24.   Appearance -> "Labeled"},
25. Style["Drag the locator.", 10], SaveDefinitions -> True
26. ]

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hbghlyj

点c可拖动.正常情况下是三圆相切.

如果圆c半径很小就不相切了???

hbghlyj

数值稳定性