VectorAngle returns complex angles

Question

The built-in VectorAngle function sometimes returns complex output when the actual angle is very close to π. The imaginary component is large enough and can't be removed by Chop with default tolerance:

VectorAngle[{0.034429687500000083, 0}, {-0.03442968749999986, 0}]
%//Chop

3.141592653589793` - 2.1073424338879928`*^-8 I
3.141592653589793- 2.1073424338879928*^-8 I

For comparison, custom implementation works as expected:

vectorAngle[v1_, v2_] := ArcCos[v1.v2/(Norm[v1] Norm[v2])]

vectorAngle[{0.034429687500000083, 0}, {-0.03442968749999986, 0}]

3.141592653589793`

From where comes imaginary part in the output? Is it a bug?

Returning complex output means that the modulus of the argument inside ArcCos is greater than 1. It may be because of precision of VectorAngle. — Anjan Kumar, Feb 18 '17 at 03:00
(VectorAngle[{0.034429687500000083, 0}, {-0.03442968749999986, 0}]/ Pi // RootApproximant)*Pi evaluates to Pi — Bob Hanlon, Feb 18 '17 at 04:48
Did you try the alternative in my answer in the linked dupe? — J. M.'s missing motivation, Mar 20 '17 at 05:46
@J.M. Yes, it works perfectly (at least in my tests). It disappoints that the built-in is so numerically unstable. — Alexey Popkov, Mar 20 '17 at 06:06

Bob Hanlon · Answer 1 · 2017-02-18T05:20:28.827

Use higher precision:

VectorAngle @@ SetPrecision[#, 20]& @ {
  {0.034429687500000083, 0}, {-0.03442968749999986, 0}}

(*  3.141592654  *)

EDIT: High precision isn't required, just setting a precision to avoid machine precision provides a real value:

Table[
 VectorAngle @@ SetPrecision[#, p] &@
  {{0.034429687500000083, 0},
   {-0.03442968749999986, 0}},
 {p, 5, 25, 5}]

(*  {3.1, 3.1416, 3.141593, 3.141592654, 3.14159265359}  *)

VectorAngle returns complex angles

1 Answers1