How to interpret situations where Math.Acos() reports invalid input?

Question

Hey all. I'm computing the angle between two vectors, and sometimes Math.Acos() returns NaN when it's input is out of bounds (-1 > input && input > 1) for a cosine. What does that mean, exactly? Would someone be able to explain what's happening? Any help is appreciated!

Here's my method:

 public double AngleBetween(vector b)
    {
        var dotProd = this.Dot(b);
        var lenProd = this.Len*b.Len;
        var divOperation = dotProd/lenProd;

        //  http://msdn.microsoft.com/en-us/library/system.math.acos.aspx
        return Math.Acos(divOperation) * (180.0 / Math.PI);
    }

Here's my implementation of Dot and Len:

public double Dot(vector b)
    {
        // x's and y's are lattitudes and longitudes (respectively)
        return ( this.From.x*b.From.x + this.From.y*b.From.y);
    }

    public double Len{
        get
        {
             // geo is of type SqlGeography (MS SQL 2008 Spatial Type) with an SRID of 4326
             return geo.STLength().Value;
        }
    }

Answer 1

NaN means "not a number". Mathematically, you can't take the arccosine of a number that is outside the range [-1, 1] (or maybe you can but the result is complex -- I don't remember) so the result of trying to do that is not any number at all.

Answer 2

Since the Cos of an angle is always between -1 and +1 there is no way to compute the inverse function (Acos) of a value outside that range OR it means you passed NaN to the ACos function.

I suspect in this case it's the latter - one of your lengths is probably zero.

Answer 3

You have vectors for which divOperation turns out to be < -1 or > 1? Then I think you should check your implementations of Dot and Len.