/// <summary>Computes unit version of this multivector given its norm <paramref name="N"/>.</summary>
/// <param name="N">The norm of the multivector.</param>
/// <returns>Normalized Multivector</returns>
/// <remarks>Throws exception when norm is 0.</remarks>
protected Multivector Unit(Multivector N)
{
// zero -> throw exception
if (N.IsZero()) throw new ArgumentException("Multivector.Unit_e(): zero norm");
// scalar: the unit scalar is 1
if (this.IsScalar()) return new Multivector(1.0);
// if no symbolic scalars, then we can actually compute:
if (!N.HasSymbolicScalars())
return Multivector.gp(this, 1.0 / N.BasisBlades[0].scale);
// otherwise: symbolically invert
else return Multivector.gp(this, RefGA.Symbolic.UnaryScalarOp.Inverse(N));
}
/// <summary>
/// Internal function. Computes the versor inverse given the reverse R and the scale of (R reverse(R))
/// Throws exception when non-invertible. Can handle symbolic scalars.
/// </summary>
/// <param name="R">Reverse of multivector</param>
/// <param name="s">scale of (R reverse(R))</param>
/// <returns>Inverse of Multivector reverse(R)</returns>
private static Multivector VersorInverseInternal(Multivector R, Multivector s)
{
if (s.IsZero()) throw new Exception("non-invertible multivector");
if (!s.HasSymbolicScalars())
{
// noni symbolic scalars: do true inverse
double ss = s.BasisBlades[0].scale;
return gp(R, 1.0 / ss);
}
else {
// do symbolic inverse:
return gp(R, RefGA.Symbolic.UnaryScalarOp.Inverse(s));
}
}
/// <summary>Computes the norm given the squared norm of this multivector.</summary>
/// <returns>Norm of 'this', given norm squared 'N2'</returns>
protected Multivector Norm(Multivector N2, bool positiveDefinite)
{
if (N2.IsZero()) return Multivector.ZERO;
if (!N2.HasSymbolicScalars())
{
double s = N2.BasisBlades[0].scale;
return new Multivector(Math.Sign(s) * Math.Sqrt(Math.Abs(s)));
}
else
{
// if metric is note positive definite, add a fabs inside the sqrt because otherwise we could do sqrt(negative value)
// TODO: maybe just also have a ABS because floating point noise can make a (small) positive or zero value negative . . .
if (!positiveDefinite) N2 = RefGA.Symbolic.UnaryScalarOp.Abs(N2);
return RefGA.Symbolic.UnaryScalarOp.Sqrt(N2); // NOTE: this is not the same result as the numerical version above!!!
}
}