std::proj(std::complex)

template< class T > 
complex<T> proj( const complex<T>& z );

std::complex<long double> proj( long double z );

template< class DoubleOrIngeter >
std::complex<double> proj( DoubleOrInteger z );

std::complex<float> proj( float z );

Returns the projection of the complex number z onto the Riemann sphere.

For most z, std::proj(z)==z, but all complex infinities, even the numbers where one component is infinite and the other is NaN, become positive real infinity, (INFINITY, 0) or (INFINITY, -0). The sign of the imaginary (zero) component is the sign of std::imag(z).

Additional overloads are provided for float, double, long double, and all integer types, which are treated as complex numbers with zero imaginary component.

Parameters

Return value

the projection of z onto the Riemann sphere.

Notes

The proj function helps model the Riemann sphere by mapping all infinities to one (give or take the sign of the imaginary zero), and should be used just before any operation, especially comparisons, that might give spurious results for any of the other infinities.

Example

#include <iostream>
#include <complex>
 
int main()
{
    std::complex<double> c1(1, 2);
    std::cout << "proj" << c1 << " = " << std::proj(c1) << '\n';
 
    std::complex<double> c2(INFINITY, -1);
    std::cout << "proj" << c2 << " = " << std::proj(c2) << '\n';
 
    std::complex<double> c3(0, -INFINITY);
    std::cout << "proj" << c3 << " = " << std::proj(c3) << '\n';
}

proj(1,2) = (1,2)
proj(inf,-1) = (inf,-0)
proj(0,-inf) = (inf,-0)

See also