  We develop a new method for constructing K3 surfaces. We construct such a K3
surface $X$ by patching two open complex surfaces obtained as the complements
of tubular neighborhoods of elliptic curves embedded in blow-ups of the
projective planes at general nine points. Our construction has $19$ complex
dimensional degrees of freedom. For general parameters, the K3 surface $X$ is
neither Kummer nor projective. By the argument based on the concrete
computation of the period map, we also investigate which points in the period
domain correspond to K3 surfaces obtained by such construction.
