How could Jesus be descended from the royal line of David if he was born of the Virgin Birth?

The first chapter of Matthew traces Jesus' lineage through his father Joseph back to King David through something like 18 generations. But as we know, Jesus was born through the agency of the Holy Spirit meaning that Joseph did *not* impregnate his mother, the Virgin Mary. How would Jesus have a bloodline back to King David? Was Mary a cousin (of some degree) to Joseph with similar bloodlines? If not, what doctrine would cause Jesus to be considered descended from King David? Or put another way, are there any sources that describe what bloodline Jesus actually had through Mary?