FWIW, I've been taught that the Luke geneology is actually through Mary, not Joseph. The argument goes that if Joseph had been the blood father of Jesus, then Jesus would be barred from the throne according to the Matthew geneology. This is because it includes the name of (Jehoiachin) Jeconiah (v. 1:11). Apparently, there was a curse on this Royal line:
Jer. 22:30 : This is what the LORD says: "Record this man as if childless, a man who will not prosper in his lifetime, for none of his offspring will prosper, none will sit on the throne of David or rule anymore in Judah."
So, this would not be the best line to use because most people would assume Joseph was the blood father. Instead, He uses the Luke geneology, which traces back through Nathan, not Jeconiah. This is Mary's line. Now, I saw in your post the requirement that all blood lines have to be traced through the male. One argument is that there is an exception spelled out in the Bible. Here is an excerpt from an article I found on the subject: The Lineage Loophole
"However, many of the people that teach on the genealogies fail to realize or address a major problem associated with the genealogical listing found in Luke's gospel, the lineage of Mary. Once you have established that the line is indeed Mary's you must deal with a second difficulty. The rights of the line are not passed through the mother, only the father. Even though Mary, through her lineage, was of the Davidic bloodline, she should be excluded from being able to pass those rights of the bloodline because of being a female (Deut 21:16). So it is not enough to prove that Mary was an unblemished descendant of David, she had to be a male to transfer the rights. Therefore she would be disqualified to transfer the rights to her son Jesus, except for a little known exception to the rule."
"In Numbers 26 we are introduced to Zelophehad. Zelophehad, we are told, had no sons, only daughters. In Numbers 27, following the death of Zelophehad, the daughters of Zelophehad came before Moses and argued their plight. Because their father had died with no sons, all of their rights of inheritance were to be lost and they felt this was unfair. So Moses prayed to God and God gave Moses an exception to the rule. The Lord told Moses that the inheritance CAN flow through a female, IF they fulfill two requirements. There must be no male offspring in the family (Num 27:8) and if the female offspring should marry, they must marry within their own tribe (Num 36:6)."
Anyway, it's just another view. I thought you all might be interested. :)
I remember being told, when I was a Protestant, that the Luke geneology was through Mary as well, but all of the early church commentators (we're talking as early as Eusebius in the 200's) reject that notion. I suppose that the idea arose among Protestants that this was the explanation, since there was no traditional account to explain the two geneologies if you don't accept the early Christian commentaries.
Both geneologies are traced to Joseph, unless you don't want to take the account in the Bible literally. :-)
One geneology says that Joseph was the son of Jacob, and the other says that he was the son of Heli. The early commentaries are universal (St. Augustine is particularly firm on this point, FWIW) in saying that it happened in this wise:
Heli (the father in the Luke account) died childless, and Jacob his brother took his wife as required by the law, and raised up seed to him.
Thus Joseph was the son of Jacob by nature, and the son of Heli by law. In either case -- whether by law or nature -- he was the son (by adoption) of David.
Given the fact that the Church has no problem making a big deal over Mary, had it been a legitimate tradition that this was her geneology. Why wouldn't the Scriptures have simply said so, had this been the case?
The name of the Theotokos' parents -- Joachim and Anna -- is a pretty ancient tradition, furthermore.
Anyway, we can be sure about our account because we have Holy Tradition -- you can't, since it isn't in the Bible. :-)