Why did St. Martin give half of his cloak, and didn't give the whole?

There was a saint called St. Martin, from the city of Tours (France) He gave half of his cloak to a poor he saw, because he felt pity for him, it was cold, and the beggar was more or less in rags. I see the lesson in this story, but what I wonder is why he only gave half of his cloak, why not the whole coat? I wonder if it was a real act, or an allegory (sharing when you have very little), but giving the whole wouldn't have been even more generous? What is the (Catholic) Church's opinion about this story? Is this story known outside the Catholic Church?