What can we learn from calculating and mapping eclipses?

People are often curious: how exactly are eclipses calculated, and how are those amazingly accurate eclipse maps produced?

And people are often surprised by how much they can learn from the answers. For instance, how looking at eclipse frequencies across the globe gives us an insight into how the amount of daylight per year varies with latitude. Or how trying to make detailed predictions of eclipse paths a few decades into the future teaches us about the unpredictability of Earth’s rotational speed. Or how the whole process is a wonderful lesson in breaking down a daunting challenge into a series of simple steps.