Sending astronauts to the moon and returning them to Earth took thousands of people solving problems no had faced before.
One of those people was Stanley Schmidt, chief of the NASA Ames Dynamic Analysis Branch in California.
"My father had been assigned the problem of navigating to the Moon and, as he told it to me, it was a very difficult problem," said Greg Schmidt, Stanley's son and director of NASA's Solar System Exploration Research Virtual Institute at Ames. "They didn't have a mathematical solution to it. It involved taking a number of different sources of information and combining them in an optimal way to get the best estimate of where your spacecraft is at any time, how fast you're going and other variables, too."
The computers on board the Apollo 11 mission would need to allow for navigating the trajectory to the moon and back, the angle of reentry, and complicated maneuvers in between. But they weren't anything like computers today and lacked processing power.
When studying the upcoming lunar mission, the elder Schmidt thought of mathematician Rudolf Kalman.
"My dad invited Rudy Kalman to give a lecture at Ames, and when he did, Dad had an epiphany," Greg Schmidt said. "Kalman had written a paper about a theoretical 'linear' solution to estimating a vehicle's location and speed. The problem was that this was a fundamentally 'nonlinear' problem; that's like the difference in complexity between floating down a lazy river and going over a waterfall, where your motion becomes chaotic and unpredictable. My dad then developed the equations for how to solve this nonlinear problem -- a major extension of Kalman's work."
The same math, known as the Schmidt-Kalman filter, is used today to help improve the efficient of air traffic control.
NASA works with the Federal Aviation Administration to research the best ways of managing takeoff, cruising and landing of aircraft around the country.
For example, the Schmidt-Kalman filter can help determine the closest estimate of a plane's position by combining the expected flight path of the aircraft with real-time measurements, the agency said. The filter essentially eliminates any other data that isn't needed.
"In air traffic management, the job is to keep aircraft safe and separated," said Jeremy Coupe, an aerospace engineer in aviation systems at Ames. "If you have a very accurate idea of where every aircraft is, you can increase the number of flights in a given area. But if you don't have a good idea, you can't be sure how to safely pack more of them into the airspace."
Like many innovations that improved space explorations during the Apollo era, this filter still has an impact today.
"I'm immensely proud of what my father did," said Greg Schmidt. "Before he passed away, I remember being at the hospital talking with him about his work. He was barely even able to talk but recounted all the equations as clearly as if it were 50 years earlier. He was a truly amazing man."