How is a parabola related to a circle?

I know this seems like an incredibly general or even silly/random question, however, please allow me to explain why I'm asking this question.

Someone who, for example, didn't understand that trigonometric functions and circles are related, would be missing out on a whole genre of important possible insights about trigonometry and circles. I feel that I am missing a similar important understanding of a connection between parabolas and circles.

Here's why.

Note that if you plot a pendulum's position against time it would form a sine/cosine function. The force that sets a pendulum in motion is gravity. That same force causes a ball to fall in a parabolic path.

Circles -> sine functions -> pendulum -> gravity -> parabola.

Something else led me to this question, but this was my way of explaining it. I feel like I'm missing an insight that is deep, beautiful, and important.

• This question is likely to be closed as too broad. That said, both parabolas and circles are conic sections. You can check that out on wikipedia: en.wikipedia.org/wiki/Conic_section – Ethan Bolker Feb 23 '18 at 21:46
• Note that the derivation of the position as a function of time for a pendulum usually makes the "small angle approximation", namely that $\sin \theta \approx \theta$. So it isn't really an exact solution, just very close to exact. – Morgan Sherman Feb 23 '18 at 21:53
• The force of gravity causes a ball to fall in an elliptical path, which for short paths is approximately a parabola. – John Wayland Bales Feb 23 '18 at 21:55
• @MorganSherman Moreover double pendulum can exhibit chaotic behaviour which is completely unrelated to circles and parabolas youtube.com/watch?v=AwT0k09w-jw – user Feb 23 '18 at 22:00
• Unfortunately for me, I fear you might be right that this question will likely get closed as too broad. If anyone is interested, this is the actual question that I'm wracking my brain over right now that led me down this path. math.stackexchange.com/questions/2663930/… – Steven2163712 Feb 23 '18 at 22:37