## Quasi-Spherical Orbits – by Author Bob Chester

The Most Interesting Curves You've Never Heard Of by Robert G. Chester [caption id="attachment_13083" align="alignleft" width="340"] Author Robert G. Chester[/caption] Quick, what simple rotations simultaneously generate the circle, the parabola, and the intersection of a cylinder and a sphere? Can these rotations also subsume the hippopede of Eudoxus [1], the limaçon [2], Viviani’s curve [3], rhodonea [4], the lemniscate of Gerono [5], and Fuller’s “great circle railroad tracks of energy” [6]? Quasi-Spherical Orbits, or QSOs, are the dynamic three-dimensional curves that result when a point rotates simultaneously about two or more axes. These intriguing curves provide insights and yield results in mathematics and physics alike. Viviani's Curve [caption id="attachment_13087" align="alignleft" width="133"] Rotation a[/caption] A point rotates in the right hand direction around the z-axis. The orbit is a circle in…

## Approximate Calculus: Area Under a Curve

Is it possible to calculate the area under a curve with any degree of accuracy? If we have a strong mathematical background, we may say, "Oh, that's easy. It's a matter of calculus." But what if you didn't take calculus? In fact, what if you never even attended high school? Is it possible to achieve an answer? Is it possible to use reason to come up with the principles of calculus? The answer is, indeed it is. I met a man who did just that. He asked me how I would figure the area under a curve. But he did more. He wanted to show how me how he had succeeded in solving the matter himself on his job in construction work. I was so impressed I exclaimed, "You've just…