Principles of Linear Algebra With Mathematica® Rolling an Ellipse ...

Chapter 1Rolling an Ellipse Along aCurve1.1 The SetupIn Section 7.2 of Principles of Linear Algebra With Mathematica ® , we discussedthe topic of “rolling” a circle along a curve. Of course, the rolling wasin quotes due to the fact that we were really just sliding the circle along thecurve. The animation have the appearance of a rolling circle, but this was dueonly to the fact a circle is a perfectly symmetric object, and no matter whichway it is rotated, there is no apparent change in orientation. So what happensif we pick an object which is not as symmetric? Take, for instance, an ellipse.Clearly if an ellipse is rotated through most angles, it will be obvious to anobserver that the ellipse was indeed rotated. The only two angles for whichthis is not the case is θ = π and θ = 2π. The question then becomes: whatadditionally must be done to rotate an ellipse versus a circle?To determine some of the math required to rotate an ellipse, we will startwith a simple setup, that of rolling the ellipse along a horizontal line, similarto rolling a football, along a level surface, end over end. We define an ellipsewith center at (x c , y c ) parametrically asx(t) = x c + a cos(t)y(t) = y c + b sin(t)(1.1)where a and b are the length of the major/minor axes corresponding, dependentupon a > b or a < b. If we graph this ellipse, starting at t = 0, then initiallywe have the point (x c + a, y c ). The bottom most point on the ellipse occurswhen t = 3 2 π, yielding the point (x c, y c − b). So in order to roll an ellipse along1