I am examining the speed of motion on curves and in the textbook i am reading , the example was showing that using a parametric equation for an ellipse will result in a regular motion (as expected) , but then the author used an altered ellipse parametric equation so that he alters the speed of motion to be 3 times faster after a certain point (alter point 'a')
so i graphed the functions and experimented with them , but i want to extract another lesson from the example , which is , how did he scale the speed after the alter point ? , i want to control the speed of each part of the curve , why and how did he choose the numbers 3 and 2 in the equation , and how did he choose the second range for 't'
i want to generalize and get an idea on how to control this , how do i split the equation into several ranges each with their altered equation and range so that each range has a different speed, in the example he scaled the speed by 3 times , how do i scale it by 2 ? or N ? and what ranges of 't' will i have then ?
i hope i made my question clear , if not please guide me to make it more clear , this is my first post
regular equation $$ P(t) = (~x(t) ~,~ y(t)~) = (~W~cos(t) ~,~ H~sin(t)~) ~~~~ \text{for $0 < t < 2\pi$} $$
altered equations $$ P'(t) = (~x(t) ~,~ y(t)~) = \left\{ \begin{array}{l l} (~W~cos(t) ~,~ H~sin(t)~) & \quad \text{for $0 < t < a$}\\ (~W~cos(3t - 2a) ~,~ H~ sin(3t - 2a)~) & \quad \text{for $a < t <\frac{2( \pi + a )}{3}$} \end{array} \right. $$