Sec1Sess18
T = theta = angle
For cycloid:
<math>\vec{OP} = \langle aT - a*sin(T), a - a*cos(T) \rangle </math>
What happens when the point measured on the cycloid reaches near the bottom?
take length unit = radius: a = 1
x(T) = T - sin(T)
y(T) = 1 - cos(T)
For small T
<math>sin(T) \approx T</math>
<math>cos(T) \approx 1</math>
Better Approximation?
Use Taylor approximation (similar to Taylor expansion seen before):
for t small,
<math>\mathrm{f(t) \approx f'(0) + t*f'(0)+(t^2)/2*f(0)+(t^3)/6*f'(0)}</math>
add higher level derivatives of f(x) for more accuracy.
Better approximation for small angles:
<math>sin(T) \approx T - T^3/6</math>
<math>cos(T) \approx 1 - T^2/2</math>
Even better approximation
<math>sin(T) \approx T - (T - T^3/6) \approx T^3/6</math>
<math>cos(T) \approx 1 - (1 - T/2) \approx T^2/2</math>
|X| << |y|
<math>\mathrm{y/x\ \approx \ T^2/T^3\ =\ 2/T\ ->inf\ (as\ it\ approaches\ infinity,\ an\ asymptotic\ convergence)}</math>
slope at origin is therefore infinite (inflection point as the left side of the plotted cycloid point falls and the right side raises)