>>2,3
CLOSED SIMPLE PLANE CURVE: BIJECTIVE MAP c: [a,b] -> R^2 SUCH THAT c[a] = c
[b]. BIJECTIVITY IMPLIES IT NEVER INSECTS AND dc(s)/ds IS NEVER ZERO, WHERE WE TAKE dc PARAMERTICISED WITH ARC LENGTH s (HENCE ds IS THE ARC LENGTH DIFFERENTIAL)
THE CURVATURE OF A CLOSED SIMPLE PLANE CURVE c PARAMERTICISED TO UNIT VELOCITY IS |c''(t)| (I.E. IT'S ACCELERATION)
PROVE THAT ALL SUCH CURVES HAVE /AT LEAST/ 4 POINTS WHERE THE CURVATURE IS ZERO (IN ANY PARAMETRISATION)