Is the following surface smooth and compact, when all its partial derivatives are continuous?
How to tell about self-intersections without visualization?
$x=\cos(a) + \cos(a + b) + \cos(a + b + c),$
$y= \sin(a) + \sin(a + b) + \cos(a + b + c)$
$z= \sin(a) + \cos(a + b) + \sin(a + b + c) $
inside the interval $ (a, -\pi, \pi), (b, -\pi, \pi)$
For Mathematica visualization:
Manipulate[ ParametricPlot3D[{Cos[a] + Cos[a + b] + Cos[a + b + c], Sin[a] + Sin[a + b] + Cos[a + b + c], Sin[a] + Cos[a + b] + Sin[a + b + c]}, {a, -Pi, Pi}, {b, -Pi, Pi}, PlotStyle -> {Yellow}, Axes -> None, Boxed -> False], {c, -Pi/2, Pi/2, Pi/24}]