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11 Nov 2019
For m epsilon [1, ], let Gamma m denote the equi-angular spiral which crosses the x-axis at Q1:=(1.0) and, one wrap later, at Qm:=(m,0), in cartesian coords, (when m=1, the "spiral" degenerates into a circle.) Let Pm - (alpha m, beta m) be the cartesian coordinate parametrization of Gamma m st. Pm(0) = Q1, Pm(2 pi ) = Qm and Pm(t) wraps once whenever t increases by 2pi. [After the second wrap, the spiral hits the x-axis at (m2,0).] Drawing Good Pictures, compute that Total-length of Gamma m going in to the origin from Q1, is Tm = [Hint: As , geometrically you expect Lm rightarrow ?? and Tm rightarrow ?? do they? (Think L'Hopital.) As , geometry tells you to expect and Lm to be asymptotic to ??. Are they?] Showing the interesting steps, compute from F() the arclength parametrization A(s) = (x(s),y(s)), of the spiral, satisfying that A(0) = F(0). Indeed, Create some interesting mathematical problem concerning these spirals. Elegantly solve the problem that you created, drawing nice pictures.
For m epsilon [1, ], let Gamma m denote the equi-angular spiral which crosses the x-axis at Q1:=(1.0) and, one wrap later, at Qm:=(m,0), in cartesian coords, (when m=1, the "spiral" degenerates into a circle.) Let Pm - (alpha m, beta m) be the cartesian coordinate parametrization of Gamma m st. Pm(0) = Q1, Pm(2 pi ) = Qm and Pm(t) wraps once whenever t increases by 2pi. [After the second wrap, the spiral hits the x-axis at (m2,0).] Drawing Good Pictures, compute that Total-length of Gamma m going in to the origin from Q1, is Tm = [Hint: As , geometrically you expect Lm rightarrow ?? and Tm rightarrow ?? do they? (Think L'Hopital.) As , geometry tells you to expect and Lm to be asymptotic to ??. Are they?] Showing the interesting steps, compute from F() the arclength parametrization A(s) = (x(s),y(s)), of the spiral, satisfying that A(0) = F(0). Indeed, Create some interesting mathematical problem concerning these spirals. Elegantly solve the problem that you created, drawing nice pictures.