Applet: A parametrized helicoid
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The function $\dlsp(\spfv,\spsv) = (\spfv\cos \spsv, \spfv\sin \spsv, \spsv)$ parametrizes a helicoid when $0 \le \spfv \le 1$ and $0 \le \spsv \le 2\pi$. You can drag the cyan and magenta points on the sliders to change the values of $\spfv$ and $\spsv$. Or, you can drag the blue point on the helix directly, which will then change $\spfv$ and $\spsv$ so that the blue point is at $\dlsp(\spfv,\spsv)$. If you leave $\spfv$ fixed and change only $\spsv$, then the blue point traces out a helix with radius given by $\spfv$. If you keep $\spsv$ constant and change only $\spfv$, the blue point traces out a straight line.
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Notation systems
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