Day: November 18, 2019

Posted on

201911181237 Example 1.1

Put the equations of the circular helix (3) in the form (8).

Background.

Eq. (3):

x=a\cos u, y=a\sin u, z=bu

is the parametric form of a circular helix.

Eq. (8):

x=x, y=F_2(x), z=F_3(x)

is another form of the equations of a curve.

Let x\stackrel{\mathrm{def}}{=} a\cos u,

or u=\displaystyle{\cos^{-1}\bigg( \frac{x}{a} \bigg)}.

\begin{aligned} x^2 & = a^2\cos^2 u \\ a^2 - x^2 & = a^2 - a^2 \cos^2 u = a^2\sin^2 u \\ \pm \sqrt{a^2-x^2} & = a\sin u \\ y & = \pm \sqrt{a^2-x^2} \end{aligned}

z=b\cos^{-1}\displaystyle{\bigg( \frac{x}{a} \bigg)}.

In sum, the circular helix is now expressed in the form

\begin{aligned} x & = a\cos u \\ y & = F_2(x)=\pm\sqrt{a^2-x^2} \\ z & =F_3(x)=b\cos^{-1}\bigg( \frac{x}{a} \bigg) \\ \end{aligned}

Remark. In this form the curve is really defined by the last two equations, or, if it be a plane curve in the xy-plane, its equation is in the customary form Eq. (9): y=f(x).

pg.3 , Chapter 1 Curves in Space