Home page for accesible maths 3 Chapter 3 contents 3.1 Polar coordinates 3.3 Tangent vector to a curve

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3.2 Parametrizing curves

Let $t$ be a real parameter, which we think of as time, and let $(x(t),y(t))$ be points that describe the curve $C$ as $t$ varies; this is a parametric form of $C$ .

Example.

Rational parameters for the circle $x^{2}+y^{2}=1$ are

x={{1-t^{2}}\over{1+t^{2}}},\qquad y={{2t}\over{1+t^{2}}}\qquad(-\infty<t<% \infty).

Equivalently, one can let $t=\tan\frac{\theta}{2}$ and obtain $x=\cos\theta$ and $y=\sin\theta$ . For then $1+t^{2}=1+\tan^{2}\frac{\theta}{2}=\sec^{2}\frac{\theta}{2}$ and so, for example:

\frac{1-t^{2}}{1+t^{2}}=\,{\frac{1-\tan^{2}\frac{\theta}{2}}{\sec^{2}\frac{% \theta}{2}}}\,{=\cos^{2}\frac{\theta}{2}-\sin^{2}\frac{\theta}{2}}\,{=\cos% \theta.}