Wand (flat curve)

Wand curve

The Rod is a flat transcendental curve defined by the equation (in the polar coordinate system):

$\rho ={\frac {a}{\sqrt {\varphi }}}$ ${\ displaystyle \ rho = {\ frac {a} {\ sqrt {\ varphi}}}}$ ${\ displaystyle \ rho = {\ frac {a} {\ sqrt {\ varphi}}}}$ ,

Where $\alpha$ ${\ displaystyle \ alpha}$ $\ alpha$ - some constant constant.

It is a special case of the Archimedean spiral. $\rho =\alpha \phi ^{1/n}$ ${\ displaystyle \ rho = \ alpha \ phi ^ {1 / n}}$ ${\ displaystyle \ rho = \ alpha \ phi ^ {1 / n}}$ at $n=-2$ ${\ displaystyle n = -2}$ $n = -2$ .

The calculation of the curvature of the spiral and the angle of inclination of the tangent are performed by the formulas:

$\kappa (\phi )=(8\phi ^{2}-2){\biggl (}{\frac {\phi }{1+4\phi ^{2}}}{\biggr )}^{3/2}$ ${\ displaystyle \ kappa (\ phi) = (8 \ phi ^ {2} -2) {\ biggl (} {\ frac {\ phi} {1 + 4 \ phi ^ {2}}} {\ biggr)} ^ {3/2}}$ ${\ displaystyle \ kappa (\ phi) = (8 \ phi ^ {2} -2) {\ biggl (} {\ frac {\ phi} {1 + 4 \ phi ^ {2}}} {\ biggr)} ^ {3/2}}$

$\tau (\phi )=\phi -\tan ^{-1}(2\phi )$ ${\ displaystyle \ tau (\ phi) = \ phi - \ tan ^ {- 1} (2 \ phi)}$ ${\ displaystyle \ tau (\ phi) = \ phi - \ tan ^ {- 1} (2 \ phi)}$ ^[one]

The curve tends from infinity (where it asymptotically approaches the horizontal axis) to the point $(0;0)$ ${\ displaystyle (0; 0)}$ $(0; 0)$ spinning around it in a spiral counterclockwise. The size of the spiral is determined by the coefficient $a$ ${\ displaystyle a}$ $a$ . It has one inflection point - $\textstyle \left({\frac {1}{2}};a{\sqrt {2}}\right)$ ${\ displaystyle \ textstyle \ left ({\ frac {1} {2}}; a {\ sqrt {2}} \ right)}$ ${\ displaystyle \ textstyle \ left ({\ frac {1} {2}}; a {\ sqrt {2}} \ right)}$ .

The curve refers to algebraic spirals .

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History

The curve was named by the ancient Romans with a rod and described by Roger Cots in a collection of works entitled Harmonic Measurements (Harmonia Mensurarum) (1722), published 6 years after his death.

Notes

↑ Weisstein, Eric W. Lituus (English) on Wolfram MathWorld .

Links

Interactive Curve Visualization with JSXGraph

Wand (flat curve)

Content

History

Notes

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see also

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