Lituus (mathematics)

The representations of the lituus spiral in polar coordinates {{math|(r, θ)}} is given by the equation

: $r\; =\; \backslash frac\{a\}\{\backslash sqrt\{\backslash theta\}\},$

where {{math|θ ≥ 0}} and {{math|k ≠ 0}}.

The lituus spiral with the polar coordinates {{math|r {{=}} {{sfrac|a|{{lower|0.2em|{{sqrt|θ}}}}}}}} can be converted to Cartesian coordinates like any other spiral with the relationships {{math|x {{=}} r cos θ}} and {{math|y {{=}} r sin θ}}. With this conversion we get the parametric representations of the curve:

: $\backslash begin\{align\}$

x &= \frac{a}{\sqrt{\theta}} \cos\theta, \\

y &= \frac{a}{\sqrt{\theta}} \sin\theta. \\

\end{align}

These equations can in turn be rearranged to an equation in {{mvar|x}} and {{mvar|y}}:

: $\backslash frac\{y\}\{x\}\; =\; \backslash tan\backslash left(\; \backslash frac\{a^2\}\{x^2\; +\; y^2\}\; \backslash right).$

{{hidden begin|title=Derivation of the equation in Cartesian coordinates|showhide=left}}

1. Divide $y$ by $x$:$\backslash frac\{y\}\{x\}\; =\; \backslash frac\{\backslash frac\{a\}\{\backslash sqrt\{\backslash theta\}\}\; \backslash sin\backslash theta\}\{\backslash frac\{a\}\{\backslash sqrt\{\backslash theta\}\}\; \backslash cos\backslash theta\}\; \backslash Rightarrow\; \backslash frac\{y\}\{x\}\; =\; \backslash tan\backslash theta.$
2. Solve the equation of the lituus spiral in polar coordinates: $r\; =\; \backslash frac\{a\}\{\backslash sqrt\{\backslash theta\}\}\; \backslash Leftrightarrow\; \backslash theta\; =\; \backslash frac\{a^2\}\{r^2\}.$
3. Substitute $\backslash theta\; =\; \backslash frac\{a^2\}\{r^2\}$: $\backslash frac\{y\}\{x\}\; =\; \backslash tan\backslash left(\; \backslash frac\{a^2\}\{r^2\}\; \backslash right).$
4. Substitute $r\; =\; \backslash sqrt\{x^2\; +\; y^2\}$: $\backslash frac\{y\}\{x\}\; =\; \backslash tan\backslash left(\; \backslash frac\{a^2\}\{\backslash left(\; \backslash sqrt\{x^2\; +\; y^2\}\; \backslash right)^2\}\; \backslash right)\; \backslash Rightarrow\; \backslash frac\{y\}\{x\}\; =\; \backslash tan\backslash left(\; \backslash frac\{a^2\}\{x^2\; +\; y^2\}\; \backslash right).$

{{hidden end}}

The curvature of the lituus spiral can be determined using the formula{{Mathworld |id=Lituus |access-date=2023-02-04}}

: $\backslash kappa\; =\; \backslash left(\; 8\; \backslash theta^2\; -\; 2\; \backslash right)\; \backslash left(\; \backslash frac\{\backslash theta\}\{1\; +\; 4\; \backslash theta^2\}\; \backslash right)^\backslash frac32.$

In general, the arc length of the lituus spiral cannot be expressed as a closed-form expression, but the arc length of the lituus spiral can be represented as a formula using the Gaussian hypergeometric function:

: $L\; =\; 2\; \backslash sqrt\{\backslash theta\}\; \backslash cdot\; \backslash operatorname\{\_2\; F\_1\}\backslash left(\; -\backslash frac\{1\}\{2\},\; -\backslash frac\{1\}\{4\};\; \backslash frac\{3\}\{4\};\; -\backslash frac\{1\}\{4\; \backslash theta^2\}\; \backslash right)\; -\; 2\; \backslash sqrt\{\backslash theta\_0\}\; \backslash cdot\; \backslash operatorname\{\_2\; F\_1\}\backslash left(\; -\backslash frac\{1\}\{2\},\; -\backslash frac\{1\}\{4\};\; \backslash frac\{3\}\{4\};\; -\backslash frac\{1\}\{4\; \backslash theta\_0^2\}\; \backslash right),$

where the arc length is measured from {{math|θ {{=}} θ₀}}.

The tangential angle of the lituus spiral can be determined using the formula

: $\backslash phi\; =\; \backslash theta\; -\; \backslash arctan\; 2\backslash theta.$

{{reflist}}

{{cc}}

• {{springer|title=Lituus|id=p/l059750}}.
• {{mathworld|title=Lituus|urlname=Lituus}}
• [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Lituus Interactive example using JSXGraph].
• {{MacTutor|class=Curves|id=Lituus|title=Lituus}}.
• https://hsm.stackexchange.com/a/3181 on the history of the lituus curve.

Category:Spirals

Category:Plane curves

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