Cross curve

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The Cross curve is a plane curve which outlines resemble a cross symbol (sign).

Fourth-degree equation

The Cross curve built according to the equation of the 4th degree

The Cross curve (Cruciform curve) is given implicitly by the equation of the fourth degree:

$\frac{a^2}{x^2}+\frac{b^2}{y^2}=1,$

where $a$, $b$ are some constants.

The same curve in parametric form:

$\begin{array}{l}x(\alpha )=a\sec \alpha ,\\ y(\alpha )=b\csc \alpha ,\end{array}$

where $\alpha$ is a real parameter.

Lapshin's equation

The curves of crosses: Greek, St. Andrew's, Brigid's, Pattee as well as solar symbols (Kolovrats) built according to the Lapshin's equations

Cross can be described by a two-dimensional piecewise-linear closed plane curve (12-sided polygon) using the following parametric equations found by R. V. Lapshin:

$\begin{array}{l}x(\alpha )=\frac{k_x{b}_x}{\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{s}}(\frac{T}{8})}\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{c}}\alpha ,\\ y(\alpha )=\frac{k_y{b}_y}{\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{s}}(\frac{T}{8})}\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{s}}\alpha =\frac{k_y{b}_y}{k_x{b}_x}x(\alpha -\frac{T}{4}),\end{array}$

where $k_x$, $k_y$ are normalization factors along axes $x$ and $y$, respectively; $b_x$, $b_y$ are half-width and half-height of the cross, respectively; $\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{s}}$, $\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{c}}$ are trapezoidal pulses; $T$ is a period of the trapezoidal pulses; $\alpha$ is a real parameter ($\alpha =0\dots T$).

The trapezoidal pulses $\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{s}}$, $\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{c}}$ are defined as follows:

$\begin{array}{l}\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{s}}\alpha =\sum _i(-1{)}^i[\frac{4}{D-d}(\alpha -\frac{T}{2}i)\mathrm{r}\mathrm{e}\mathrm{c}{\mathrm{t}}_{\mathrm{1}}(\alpha ,i)+\mathrm{r}\mathrm{e}\mathrm{c}{\mathrm{t}}_{\mathrm{2}}(\alpha ,i)],\\ \mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{c}}\alpha =\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{s}}(\alpha +\frac{T}{4}),\end{array}$

where $d$ and $D$ are the upper and the lower bases of the trapezoidal pulses, respectively; $\mathrm{r}\mathrm{e}\mathrm{c}{\mathrm{t}}_{\mathrm{1}}$ and $\mathrm{r}\mathrm{e}\mathrm{c}{\mathrm{t}}_{\mathrm{2}}$ are rectangular pulses.

The rectangular pulses $\mathrm{r}\mathrm{e}\mathrm{c}{\mathrm{t}}_{\mathrm{1}}$ and $\mathrm{r}\mathrm{e}\mathrm{c}{\mathrm{t}}_{\mathrm{2}}$, in turn, are determined by a step function $\mathrm{H}(\alpha )$ (Heaviside function):

$\begin{array}{l}\mathrm{r}\mathrm{e}\mathrm{c}{\mathrm{t}}_{\mathrm{1}}(\alpha ,i)=\mathrm{H}(\alpha -\frac{T}{2}i+\frac{D-d}{4})-\mathrm{H}(\alpha -\frac{T}{2}i-\frac{D-d}{4}),\\ \mathrm{r}\mathrm{e}\mathrm{c}{\mathrm{t}}_{\mathrm{2}}(\alpha ,i)=\mathrm{H}(\alpha -\frac{T}{2}i-\frac{D-d}{4})-\mathrm{H}(\alpha -\frac{T}{2}i-\frac{D+3d}{4}).\end{array}$

With certain values of $d$, $D$, and $T$ ($T\ne d+D$), the equations describe a number of closed curves which look like crosses.

The crosses tilted by 45${\circ }^{}$ are defined by the following equations:

$\begin{array}{l}x(\alpha )=\frac{k_x{b}_x}{\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{s}}(\frac{T}{8})}(\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{s}}\alpha +\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{c}}\alpha ),\\ y(\alpha )=\frac{k_y{b}_y}{\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{s}}(\frac{T}{8})}(\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{s}}\alpha -\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{c}}\alpha )=\frac{k_y{b}_y}{k_x{b}_x}x(\alpha -\frac{T}{4}).\end{array}$

As an example, the figure shows the piecewise-linear curves: Greek cross, St. Andrew's cross, Brigid's cross, and cross Pattee. The upright crosses are demonstrated in the 1st column, the tilted crosses – in the 2nd one, and the solar symbols – in the 3rd one. The solar symbols (Kolovrats) are obtained by overlapping an upright cross upon its copy rotated by 45${\circ }^{}$. Beside the curves of the Cross type, the curves of the Swastika type can be built by the similar equations.

See also

• Quartic plane curve
• Cross

External links

• "Cruciform". Mathworld.

This article "Cross curve" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Cross curve. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

📰 Article(s) of the same category(ies)

• Square (parametric form)
• Engare
• Space cardioid

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