Square (parametric form)

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Squares with side $a=1$ set in parametric form. $x(\alpha )$, $y(\alpha )$ are generating functions

A square in parametric form is given by the Lapshin's formula:

$\begin{array}{l}x(\alpha )=\frac{a}{\sqrt{2}}\mathrm{t}\mathrm{r}{\mathrm{i}}_{\mathrm{c}}\alpha ,\\ y(\alpha )=\frac{a}{\sqrt{2}}\mathrm{t}\mathrm{r}{\mathrm{i}}_{\mathrm{s}}\alpha =x(\alpha -\frac{T}{4}),\end{array}$

where $a$ is the length of the square side; $\mathrm{t}\mathrm{r}{\mathrm{i}}_{\mathrm{s}}$, $\mathrm{t}\mathrm{r}{\mathrm{i}}_{\mathrm{c}}$ are triangular pulses; $T$ is a period of the triangular pulses; $\alpha$ is a real parameter ($\alpha =0\dots T$).

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

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

where $\mathrm{rect}$ are rectangular pulses.

The rectangular pulses $\mathrm{rect}$ are determined by a step function $\mathrm{H}(\alpha )$ (Heaviside function):

$\mathrm{rect}(\alpha ,i)=\mathrm{H}(\alpha -\frac{T}{2}i+\frac{T}{4})-\mathrm{H}(\alpha -\frac{T}{2}i-\frac{T}{4}).$

The following equations describe a square whose sides are parallel to the coordinate axes:

$\begin{array}{l}x(\alpha )=\frac{a}{2}\left(\mathrm{t}\mathrm{r}{\mathrm{i}}_{\mathrm{c}}(\alpha +\frac{T}{8})+\mathrm{t}\mathrm{r}{\mathrm{i}}_{\mathrm{c}}(\alpha -\frac{T}{8})\right)=\frac{a}{2}\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{c}}\alpha ,\\ y(\alpha )=\frac{a}{2}\left(\mathrm{t}\mathrm{r}{\mathrm{i}}_{\mathrm{s}}(\alpha +\frac{T}{8})+\mathrm{t}\mathrm{r}{\mathrm{i}}_{\mathrm{s}}(\alpha -\frac{T}{8})\right)=\frac{a}{2}\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{s}}\alpha =x(\alpha -\frac{T}{4}),\end{array}$

where $\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{s}}$, $\mathrm{t}\mathrm{r}{\mathrm{p}}_{\mathrm{c}}$ are trapezoidal pulses with period $T$.

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

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

where $d$ and $D$ are the upper and lower bases of the trapezoidal pulses, respectively ($D=3d$, $T=d+D$); $\mathrm{r}\mathrm{e}\mathrm{c}{\mathrm{t}}_{\mathrm{1}}$ 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 $\mathrm{r}\mathrm{e}\mathrm{c}{\mathrm{t}}_{\mathrm{2}}$ are determined as

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

The figure shows squares and their generating functions $x(\alpha )$, $y(\alpha )$.

See also

• Square

References

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

• Engare
• Space cardioid
• Cross curve

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