Polygon Renderers

This page documents library functions that render polygons.

$n$-Sided Regular Polygon

$$d_ ext{polygon} left(, n, p_ ext{centre}, r, d , ight)$$

Generate vertices of an $n$-sided regular polygon.

Arguments

Argument	Description	Type	Constraints	Notes
$n$	number of sides	$\mathbb{Z}^{+}$	$3 \leq n$ (required)
$p_\text{centre}$	centre	$(\mathbb{R}, \mathbb{R})$
$r$	radius of the polygon	$\mathbb{R}$	$r \neq 0$ (required) $0 < r$	“Radius” refers to the distance from the centre of the polygon to any vertex.
$d$	direction (rotation)	$\mathbb{R}$	$0 \leq d < 2\pi$

Return

Value	Description	Type	Constraints	Notes
$\omega$	polygon vertices	$[(\mathbb{R}, \mathbb{R}), ...]$		Use $\operatorname{polygon}(\omega)$ to render the polygon.

Usage

f_{nrange}\left(s_{start},\ s_{stop},\ n\right)=\left[\left(s_{start}+i\left(\frac{s_{stop}-s_{start}}{n-1}\right)\right)\operatorname{for}i=\left[0...\left(n-1\right)\right]\right]
d_{polygon}\left(n,\ p_{centre},\ r,\ d\right)=\left[\left(p_{centre}.x+r\cos\theta,\ p_{centre}.y+r\sin\theta\right)\operatorname{for}\theta=f_{nrange}\left(d,\ d+2\pi,\ n+1\right)\right]
\operatorname{polygon}\left(d_{polygon}\left(6,\ \left(0,\ 0\right),\ 4,\ \frac{\pi}{6}\right)\right)

Implementation

$$d_ ext{polygon} left(, n, p_ ext{centre}, r, d , ight) = left[, left(p_{centre}.x+rcos heta, p_{centre}.y+rsin heta ight)operatorname{for} heta=f_{nrange}left(d, d+2pi, n+1 ight) , ight]$$

Dependencies

• $f_\text{nrange}()$

Rectangle

$$d_ ext{rect} left(, p_ ext{centre}, x, y , ight)$$

Draw a rectangle at centre $p_\text{centre}$ with dimensions $x \times y$.

Arguments

Argument	Description	Type	Constraints	Notes
$p_\text{centre}$	rectangle centre	$(\mathbb{R}, \mathbb{R})$
$x$	rectangle width	$\mathbb{R}^{+}$
$y$	rectangle height	$\mathbb{R}^{+}$

Return

None

Usage

d_{rect}\left(p_{centre},\ s_{width},\ s_{height}\right)=\operatorname{polygon}\left(p_{centre}+\left(-\frac{s_{width}}{2},\ -\frac{s_{height}}{2}\right),\ p_{centre}+\left(\frac{s_{width}}{2},\ -\frac{s_{height}}{2}\right),\ p_{centre}+\left(\frac{s_{width}}{2},\ \frac{s_{height}}{2}\right),\ p_{centre}+\left(-\frac{s_{width}}{2},\ \frac{s_{height}}{2}\right)\right)
d_{rect}\left(\left(3,\ 2\right),\ 4,\ 1\right)

Implementation

$$d_ ext{rect} left(, p_ ext{centre}, x, y , ight) = operatorname{polygon}left(p_{centre}+left(-rac{x}{2}, -rac{y}{2} ight), p_{centre}+left(rac{x}{2}, -rac{y}{2} ight), p_{centre}+left(rac{x}{2}, rac{y}{2} ight), p_{centre}+left(-rac{x}{2}, rac{y}{2} ight) ight)$$

Dependencies

None

Aligned Rectangle

$$d_ ext{rectaligned} left(, p_ ext{pivot}, p_{xy}, p_ ext{align} , ight)$$

Draw a rectangle at pivot $p_\text{pivot}$ with dimensions $p_{xy}$.

Arguments

Argument	Description	Domain	Constraints	Notes
$p_\text{pivot}$	pivot point	$(\mathbb{R}, \mathbb{R})$
$p_\text{xy}$	rectangle dimensions	$(\mathbb{R}, \mathbb{R})$
$p_\text{align}$	alignment	$(\mathbb{R}, \mathbb{R})$	$(-1, -1) \leq p_\text{align} \leq (1, 1)$	$(-1, -1)$ means the upper-right corner is used as the pivot (the rectangle is drawn in the negative quadrant).

Return

None

Usage

d_{rectaligned}\left(p_{pivot},\ p_{xy},\ p_{align}\right)=\operatorname{polygon}\left(p_{pivot}+\left(\left(p_{align}.x-1\right)\frac{p_{xy}.x}{2},\ \left(p_{align}.y-1\right)\frac{p_{xy}.y}{2}\right),\ p_{pivot}+\left(\left(p_{align}.x+1\right)\frac{p_{xy}.x}{2},\ \left(p_{align}.y-1\right)\frac{p_{xy}.y}{2}\right),\ p_{pivot}+\left(\left(p_{align}.x+1\right)\cdot\frac{p_{xy}.x}{2},\ \left(p_{align}.y+1\right)\frac{p_{xy}.y}{2}\right),\ p_{pivot}+\left(\left(p_{align}.x-1\right)\cdot\frac{p_{xy}.x}{2},\ \left(p_{align}.y+1\right)\frac{p_{xy}.y}{2}\right)\right)
d_{rectaligned}\left(\left(0,\ 0\right),\ \left(3,\ 4\right),\ \left(-1,\ -1\right)\right)
d_{rectaligned}\left(\left(2,\ 2\right),\ \left(5,\ 5\right),\ \left(1,\ -1\right)\right)

Implementation

$$d_ ext{rectaligned} left(, p_ ext{pivot}, p_{xy}, p_ ext{align} , ight) = operatorname{polygon}left( p_{pivot}+left(left(p_{align}.x-1 ight)rac{p_{xy}.x}{2}, left(p_{align}.y-1 ight)rac{p_{xy}.y}{2} ight), p_{pivot}+left(left(p_{align}.x+1 ight)rac{p_{xy}.x}{2}, left(p_{align}.y-1 ight)rac{p_{xy}.y}{2} ight), p_{pivot}+left(left(p_{align}.x+1 ight)cdotrac{p_{xy}.x}{2}, left(p_{align}.y+1 ight)rac{p_{xy}.y}{2} ight), p_{pivot}+left(left(p_{align}.x-1 ight)cdotrac{p_{xy}.x}{2}, left(p_{align}.y+1 ight)rac{p_{xy}.y}{2} ight) ight)$$

Dependencies

None