B.1 Triangular Patterns

An arbitrary triangle consisting of three points {(x₁, y₁),(x₂, y₂),(x₃, y₃)} in $\mathbb{R}$² can always be transformed to the unit triangle {(0, 0),(1, 0),(0, 1)} by

 

$$\begin{pmatrix}x \\ y \end{pmatrix} = \underline{\mathbf{A}} \beg... ...thbf{A}} = \begin{pmatrix}x_2-x_1 & x_3-x_1 \\ y_2-y_1 & y_3-y_1 \end{pmatrix}.$$ (B.2)

The module of the determinant of the transformation matrix $\underline{\mathbf{A}}$ equals

$$\vert\det\underline{\mathbf{A}}\,\vert = \vert(x_2-x_1)(y_3-y_1) - (y_2-y_1)(x_3-x_1)\vert.$$ (B.3)

Using (B.2), the integral (B.1) writes for triangular patterns to

$$\begin{aligned}\mathcal{T}_{nm} &= \frac{1}{ab} \iint\limits_{(x,... ... d\eta} _{\displaystyle I_{\mathcal{T}}(\alpha_{nm},\beta_{nm})}, \end{aligned}$$

whereby TRI-1 refers to the unit triangle and the coefficients $\alpha_{nm}^{}$, $\beta_{nm}^{}$ and $\gamma_{nm}^{}$ are given by

$$\renewcommand{\arraystretch}{2}\alpha_{nm} = n(x_2-x_1)/a + m(y_2-y_1)/b$$

$$\beta_{nm} = n(x_3-x_1)/a + m(y_3-y_1)/b$$ (B.4)

$$\gamma_{nm} = nx_1/a + my_1/b.$$

The function I_{$\mathcal {T}$}($\alpha_{nm}^{}$,$\beta_{nm}^{}$) is calculated as follows:^a

 

$$I_{\mathcal{T}}(\alpha_{nm},\beta_{nm}) = \begin{cases}\displayst... ...a_{nm}-\alpha_{nm}} e^{-j\pi\beta_{nm}} \right] & \text{otherwise.} \end{cases}$$ (B.5)

Footnotes

... follows:^a

The function si(x) is defined as si(x)$\overset{\mathrm{def}}{=} \sin x/x$.