5.165. inside_sboxes

Origin

Geometry, derived from [RandellCuiCohn92]

Constraint

$\mathrm{𝚒𝚗𝚜𝚒𝚍𝚎}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}\left(𝙺,\mathrm{𝙳𝙸𝙼𝚂},\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂},\mathrm{𝚂𝙱𝙾𝚇𝙴𝚂}\right)$

Synonym

$\mathrm{𝚒𝚗𝚜𝚒𝚍𝚎}$.

Types
 $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(𝚟-\mathrm{𝚍𝚟𝚊𝚛}\right)$ $\mathrm{𝙸𝙽𝚃𝙴𝙶𝙴𝚁𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(𝚟-\mathrm{𝚒𝚗𝚝}\right)$ $\mathrm{𝙿𝙾𝚂𝙸𝚃𝙸𝚅𝙴𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(𝚟-\mathrm{𝚒𝚗𝚝}\right)$
Arguments
 $𝙺$ $\mathrm{𝚒𝚗𝚝}$ $\mathrm{𝙳𝙸𝙼𝚂}$ $\mathrm{𝚜𝚒𝚗𝚝}$ $\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚘𝚒𝚍}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚒𝚍}-\mathrm{𝚒𝚗𝚝},𝚡-\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)$ $\mathrm{𝚂𝙱𝙾𝚇𝙴𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚜𝚒𝚍}-\mathrm{𝚒𝚗𝚝},𝚝-\mathrm{𝙸𝙽𝚃𝙴𝙶𝙴𝚁𝚂},𝚕-\mathrm{𝙿𝙾𝚂𝙸𝚃𝙸𝚅𝙴𝚂}\right)$
Restrictions
 $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},𝚟\right)$ $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|=𝙺$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝙸𝙽𝚃𝙴𝙶𝙴𝚁𝚂},𝚟\right)$ $|\mathrm{𝙸𝙽𝚃𝙴𝙶𝙴𝚁𝚂}|=𝙺$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝙿𝙾𝚂𝙸𝚃𝙸𝚅𝙴𝚂},𝚟\right)$ $|\mathrm{𝙿𝙾𝚂𝙸𝚃𝙸𝚅𝙴𝚂}|=𝙺$ $\mathrm{𝙿𝙾𝚂𝙸𝚃𝙸𝚅𝙴𝚂}.𝚟>0$ $𝙺>0$ $\mathrm{𝙳𝙸𝙼𝚂}\ge 0$ $\mathrm{𝙳𝙸𝙼𝚂}<𝙺$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂},\left[\mathrm{𝚘𝚒𝚍},\mathrm{𝚜𝚒𝚍},𝚡\right]\right)$ $\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}.\mathrm{𝚘𝚒𝚍}\ge 1$ $\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}.\mathrm{𝚘𝚒𝚍}\le |\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}|$ $\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}.\mathrm{𝚜𝚒𝚍}\ge 1$ $\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}.\mathrm{𝚜𝚒𝚍}\le |\mathrm{𝚂𝙱𝙾𝚇𝙴𝚂}|$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚂𝙱𝙾𝚇𝙴𝚂},\left[\mathrm{𝚜𝚒𝚍},𝚝,𝚕\right]\right)$ $\mathrm{𝚂𝙱𝙾𝚇𝙴𝚂}.\mathrm{𝚜𝚒𝚍}\ge 1$ $\mathrm{𝚂𝙱𝙾𝚇𝙴𝚂}.\mathrm{𝚜𝚒𝚍}\le |\mathrm{𝚂𝙱𝙾𝚇𝙴𝚂}|$
Purpose

Holds if, for each pair of objects $\left({O}_{i},{O}_{j}\right)$, $i, ${O}_{i}$ is inside ${O}_{j}$ with respect to a set of dimensions depicted by $\mathrm{𝙳𝙸𝙼𝚂}$. ${O}_{i}$ and ${O}_{j}$ are objects that take a shape among a set of shapes. Each shape is defined as a finite set of shifted boxes, where each shifted box is described by a box in a $𝙺$ -dimensional space at a given offset (from the origin of the shape) with given sizes. More precisely, a shifted box is an entity defined by its shape id $\mathrm{𝚜𝚒𝚍}$, shift offset $𝚝$, and sizes $𝚕$. Then, a shape is defined as the union of shifted boxes sharing the same shape id. An object is an entity defined by its unique object identifier $\mathrm{𝚘𝚒𝚍}$, shape id $\mathrm{𝚜𝚒𝚍}$ and origin $𝚡$.

An object ${O}_{i}$ is inside an object ${O}_{j}$ with respect to a set of dimensions depicted by $\mathrm{𝙳𝙸𝙼𝚂}$ if and only if, for all shifted boxes ${s}_{i}$ associated with ${O}_{i}$, there exists a shifted box ${s}_{j}$ of ${O}_{j}$ such that ${s}_{j}$ is inside ${s}_{i}$. A shifted box ${s}_{j}$ is inside a shifted box ${s}_{i}$ if and only if, for all dimensions $d\in \mathrm{𝙳𝙸𝙼𝚂}$, (1) the start of ${s}_{j}$ in dimension $d$ is strictly less than the start of ${s}_{i}$ in dimension $d$, and (2) the end of ${s}_{i}$ in dimension $d$ is strictly less than the end of ${s}_{j}$ in dimension $d$.

Example
$\left(\begin{array}{c}2,\left\{0,1\right\},\hfill \\ 〈\begin{array}{ccc}\mathrm{𝚘𝚒𝚍}-1\hfill & \mathrm{𝚜𝚒𝚍}-1\hfill & 𝚡-〈3,3〉,\hfill \\ \mathrm{𝚘𝚒𝚍}-2\hfill & \mathrm{𝚜𝚒𝚍}-2\hfill & 𝚡-〈2,2〉,\hfill \\ \mathrm{𝚘𝚒𝚍}-3\hfill & \mathrm{𝚜𝚒𝚍}-3\hfill & 𝚡-〈1,1〉\hfill \end{array}〉,\hfill \\ 〈\begin{array}{ccc}\mathrm{𝚜𝚒𝚍}-1\hfill & 𝚝-〈0,0〉\hfill & 𝚕-〈1,1〉,\hfill \\ \mathrm{𝚜𝚒𝚍}-2\hfill & 𝚝-〈0,0〉\hfill & 𝚕-〈3,3〉,\hfill \\ \mathrm{𝚜𝚒𝚍}-3\hfill & 𝚝-〈0,0〉\hfill & 𝚕-〈5,5〉\hfill \end{array}〉\hfill \end{array}\right)$

Figure 5.165.1 shows the objects of the example. Since ${O}_{1}$ is inside ${O}_{2}$ and ${O}_{3}$, and since ${O}_{2}$ is also inside ${O}_{3}$, the $\mathrm{𝚒𝚗𝚜𝚒𝚍𝚎}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}$ constraint holds.

Typical
$|\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}|>1$
Symmetries
• Items of $\mathrm{𝚂𝙱𝙾𝚇𝙴𝚂}$ are permutable.

• Items of $\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}.𝚡$, $\mathrm{𝚂𝙱𝙾𝚇𝙴𝚂}.𝚝$ and $\mathrm{𝚂𝙱𝙾𝚇𝙴𝚂}.𝚕$ are permutable (same permutation used).

Remark

One of the eight relations of the Region Connection Calculus [RandellCuiCohn92]. The constraint $\mathrm{𝚒𝚗𝚜𝚒𝚍𝚎}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}$ is a restriction of the original relation since it requires that each box of an object is contained by one box of the other object.

 $•\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\left(𝙾\mathtt{1},𝚂\mathtt{1},𝙳\right)\stackrel{\mathrm{def}}{=}𝙾\mathtt{1}.𝚡\left(𝙳\right)+𝚂\mathtt{1}.𝚝\left(𝙳\right)$ $•\mathrm{𝚎𝚗𝚍}\left(𝙾\mathtt{1},𝚂\mathtt{1},𝙳\right)\stackrel{\mathrm{def}}{=}𝙾\mathtt{1}.𝚡\left(𝙳\right)+𝚂\mathtt{1}.𝚝\left(𝙳\right)+𝚂\mathtt{1}.𝚕\left(𝙳\right)$ $•\begin{array}{c}\mathrm{𝚒𝚗𝚜𝚒𝚍𝚎}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}\left(\mathrm{𝙳𝚒𝚖𝚜},𝙾\mathtt{1},𝚂\mathtt{1},𝙾\mathtt{2},𝚂\mathtt{2}\right)\stackrel{\mathrm{def}}{=}\hfill \\ \begin{array}{c}\forall 𝙳\in \mathrm{𝙳𝚒𝚖𝚜}\hfill \\ \bigwedge \left(\begin{array}{c}\begin{array}{c}\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\left(𝙾\mathtt{2},𝚂\mathtt{2},𝙳\right)<\hfill \\ \mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\left(𝙾\mathtt{1},𝚂\mathtt{1},𝙳\right)\hfill \end{array},\hfill \\ \begin{array}{c}\mathrm{𝚎𝚗𝚍}\left(𝙾\mathtt{1},𝚂\mathtt{1},𝙳\right)<\hfill \\ \mathrm{𝚎𝚗𝚍}\left(𝙾\mathtt{2},𝚂\mathtt{2},𝙳\right)\hfill \end{array}\hfill \end{array}\right)\hfill \end{array}\hfill \end{array}$ $•\begin{array}{c}\mathrm{𝚒𝚗𝚜𝚒𝚍𝚎}_\mathrm{𝚘𝚋𝚓𝚎𝚌𝚝𝚜}\left(\mathrm{𝙳𝚒𝚖𝚜},𝙾\mathtt{1},𝙾\mathtt{2}\right)\stackrel{\mathrm{def}}{=}\hfill \\ \begin{array}{c}\forall 𝚂\mathtt{1}\in \mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}\left(\left[𝙾\mathtt{1}.\mathrm{𝚜𝚒𝚍}\right]\right)\hfill \\ \begin{array}{c}\exists 𝚂\mathtt{2}\in \mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}\left(\begin{array}{c}\left[\begin{array}{c}𝙾\mathtt{2}.\mathrm{𝚜𝚒𝚍}\hfill \end{array}\right]\hfill \end{array}\right)\hfill \\ \mathrm{𝚒𝚗𝚜𝚒𝚍𝚎}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}\left(\begin{array}{c}\mathrm{𝙳𝚒𝚖𝚜},\hfill \\ 𝙾\mathtt{1},\hfill \\ 𝚂\mathtt{1},\hfill \\ 𝙾\mathtt{2},\hfill \\ 𝚂\mathtt{2}\hfill \end{array}\right)\hfill \end{array}\hfill \end{array}\hfill \end{array}$ $•\begin{array}{c}\mathrm{𝚊𝚕𝚕}_\mathrm{𝚒𝚗𝚜𝚒𝚍𝚎}\left(\mathrm{𝙳𝚒𝚖𝚜},\mathrm{𝙾𝙸𝙳𝚂}\right)\stackrel{\mathrm{def}}{=}\hfill \\ \begin{array}{c}\forall 𝙾\mathtt{1}\in \mathrm{𝚘𝚋𝚓𝚎𝚌𝚝𝚜}\left(\mathrm{𝙾𝙸𝙳𝚂}\right)\hfill \\ \begin{array}{c}\forall 𝙾\mathtt{2}\in \mathrm{𝚘𝚋𝚓𝚎𝚌𝚝𝚜}\left(\mathrm{𝙾𝙸𝙳𝚂}\right)\hfill \\ \begin{array}{c}\begin{array}{c}𝙾\mathtt{1}.\mathrm{𝚘𝚒𝚍}<\hfill \\ 𝙾\mathtt{2}.\mathrm{𝚘𝚒𝚍}\hfill \end{array}⇒\hfill \\ \mathrm{𝚒𝚗𝚜𝚒𝚍𝚎}_\mathrm{𝚘𝚋𝚓𝚎𝚌𝚝𝚜}\left(\begin{array}{c}\mathrm{𝙳𝚒𝚖𝚜},\hfill \\ 𝙾\mathtt{1},\hfill \\ 𝙾\mathtt{2}\hfill \end{array}\right)\hfill \end{array}\hfill \end{array}\hfill \end{array}\hfill \end{array}$ $•\mathrm{𝚊𝚕𝚕}_\mathrm{𝚒𝚗𝚜𝚒𝚍𝚎}\left(\mathrm{𝙳𝙸𝙼𝙴𝙽𝚂𝙸𝙾𝙽𝚂},\mathrm{𝙾𝙸𝙳𝚂}\right)$