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For basic information on 2-D geometry construction, see Three Ways to Create 2-D Geometry

To describe your geometry through Constructive Solid Geometry (CSG) modeling, use three data structures.

A matrix whose columns describe the basic shapes. When you export geometry from the PDE Modeler app, this matrix has the default name

`gd`

(geometry description). See Create Basic Shapes.A matrix whose columns contain names for the basic shapes. Pad the columns with zeros or 32 (blanks) so that every column has the same length. See Create Names for the Basic Shapes.

A set of characters describing the unions, intersections, and set differences of the basic shapes that make the geometry. See Set Formula.

To create basic shapes at the command line, create a matrix whose columns each describe a basic shape. If necessary, add extra zeros to some columns so that all columns have the same length. Write each column using the following encoding.

**Circle**

Row | Value |
---|---|

`1` | `1` (indicates a circle) |

`2` | x-coordinate of circle center |

`3` | y-coordinate of circle center |

`4` | Radius (strictly positive) |

**Polygon**

Row | Value |
---|---|

`1` | `2` (indicates a polygon) |

`2` | Number of line segments `n` |

`3` through `3+n-1` | x-coordinate of edge starting points |

`3+n` through `2*n+2` | y-coordinate of edge starting points |

Your polygon cannot contain any self-intersections. To check whether your polygon
satisfies this restriction, use the `csgchk`

function.

**Rectangle**

Row | Value |
---|---|

`1` | `3` (indicates a rectangle) |

`2` | `4` (number of line segments) |

`3` through `6` | x-coordinate of edge starting points |

`7` through `10` | y-coordinate of edge starting points |

The encoding of a rectangle is the same as that of a polygon, except that the first
row is `3`

instead of `2`

.

**Ellipse**

Row | Value |
---|---|

`1` | `4` (indicates an ellipse) |

`2` | x-coordinate of ellipse center |

`3` | y-coordinate of ellipse center |

`4` | First semiaxis length (strictly positive) |

`5` | Second semiaxis length (strictly positive) |

`6` | Angle in radians from x axis to first
semiaxis |

For example, specify a matrix that has a rectangle with a circular end cap and another circular excision. First, create a rectangle and two adjoining circles.

rect1 = [3 4 -1 1 1 -1 0 0 -0.5 -0.5]; C1 = [1 1 -0.25 0.25]; C2 = [1 -1 -0.25 0.25];

Append extra zeros to the circles so they have the same number of rows as the rectangle.

C1 = [C1;zeros(length(rect1) - length(C1),1)]; C2 = [C2;zeros(length(rect1) - length(C2),1)];

Combine the shapes into one matrix.

gd = [rect1,C1,C2];

In order to create a formula describing the unions and intersections of basic shapes, you need a name for each basic shape. Give the names as a matrix whose columns contain the names of the corresponding columns in the basic shape matrix. Pad the columns with 0 or 32 if necessary so that each has the same length.

One easy way to create the names is by specifying a character array whose rows contain
the names, and then taking the transpose. Use the `char`

function to create the array. `char`

pads the
rows as needed so all have the same length. Continuing the example, give names for the
three shapes.

ns = char('rect1','C1','C2'); ns = ns';

Obtain the final geometry by writing a set of characters that describes the unions and
intersections of basic shapes. Use `+`

for union, `*`

for intersection, `-`

for set difference, and parentheses for grouping.
`+`

and `*`

have the same grouping precedence.
`-`

has higher grouping precedence.

Continuing the example, specify the union of the rectangle and `C1`

,
and subtract `C2`

.

`sf = '(rect1+C1)-C2';`

After you have created the basic shapes, given them names, and specified a set
formula, create the geometry using `decsg`

. Often, you also remove some or all of the resulting face
boundaries. Completing the example, combine the basic shapes using the set
formula.

[dl,bt] = decsg(gd,sf,ns);

View the geometry with and without boundary removal.

pdegplot(dl,'EdgeLabels','on','FaceLabels','on') xlim([-1.5,1.5]) axis equal

Remove the face boundaries.

[dl2,bt2] = csgdel(dl,bt); % removes face boundaries figure pdegplot(dl2,'EdgeLabels','on','FaceLabels','on') xlim([-1.5,1.5]) axis equal

A decomposed geometry matrix has the following encoding. Each column of the matrix corresponds to one boundary segment. Any 0 entry means no encoding is necessary for this row. So, for example, if only line segments appear in the matrix, then the matrix has 7 rows. But if there is also a circular segment, then the matrix has 10 rows. The extra three rows of the line columns are filled with 0.

Row | Circle | Line | Ellipse |
---|---|---|---|

`1` | `1` | `2` | `4` |

`2` | Starting x coordinate | Starting x coordinate | Starting x coordinate |

`3` | Ending x coordinate | Ending x coordinate | Ending x coordinate |

`4` | Starting y coordinate | Starting y coordinate | Starting y coordinate |

`5` | Ending y coordinate | Ending y coordinate | Ending y coordinate |

`6` | Region label to left of segment, with direction induced by start and
end points (`0` is exterior label) | Region label to left of segment, with direction induced by start and
end points (`0` is exterior label) | Region label to left of segment, with direction induced by start and
end points (`0` is exterior label) |

`7` | Region label to right of segment, with direction induced by start and
end points (`0` is exterior label) | Region label to right of segment, with direction induced by start and
end points (`0` is exterior label) | Region label to right of segment, with direction induced by start and
end points (`0` is exterior label) |

`8` | x coordinate of circle center | 0 | x coordinate of ellipse center |

`9` | y coordinate of circle center | 0 | y coordinate of ellipse center |

`10` | Radius | 0 | Length of first semiaxis |

`11` | 0 | 0 | Length of second semiaxis |

`12` | 0 | 0 | Angle in radians between x axis and first
semiaxis |