Partial Differential Equation Toolbox

The CSG Model and the Set Formula

The PDE Toolbox uses the Constructive Solid Geometry (CSG) model paradigm for the modeling. You can draw solid objects that can overlap. There are four types of solid objects:

• Circle object -- Represents the set of points inside and on a circle.
• Polygon object -- Represents the set of points inside and on a polygon given by a set of line segments.
• Rectangle object -- Represents the set of points inside and on a rectangle.
• Ellipse object -- Represents the set of points inside and on an ellipse. The ellipse can be rotated.

Each solid object is automatically given a unique name by the GUI. The default names are C1, C2, C3, etc., for circles; P1, P2, P3, etc. for polygons; R1, R2, R3, etc., for rectangles; E1, E2, E3, etc., for ellipses. Squares, although a special case of rectangles, are named SQ1, SQ2, SQ3, etc. The name is displayed on the solid object itself. You can use any unique name, as long as it contains no blanks. In Draw mode you can alter the names and the geometries of the objects by double-clicking them, which opens a dialog box. The following figure shows an object dialog box for a circle.

You can use the name of the object to refer to the corresponding set of points in a set formula. The operators +, *, and - are used to form the set of points in the plane over which the differential equation is solved. The operators +, the set union operator, and *, the set intersection operator, have the same precedence. The operator -, the set difference operator, has higher precedence. The precedence can be controlled by using parentheses. The resulting geometrical model, , is the set of points for which the set formula evaluates to true. By default, it is the union of all solid objects. We often refer to the area as the decomposed geometry.

The GUI Modes

Creating Rounded Corners