### Calculations in TeaCat

• You can create mathematical expressions by simply typing them in the worksheet. The characters are displayed in the worksheet just as you would see them on paper.

Type numbers, letters and math operators, or insert them by clicking the appropriate buttons on the Math Toolbar. TeaCat by default understands anything you type as math.

In TeaCat, you can work with variables, parameters, constants, functions, matrices and equations. Numbers may belong to `ℕ`, `ℤ`, `ℝ` or `ℂ`.

TeaCat provides a set of built-in operators:
• Common ones (+,-,*,/), Absolute Value, Power, Square, Square root, Root, Sign, Exponential, Natural Logarithm, Common Logarithm, Evaluation, Last Result.
• Derivative, Indefinite Integral, Definite Integral, Limit, Left Limit, Right Limit.
• The trigonometric functions, the hyperbolic and their reciprocal functions.
• Matrix operations: Determinant, Diagonalize (which returns eigenvectors and eigenvalues), Inverse, Transpose, Norm, Trace, Sum, Cross Product, Diagonal.
• Geometric operators as Equation (which returns the equation of a geometric object), Perimeter, Area.
• The Arrange operator is essentially a symbolical operator which re-arranges an expression.
• The Find Root operator yields the root(s) of an expression.
• The Solve Equation operator yields the solution(s) of an equation or a set of equations.

The result of the Evaluation operator is context-keyword sensitive. The keywords determine the way the result is presented in the worksheet (numerical, symbolical, fraction, real part, ...).

In TeaCat each calculation is initiated by the Evaluation Operator (→). The kind of result depends on the kind of the operator on the left (det(M), g1(x), froot(B), solve(exp1=exp2), ...). For example: (M is a matrix) det(M) →("SYMB") computes the determinant symbolically and det(M) →("Q") gives the determinant as a fraction. In short, we say that the result of the Evaluation Operator is keyword sensitive.

You can easily change or remove the evaluation keywords by means of a popup menu.

Visualisation vs. mathematical model. The way your worksheet looks like, doesn't matter at all. Your worksheet's layout doesn't influence the order in which the commands are executed.

Your worksheet only represents those expressions you want to be depicted. The order of the calculations is fully controlled by the Command List Field of the Script Panel. TeaCat just scans the Command List Field from top to bottom and executes these commands, line by line. Although all the definitions and calculations contribute to the final result, not all of them are required to be visible on paper. Therefore some of them can be hidden from the worksheet.

### 2D Plots

• TeaCat recognises three types of plotable mathematical expressions: functions in one variable, parameter equations (in two variables) and polar equations.

These expressions are automatically identified in TeaCat amongst all commands that you've defined in the worksheet.

Besides plotting functions and equations, TeaCat provides a way to include geometric objects in a plot box.

You can insert Points, Lines, Conic Sections and Polygons.

Furthermore, TeaCat provides the Equation operator to find the equations of the inserted objects (lines, circles and ellipses) in parametric or Cartesian form. TeaCat also includes the Perimeter and Area operators to find the perimeter and the area of closed objects, such as circles, ellipses and polygons.

To find the intersecting points of two or more geometric objects or functions (lines, circles, functions, ...) one can use the Solve Equation operator.
• Plot boxes can be formatted by changing the size, the box properties or the axis/grid/background properties; they can be zoomed in or out. The curve properties can be modified as well (colour, thickness, fill, opacity) and labels can be inserted, etc. On top of that, you can use TeaCat to create animations and trace the moving objects.

Anything that can be made to dependent on a parameter can be animated. The parameter automatically runs through a range of numbers when you start the animation. The minimum and the maximum value of the parameter are adjustable, as are the step and the speed of the animation.

### Text & Images

• To illustrate your TeaCat calculations, you can add images to your worksheet. TeaCat can create a picture from a number of image file formats, including JPEG, BMP, TIFF, GIF, PNG and PSD. Each picture appears in an image box in your worksheet.

You can link the imported image to your worksheet. When the image is linked, the picture in your document is updated to the current version. This causes the TeaCat file to be smaller.

You can modify the view (zoom and size), the HSV values (hue, saturation, value or brightness), the gamma, the contrast and the opacity of a picture in TeaCat by using the Image Toolbar.

The (modified) picture can be imported into other applications.

As with plot boxes, also image boxes can be formatted by changing the size and the box properties.

TeaCat allows documenting the calculations with text – it combines the computational engine with a well-featured word processor. The text appears in a text box. You can change the text style (font, size, colour and appearance, the paragraph alignment, numbering). As with plot and image boxes, also text boxes can be formatted by changing the size and the box properties.