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# «Geometry Expressions Manual TM Saltire Software PO Box 230755 Tigard, OR 97281 0755 © 2013 Saltire Software. All rights reserved. Information in ...»

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Pictures are always inserted under your drawing objects, so you can add a picture at any time.

In Select mode you can move a picture, as with any drawing object.

The inserted picture can be rotated and the transparency level can be set in the Display Properties dialog box.

Right click on a highlighted picture and then select All Properties · Or select Properties from Edit menu.

· © 2013 Saltire Software 60 Geometry Expressions Manual Adding Expressions You can type an algebraic expression in the drawing window and Geometry Expressions will solve it with whatever information it has

available. Here are the steps:

1. From the Draw toolbox click the Expression icon or select Expression from the Draw menu.

2. Move the expression cursor to the position where you want it to appear in the drawing window and click to display the data entry box.

3. Enter the expression using numbers, variables, and output expression names. Use the Symbols toolbox to help you enter mathematics.

Adding an expression to the diagram can help you work out relationships in the geometry. Here is an example making use of the Expression tool to see the relationship between similar triangles.

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Sides a and b are perpendicular, and AC is perpendicular to BD. We obtained the Output of lengths AD and BD using the Show Name button.

The expression to solve is the relationship of the two lengths: z0 / z1. The system immediately evaluates the expression as = a/b.

Creating Functions

1. Click the Function icon in the Draw toolbox or select Function from the Draw menu

2. Click the drop-down list button to select the function type that you want to use in your drawing.

© 2013 Saltire Software 62 Geometry Expressions Manual Cartesian Function When you select Cartesian from the Function Type dialog, the next line contains a general form of the function in terms of Y.

You can define a domain of the function by enter values for Start and End.

If you want to draw function with indefinite domain, leave these fields blank.

You can define this function in these ways:

Modify the formula directly:

·

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Modify the function interactively using "handles". In the drawing · window, click the function to select it. Click and drag it and a circle appears on the curve. This handle represents one of the variables in the equation. You can click and drag this handle around the drawing.

Click and drag another place on the function and another moveable handle appears if there is another variable in the equation.

This feature is a wonderful way to understand exactly how the equation represents the function.

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© 2013 Saltire Software 64 Geometry Expressions Manual The Generic Function f(X) To use the generic form of a function, Y=f(X), select Cartesian from the

Function Type dialog, enter f(X) in the edit window and click OK:

Click the Functions tab of the Variables toolbox to see the values used in the function.

You can define this function in these ways:

Modify the edit line at the bottom of the Functions tab. (Use your · keyboard arrow keys to move through the whole function.) Modify the function interactively using "handles". In the drawing · window, click the function to select it. Click and drag it and a circle appears on the curve. This handle represents one of the variables in the equation. You can click and drag a handle around the drawing to change the curve. Click and drag another place on the curve and another moveable handle appears if there is another function variable in the equation. In the general function in this example we have 5 possible handles: f(a), f(b), f(k), f(u), and f(v).

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Polar Function When you select Polar from the Function Type dialog, the next line contains the general form of the function in terms of the radial coordinate, r and the polar angle, T.

You can define this function in these ways:

Modify the formula and the curve domain directly:

· © 2013 Saltire Software 66 Geometry Expressions Manual Modify the function interactively using "handles". In the drawing · window, click the function to select it. Click and drag it and a circle appears on the curve. This handle represents one of the variables in the equation. You can click and drag this handle around the drawing.

Click and drag another place on the function and another moveable handle appears if there is another variable in the equation.

This feature is a wonderful way to understand exactly how the equation represents the function.

In the example above, the second click gives you the a handle; click and drag another place on the curve and you get the b handle to change the shape of the curve.

Notice the Variable toolbox displays the changing values as you move the handles.

Modify the function after it's drawn by double clicking the function · tag.

Modify the domain of the function after it's drawn by double clicking ·

the curve:

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Parametric Function When you select Parametric from the Function Type dialog, the next line contains the general form of the function in terms of X and Y and a parameter, T.

You can define this function in these ways:

Modify the formula and its domain directly:

· © 2013 Saltire Software 68 Geometry Expressions Manual Modify the function interactively using "handles". In the drawing · window, click the function to select it. Click and drag it and a circle appears on the curve. This handle represents one of the variables in the equation. You can click and drag this handle around the drawing.

Click and drag another place on the function and another moveable handle appears if there is another variable in the equation.

This feature is a wonderful way to understand exactly how the equation represents the function.

In the example above, X=T-asin(T), Y=b-cos(T), a click and drag gives you the b handle, and lets you drag the function up and down.

Click and drag another place on the curve and you get the a handle to change the shape of the curve.

Notice the Variable toolbox displays the changing values as you move the handles.

Modify the function after it's drawn by double clicking the function · tag.

Modify the domain of the function after it's drawn by double clicking ·

the curve:

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Piecewise Function A piecewise function or expression can be created using the Piecewise

symbol:

or the built-in function - piecewise({expression1, domain1},{expression2, domain2}...,{last expression, otherwise}). The reserved word, "otherwise" is an option available for the last condition.

Here we show how to enter the sequence of values, followed by

conditions:

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Here's the finished piecewise function:

Piecewise Parametric Example Any of the functions available in Geometry Expressions can be piecewise, including parametric functions. Take this square for example.

1. Click the Draw / Function tool

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Here is the function:

The Selection Arrow When you are finished with the drawing functions, click the selection arrow to enable other functions or adjust your drawing.

With the arrow, you must first select elements of the drawing in order to enter constraints and constructions and to output calculations.

Many of the tools require you to select multiple objects.

The Selection arrow is also available form the icon bar at the top of the

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screen.

Selecting Multiple Objects Many of the Constrain, Construct, and Calculate tools require that you select more than one object.

To select more than one object:

–  –  –

If you click the wrong object while holding down the CTRL key, click it again and it will become unselected.

If all of the objects you wish to select fall within a rectangular region, you

can use the selection box:

Click on the selection arrow on the tool bar or the Drawing · toolbox.

Draw a rectangle around all of the objects by clicking and dragging.

· If you are using Geometry Expressions on a Smartboard, turn on Smartboard mode to make multiple selections without using the Ctrl or Shift keys. In this mode selections accumulate. Clicking a blank space in the drawing does not clear the selections. Clicking on an item a second time unselects only that item.

© 2013 Saltire Software 74 Geometry Expressions Manual Smartboard Mode To turn on the Smartboard mode select Edit / Preferences / Selection;

in the Selection group set Smartboard mode to True.

When set to True, selections accumulate until you click a command.

Clicking a selected item a second time unselects that item, but any other selected items stay selected. To clear all the selections, select Clear Selection under the Edit menu or click all the selected objects again.

Adjusting the Drawing Click the Select arrow, either from the Drawing toolbox or the icon bar, to move, rotate or delete selected object(s) in the drawing.

You can change a constraint value, annotation or label by double clicking it, retyping the value or variable and then pressing the enter key.

The Scale, Zoom and Move / Pan functions have a mouse shortcut:

The scroll wheel on your mouse can be used to Scale the drawing up · or down. Hold down the ctrl key while moving the wheel and the operation becomes a Zoom.

Right-click and drag the mouse anywhere in the drawing window to · Move or Pan.

The Move geometry icon is a modal command. It stays active until you select or choose another mode (e.g. any Draw tool).

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Constraints Using Drawing Constraints After sketching the geometry of a problem, constrain it with measurements, coordinates and implicit equations in real or symbolic terms. The drawing responds automatically to the assigned input constraints. Geometry Expressions will automatically add any constraints you leave out.

Since annotations may look identical to constraints, use the icon,, to Distinguish Constraints / Annotations. The icon is a toggle; to turn off the marks, click it again.

Initially, all the constraints in the toolbox are inactive. You must first select the parts for your drawing that will be constrained. Constraint choices are listed below along with the drawing elements that must be preselected. Be careful when selecting geometry objects, if extra things are selected that are not related to the constraint (like other constraints) the constraints will remain inactive. This can happen by mistake, especially when using the selection box tool.

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Occasionally you may try to add too many constraints to the geometry, causing a conflict. The system will help you correct this problem in the Resolve Constraint Conflict dialog.

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Changing a Constraint To change a constraint, double click it with the selection arrow, retype the value or variable and press the enter key.

Distinguishing Constraints from Annotations Your drawing is defined by Constraints on geometry elements.

Annotations are like labels and have no influence on the geometry measurements. The original default settings (Preferences / Math) fix Constraint lines and text in blue and Annotation lines and text in black.

To further distinguish these use the Distinguish Constraints /

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When active, the button is highlighted -, and a is placed next to all constraints and a is placed next to all annotations.

As with all toggles, you can turn these marks off with another click of the

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© 2013 Saltire Software 78 Geometry Expressions Manual System Added Constraints In Geometry Expressions, you do not need to fully constrain your model.

Any parameters you have left unconstrained are filled in by the geometry engine.

For example, the following drawing is constrained only modulo a rigid transformation (typical for many geometry problems).

So if you ask for the coordinates of any of the points, they will be given in

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Clearly the system has added variables for the location of point A, and for the direction of line AC.

To display these variables, check the default setting, Show System Variables, found in the Edit / Preferences menu selection on the Math page (left panel) under Output.

© 2013 Saltire Software 80 Geometry Expressions Manual When the setting is True, you will see the system variables when you

select the output:

–  –  –

Constraint Conflicts If you enter a constraint for some geometry which is already constrained by another constrained object you will see a message like the one below.

In this case, the Coordinate constraint was already determined by the other two sides and an angle constraint on the triangle. There are three

ways of dealing with this problem:

1. Hit the Cancel button to leave the drawing as it was without the new © 2013 Saltire Software 82 Geometry Expressions Manual constraint.

2. Click the first button, "Calculate the coordinates from other constraints" (the default choice). This is similar to the first choice, it eliminates the new constraint, but also calculates the selected geometry’s value. In this case, the coordinates of point C would be calculated and displayed.

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