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

All mathematical expressions in Geometry Expressions are written in MathML, so you can copy and paste directly to or from any other MathML program.

Usually the standard Edit / Copy or Ctrl-C command should work. If this doesn't give you the desired result, try the Edit / Copy As command.

Select from the submenu of choices appropriate to your application.

If you copy an expression that uses Intermediate Variables, the algebra system will create a function from the pasted expression.

To Paste math into Geometry Expressions it must be Content MathML.

Maple creates Content MathML by default.

· To create Content MathML in Mathematica v6.0, execute the following ·

**commands:**

SetOptions[XML`MathML`ExpressionToMathML, "NamespacePrefixes" {}, "Formats" {"ContentMathML"}] SetOptions[XML`MathML`BoxesToMathML, "NamespacePrefixes" {}, "Formats" {"ContentMathML"}] SetOptions[Export, ConversionOptions {"NamespacePrefixes" {}, "Formats" {"ContentMathML"}}] SetOptions[ExportString, ConversionOptions {"NamespacePrefixes" {}, "Formats" {"ContentMathML"}}] © 2013 Saltire Software VII Some Opening Examples 38 Geometry Expressions Manual Your installation comes with several examples - look in the Examples subdirectory. For step by step instructions through some example sets, see the Geometry Expressions Tutorial.pdf and Geometry Expressions Examples.pdf which can be found under the Doc subdirectory in your Gx installation.

Drawing Using the Drawing Tools The Draw toolbox contains the drawing commands and the Selection Arrow. The drawing commands can also be invoked from the Draw menu.

Unlike commands in the other toolboxes, Drawing commands are always available and require no pre-selection. Many of the commands are active until you change to a different one or press the Esc key or click the selection arrow, e.g. you can make multiple points until you select another tool or press esc. The selection mode, indicated by the selection arrow cursor,, is used for invoking all other commands. You can find the active command mode by noting which button is pressed or looking at the current mode on the status bar.

Before you start your drawing, select Edit / Preferences / Geometry to set various attributes, such as line color / style, fill color / style and transparency level, of the various drawing elements.

You can change the attributes of individual drawing elements by selecting them, right click the mouse, and choose All Properties from the context menu.

Adding a Point

**To insert a point in your drawing, follow these steps:**

2. Move the crosshairs into position.

3. Click the mouse to place the point under the crosshair.

When the crosshairs are positioned over some geometry an incidence symbol (bowtie) is displayed around the point and the geometry is highlighted. A click of the mouse will create the point incident to the highlighted geometry.

© 2013 Saltire Software 42 Geometry Expressions Manual Each point is displayed with a letter label. You can change the label from the Select mode.

Point is a modal command. You can continue making points until you choose the select arrow or another drawing tool.

Adding Line Segments

**To add line segments to your geometry follow these steps:**

1. Click on the Line Segment icon in the Draw toolbox or select Line Segment from the Draw menu.

2. Position the cursor in the drawing window.

3. Click the mouse to place each endpoint.

Each line segment is displayed with a letter label for each endpoint.

You can change the label from the Select mode.

To abort a line segment in the middle of the drawing operation, hit the "esc" key.

Line Segment is a modal command. You can continue making segments until you choose the select arrow or another drawing tool.

Drawing Lines Lines are similar to line segments except they have infinite length.

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

2. Position the line cursor in the drawing window.

3. Click the cursor to anchor the line at the cursor position. The anchor point will be displayed on the line.

4. Move the cursor in the drawing window to position the line and click the cursor when you get the line in the desired orientation.

Lines are infinite and do not have points associated with them unless you specifically place one on the line.

Line is a modal command. You can continue making lines until you choose the select arrow or another drawing tool.

Active Axes The x and y axes have the properties of perpendicular infinite lines. When the crosshairs are positioned over an axis, the incidence symbol (bow tie) is displayed at the intersection and the axis is highlighted. When © 2013 Saltire Software 44 Geometry Expressions Manual the cursor is at the origin, both axes are highlighted. Points and End points of line segments can be placed directly on the axes without using the Constrain / Incident tool when the bow tie is displayed.

Drawing Vectors

**To add vectors to your geometry follow these steps:**

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

2. Position the cursor in the drawing window.

3. Click the mouse to place each endpoint.

Each vector is displayed with a letter label for each endpoint.

Drawing vectors is similar to drawing line segments, but vectors are

**constrained with coefficients of the form:**

Vector is a modal command. You can continue making vectors until you choose the select arrow or another drawing tool.

© 2013 Saltire Software 46 Geometry Expressions Manual Drawing Circles

**To add a circle to your diagram, follow these steps:**

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

2. Move the cursor in the drawing window to the position of the center of the circle and click once.

3. Move the cursor to draw the circle in the desired size and click again.

Notice the circle is displayed with 2 points, the center and a point on the perimeter.

Circle is a modal command. You can continue making circles until you choose the select arrow or another drawing tool.

You can adjust the circle in Select mode.

Circles can be filled with the color / transparency level of your choice. See the Edit /Preferences menu, Geometry tab to set the default. To change the color / transparency level of selected circles, select the circle (s), right-click, and select All Properties from the selection context menu (or Edit / Properties from the menu bar).

Drawing Ellipses

**To create an ellipse in your diagram, follow these steps:**

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

2. Move the cursor in the drawing window to the position of one focal point. Click to place the first focus point. Move the cursor and click again to place the second focal point.

3. Then move the cursor to open the ellipse to the desired shape and click the mouse a third time.

The ellipse will appear with three labeled points, the two foci and a point on the ellipse.

The Ellipse tool is a modal command. You can continue making ellipses until you choose the select arrow or another drawing tool.

Ellipses can be filled with the color / transparency level of your choice. See the Edit /Preferences menu, Geometry tab, Conic properties group to set the default. To change the color / transparency level of selected ellipses, select the ellipse(s), right-click, and select All Properties from the selection context menu (or Edit / Properties from the menu bar).

© 2013 Saltire Software 48 Geometry Expressions Manual Drawing Parabolas

**To create a parabola in your diagram, follow these steps:**

2. Move the cursor in the drawing window to the position of the parabola's vertex. Click and drag the mouse along the major axis.

Release the mouse at the focus.

After sketching the general parabola, you can constrain it in the following

**ways:**

1. Click the parabola and select Implicit equation from the Constrain toolbox and type or paste the formula.

2. Constrain the vertex and focus points to some coordinate values.

3. You can also adjust the parabola with the Translation, Rotation and Dilation tools.

Drawing Hyperbolas

**To create a hyperbola in your diagram, follow these steps:**

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

2. Move the cursor in the drawing window to the position of one focal point. Click to place the first focus. Move the cursor and click again to place the second focal point.

3. Then move the cursor to open the hyperbola to the desired shape and click the mouse a third time.

© 2013 Saltire Software 50 Geometry Expressions Manual The hyperbola will appear with three labeled points, the two foci and a point on the hyperbola.

The Hyperbola tool is a modal command. You can continue making hyperbolas until you choose the select arrow or another drawing tool.

Drawing Arcs Arcs can be placed on any of the conics - circle, ellipse, parabola, hyperbola - or any function. Points are automatically placed at the ends of the arc.

**Here are the steps:**

1. First draw the conic or function which will be the basis for the arc.

2. Choose the Arc tool and move the cursor over the section of the existing curve where the arc will be defined. Click and drag the cursor over the curve.

Endpoints are automatically inserted on the arc.

Curvilinear Polygons The Arc drawing tool lets you make curvilinear polygons for which you can assign fill properties and find areas. There are some limitations, however.

Since you can't construct a point on two intersecting curves (except for circles), you have to connect curves with line segments. If you want to connect two arcs, you have to first connect them with a line and move it

**to the intersection like this:**

**© 2013 Saltire Software 52 Geometry Expressions Manual**

To make a curvilinear polygon of the intersecting parabolas here are the

**steps:**

· Attach two lines.

· Make endpoints C and D -t point proportional along the curves.

· Make endpoints A and B t point proportional along the curves.

· Draw the two arcs - select Draw / Arc, from C to A and D to B.

· Select the sides and arcs of the polygon in order and click Construct / Polygon.

· Set t=2 in the Variables toolbox and the lines will become the intersection points.

Drawing Regular Polygons The N-gon tool lets you quickly draw any regular polygon. You can even work on problems where the number of sides is n, or whatever variable you choose.

**Here are the steps:**

1. Click the N-gon button in the Draw toolbox.

2. Similar to drawing a circle, position the cursor to place the center of the n-gon and click to the desired size. The n-gon at first appears to be a pentagon.

3. In the data entry box, enter the number of sides you want or a variable to represent the number of sides and press Enter.

© 2013 Saltire Software 56 Geometry Expressions Manual Drawing Curve Approximations The Curve Approximation tool will insert a specified number of points and edges evenly spaced on a selected section of a curve or conic. This is a great tool for introducing problems using the Trapezoidal Method of integration. (Take a look at our Gx book, Calculus Explorations.)

**Here are the steps:**

1. Click Draw / Curve Approximation.

2. Select any function, circle, ellipse or parabola and drag the cursor over the curve.

3. In the data entry box type the number of points you want on the arc.

Note: It is best to draw your curve independent of existing points on the curve. Then connect other geometry to the approximation points. If you start or finish the arc with points lying on the curve (e.g. point C in the ellipse above) and later decide to delete the curve approximation, attached geometry may also get deleted.

**Here is the area of a function using the Trapezoidal Method:**

Adding Text to the Drawing

**To add titles or other annotation to the drawing follow these steps:**

2. Position the text cursor at the upper left corner where you would like your window of text located.

3. Click and drag to form your text box to the appropriate size.

© 2013 Saltire Software 58 Geometry Expressions Manual

4. Enter and format your text in the Edit Text dialog.

Enter mathematical statements using the Annotation / Expression tool.

Inserting and Editing Text In the Edit Text dialog you can enter and format the text that will be displayed in your defined text window.

The default text formatting for this dialog is set in Edit / Preferences / Text & Pictures.

Using Pictures in the Drawing Liven up your examples with a picture or two, or use an image for

**reference points in your drawing. Here's how:**

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

2. Click and drag the cursor to delimit the area where you want to place the picture.

3. Find your image in the Select Image File dialog. Image formats include:.bmp,.gif,.jpg,.pcx,.png, and.tif