Geometria 4.0

User's Guide


1. General
1. Quick Start
1.1. Installation
1.2. Troubleshooting
2. License
3. Acknowledgements
4. Overview
4.1. Editors and Solvers
5. Variables
5.1. Creating a Variable
5.2. Value of a Variable
5.3. Pinned Variable
5.4. Lifetime of a Variable
5.5. Rename Variable
6. Calculator
6.1. Expressions
6.2. Operators
6.3. Functions
6.4. Examples
7. Notepad
7.1. Clear Notepad
8. Undo Actions
9. Redo Actions
10. Language
11. Help
12. Known Limitations
12.1. Inordinate Size or Shape
12.2. Previous Versions of Geometria
12.3. Solution Language
2. Documents
1. Problem Text
2. Document Properties
3. Answer
3.1. Editor's Answer
3.2. Solver's Answer
4. Navigator
4.1. New Folder
4.2. Rename File
4.3. Move File
4.4. Delete File
5. New Problem
6. Open Problem
7. Save Problem
8. Lock Problem
9. Solve Problem
10. Open Solution
11. Save Solution
12. Solution Log
12.1. Playback
12.2. Solution Comments
13. Share Document
13.1. Export Document
13.2. Import Document
3. Figures
1. New Figure
2. Open Figure
3. Clone Figure
4. Save Figure
5. Print Figure
6. Rename Figure
7. Rename Point
8. Remove Figure
9. Erase Elements
9.1. Erase Segment
9.2. Erase Selection
10. View
10.1. Attitude
10.2. Rotate Figure
10.3. Select Elements
10.4. View Modes
10.5. Zoom Figure
11. Measure
11.1. Measure Distance
11.2. Measure Angle
11.3. Compute Area
11.4. Compute Total Area
11.5. Compute Volume
12. Draw
12.1. Join Points
12.2. Draw Perpendicular
12.3. Divide Segment
12.4. Draw Midpoint
12.5. Divide Angle
12.6. Draw Bisector
12.7. Draw Intersection
12.8. Lay Distance
12.9. Lay Angle
13. Transform
13.1. Scale Figure
13.2. Shear Figure
13.3. Cut Figure
13.4. Join Figures

Chapter 1. General

1. Quick Start

1.1. Installation

Geometria runs in a web browser on your desktop or laptop. Supported web browsers are:

  • Firefox
  • Google Chrome
  • Internet Explorer 10+

To install Geometria on your computer, download it from: http://geocentral.net/geometria/download and unzip the file into a folder. Open that folder.

1.1.1. Windows Users

To start Geometria:

  • Double click on Startup.exe.
  • Open a web browser and point it to localhost:8080 .

To stop Geometria, find the icon on your system tray, click it and select Exit.

1.1.2. Linux Users

To start Geometria:

  • Open a terminal.
  • Locate the startup file corresponding to your operating system (32-bit or 64-bit).
  • Make the startup file executable. For example:
    $ chmod +x Startup_Linux64
    
  • Execute the startup file. For example:
    $ ./Startup_Linux64
    
  • Open a web browser and point it to localhost:8080 .

To stop Geometria, go back to the terminal and stop the process with Ctrl + C.

1.2. Troubleshooting

1.2.1. Port

By default, Geometria starts a tiny web server listening on port 8080. If the browser shows an error or unexpected content, there might be another process on your computer listening to port 8080. In that case, stop Geometria as shown above and try another port by changing the listening_ports (Windows) or, respectively, listening_port (Linux) value in the mongoose.conf file. For example:

listening_ports 8081
then point your web browser to localhost:8081.

1.2.2. Access Denied

Despite the fact, that Geometria needs no Internet connection to run, you may get an Access Denied message if you are behing a firewall. That is because your system administrator has configured a web proxy that blocks access to port 8080 even for a web server running from your own computer. Try disabling the proxy for the duration of your working with Geometria, then enabling it again when you are done. Note that the external web links present in Geometria documentation, problem text and document properties may stop working while the proxy is disabled.

1.2.3. Font

Geometria employs the default sans-serif font for its user interface. The font size can be control through the browser's settings as follows:

  • On Firefox, check Zoom Text Only in the View/Zoom menu. To make the font larger, press + while holding the Ctrl key down. To make the font smaller, press - while holding the Ctrl key down. To revert to the default font size, select Reset in the View/Zoom menu.
  • On Chrome, open the Settings, search for Font and change Font size as desired.
  • On Internet Explorer, select the desired font in the View/Text Size menu.

1.2.4. Firebug

If your web browser is Firefox, make sure Firebug is turned off or, at least, its Script Panel is disabled. When the Script Panel is on, Firebug reports certain scripting errors, which are a known issue unrelated to Geometria.

2. License

Geometria is free software released under the MIT License.

3. Acknowledgements

Geometria makes use of the following open-source projects:

4. Overview

Geometria provides a graphic interface for creating and solving problems in solid geometry.

4.1. Editors and Solvers

At any given time, the Geometria user is either a problem Editor (a.k.a. Teacher) or a problem Solver (a.k.a. Student).

Both problems and solutions are called documents. Documents are saved in files. Users engage in a Geometria process as follows:

Note that Geometria does not perform any semantic analysis of the problem in order to determine if the Editor's answer is actually correct. The answer may be incorrect or even make no sense at all. The problem may make no sense either. Thus, a problem might ask:

What is the highest point in Arizona?

and be answered as:

Point Z in figure MyPyramid.

If that is what the Editor considers a correct answer to the problem, it's fine with Geometria, although it is unlikely that such a problem will get much attention from solvers.

Every document contains the problem text, document properties, a number of figures, and a number of notepad records. Additionaly, a solution contains the solution log.

Figures are limited to convex polyhedra. Figures can be measured, drawn upon, transformed, rotated, viewed as wireframe or labeled, printed.

Drawings can only be made on the surface of a figure. Drawings in the figure's interior are possible by, first, cutting the figure with a suitable plane, then drawing on a fragment's surface.

Labels and variables provide a convenient abstraction over numeric coordinates and measurements. Variables are used in calculations and referenced in measurements and drawings. Labels and variables are also used to answer the problem.

The answer is not necessarily numeric. In fact, some of the most challenging problems cannot be answered with a number. The problem may ask for a certain point, segment, path or plane to be constructed. For example:

Find a plane that is parallel to the diagonal of the cube and cuts its volume in the ratio 2:3.

See also:

5. Variables

5.1. Creating a Variable

A variable is created as a result of a measurement or a calculation. A newly created variable has a name and a definition. For instance, the variable myDistance in the example below is defined as the distance between points P and W in figure MyCube:

myDistance = |PW| : MyCube

In another example, variable myArea is a result of a calculation upon previously defined variables myVolume and myDistance:

myArea = 2 * myVolume / myDistance

Newly created variables are automatically recorded, along with their definitions, on the notepad.

A variable name must consist of alphanumeric characters and begin with a letter. Special and accented characters are not allowed. A variable name cannot contain spaces.

Variable names are case sensitive. Thus, myArea and MyArea will reference two distinct variables.

PI is reserved for the ratio of a circle's circumference to its diameter and cannot be used for newly created variables. However, it can be used in calculations.

5.2. Value of a Variable

Each variable has a numeric value. However, in most cases, this value remains transparent to the user. One can create and solve Geometria problems without ever knowing the underlying values of the variables one has defined. To see the numerical value of a variable, type the variable in the calculator and press the Enter key. Due to roundup errors, the numeric values may slightly vary in time, as you go on editing or solving the problem.

5.3. Pinned Variable

After certain actions, the definition of a variable may become obsolete. For example, if point P or point W is erased, the definition of the variable myDistance above will become obsolete. Same if figure MyCube is transformed or removed. Such a variable is called pinned.

A pinned variable has a fixed value no longer dependent upon its original definition. For example, erasing point P in figure MyCube will pin variable myDistance. Likewise, scaling MyCube will pin variable myDistance. Its value will be unaffected by scaling.

If the action that pinned a variable is undone, the definition of the variable is restored.

5.4. Lifetime of a Variable

Once a variable is defined, it can be used in calculations and referenced in actions, such as measuring, drawing and transforming figures. A variable can also be used to answer a problem. A pinned variable can still be used in calculations and references.

A variable exists as long as it is recorded on the notepad. Clearing the notepad will remove all variables. Use caution since this action cannot be undone.

When the document, such as a problem or a solution, is saved, all the variables recorded on the notepad are saved, too.

See also:

5.5. Rename Variable

To rename a variable, select Rename Variable from the Edit menu. A dialog will pop up. Select the variable you wish to rename from the drop-down list, enter the new name for the variable and click OK. The new name must be a valid variable name.

Renaming a variable will result in some of the notepad records being updated accordingly.

This action can be undone.

Variables cannot be renamed during the solution playback. You have to stop the playback first.

See also:

6. Calculator

The calculator is located at the bottom of the window, just under the notepad.

6.1. Expressions

The calculator operates in the context of the current document. Calculations are performed upon valid expressions. Expressions can contain numbers, variables, the constant PI, operators, functions and parentheses.

6.1.1. Assign

To evaluate an expression myExpression and assign the result to a new variable myVariable, type in:

myVariable = myExpression

and press Enter. The variable name must be unique in the context of the current document. A new record will be added to the notepad. The newly created variable can be used in subsequent calculations and references.

6.1.2. Evaluate

To evaluate the expression without assigning it to a variable, type in:

myExpression

and press Enter. The numeric value of myExpression will be displayed. No new variable will be created.

6.2. Operators

The following operators are supported:

  • + (add)
  • - (negate, subtract)
  • * (multiply)
  • / (divide)
  • ^ (power operator)

6.3. Functions

The following functions are supported:

  • sqrt (square root)
  • sin
  • cos
  • tan
  • asin
  • acos
  • atan

Function arguments must be taken into parentheses.

6.4. Examples

Calculate the side of a cube whose volume is myVolume:

mySide = myVolume ^ (1/3)

Calculate the side adjacent to angle myAngle in a right triangle whose hypothenuse is myHypothenuse:

mySide = myHypothenuse * cos(myAngle)

See also:

7. Notepad

The notepad is located under the problem text.

Each time a new variable is created as a result of a measurement or assigned in a calculation, a record is added to the notepad automatically. Each record consists of a unique variable name and an expression. Measurement records also contain the name of the figure where the measurement has been taken. For example, the following record:

myDistance = |AB| : MyCube

is the result of measuring the distance between points A and B in a figure named MyCube. The distance has been assigned to variable myDistance. The record:

myAngle = <PLX : MyPyramid

is a result of measuring an angle in a figure named MyPyramid. The record:

myVolume = volume : Figure1

is the result of computing the volume of a figure named Figure1. The following record was added as the result of an assignment:

h = 2 * myVolume / myArea

If a variable is pinned, its definition no longer shows on the notepad. For example, if point X is erased, variable myAngle will be stripped of its definition and the second example above will read:

myAngle

Notepad records are owned by the current document, whether it is a problem or a solution. When the document is saved, the notepad records will be saved with it.

When the current document is closed, the notepad will be cleared of records. If another document is opened, its notepad records will be displayed instead.

7.1. Clear Notepad

To erase all records from the notepad, the Editor selects Clear Notepad from the Edit menu. Use caution since this action cannot be undone.

See also:

8. Undo Actions

To undo the last action performed on the current document, select Undo from the Edit menu or click the button on the tool bar. In a solution, the action will be also erased from the log. The actions are undone in the reverse order as they have been performed.

To view a short description of the action that would be undone, pause the mouse over the button.

Any undone action can be redone.

Some of the actions cannot be undone. A warning is displayed when such an action is about to take place.

Actions cannot be undone during the solution playback. You have to stop the playback first.

See also:

9. Redo Actions

To redo the last undone action on the current document, select Redo from the Edit menu or click the button on the tool bar. In a solution, the action will be added back to the log. The actions are redone in the reverse order as they have been undone.

To view a short description of the action that would be redone, pause the mouse over the button.

Actions cannot be redone during the solution playback. You have to stop the playback first.

See also:

10. Language

To switch to a different language, select Language from the Document menu, then select the desired language.

The language cannot be changed during the solution playback. You have to stop the playback first.

11. Help

Many action dialogs feature a Help button. Click that button for context-sensitive help. The relevant topic will be displayed on the right side of the window. Click the and buttons to navigate the previously viewed help topics. Click the button to go to the table of contents.

To hide the help pane, click the button. To bring it back, click the button again.

12. Known Limitations

12.1. Inordinate Size or Shape

Geometria may behave erratically if the figures contain excessively sized or oddly shaped elements. Here are some guidelines for keeping the figures safe:

  • Avoid both very large (over 100) and very small (under 0.01) scaling factors.
  • When creating a new, prism or pyramid, keep the number of sides within reasonable limit. By design, Geometria cannot handle any solids but convex polyhedra. However prisms and pyramids with a large number of sides do look like cylinders and cones respectively when the wireframe is off. Unfortunately, actions upon such figures may be slow and unimpressive. The application may become unstable and fail to respond to user actions.

12.2. Previous Versions of Geometria

Problems, solutions and figures created with Geometria versions prior to 4.0 are no longer supported. If you need help, please visit the Geometria home page and email the development team.

12.3. Solution Language

Solution files must be opened using the same language in which they have been saved.

Chapter 2. Documents

1. Problem Text

The text of the problem is located under the tool bar, on the right side of the Navigator.

The Editor can edit the problem text by double-clicking the text area. The problem text supports HTML markup. Click away to preview the changes to the problem text.

2. Document Properties

To view the properties of the current document, select Document Properties from the Document menu or click the button on the tool bar. A dialog will pop up.

To edit the properties of the problem, the Editor double clicks the problem properties area. When done editing, click OK. Click Cancel to ignore the changes.

The Solver will see both the properties of the problem, at the top, and the properties of the solution, at the bottom. To edit the properties of the solution, the Solver double clicks the solution properties area. When done editing, click OK. Click Cancel to ignore the changes. The Solver cannot edit the properties of the problem.

Both the problem and the solution properties support HTML markup.

Changes to the document properties cannot be undone.

Solution properties cannot be edited during the solution playback. You have to stop the playback first.

See also:

3. Answer

A problem is not ready to be solved until the Editor supplies it with an answer. The Solver's answer is considered correct if it matches the Editor's. See the Overview for details.

3.1. Editor's Answer

To edit the answer to the current problem, select Answer from the Document menu.

3.1.1. Answer Types

Select the answer type under the Type tab:

  • Number
  • Multiple choice
  • Point Set
  • Segment Set
  • Fixed Plane, or
  • Condition Plane

3.1.2. Answer Values

Click the Value tab. Depending on the answer type, proceed as directed in the sections below.

Note that although, as a rule, the Editor enters the answer symbolically, Geometria stores answers numerically. This refers to both variables and points. Variables are stored as numbers, points are stored as coordinates, that is triplets of numbers. Once entered, the answer will be unaffected by all subsequent actions upon figures. Therefore it is a good idea to (re)enter the answer as a last action before saving the problem and making it available to solvers.

3.1.2.1. Number

Enter a valid expression. Click OK to keep the changes to the answer. Click Cancel to discard the changes.

3.1.2.2. Point Set

The term point set refers to a single point, end points of a segment (2 points) or a path (2 or more points).

Enter one or more labels separated by commas. To minimize the input, have the points selected ahead of time in the selected figure.Click OK to keep the changes to the answer. Click Cancel to discard the changes.

3.1.2.3. Segment Set

The segments must satisfy a certain condition. Currently, the following conditions are supported:

Select the target figure, then select a condition. Click OK to keep the changes to the answer. Click Cancel to discard the changes.

3.1.2.4. Fixed Plane

Reference the plane by 3 comma-separated labels. To minimize the input, have some reference elements, such as 3 points, a point and a segment, two segments, or a face selected ahead of time in the selected figure. Click OK to keep the changes to the answer. Click Cancel to discard the changes.

3.1.2.5. Condition Plane

Unlike a fixed plane, which is a plane occupying a fixed position in space, a condition plane must satisfy a number of conditions. Generally, a condition plane is not unique.

First, select the target figure. Then select the conditions. Select a condition in at least one of the groups.Click OK to keep the changes to the answer. Click Cancel to discard the changes.

There are three groups of conditions.

The first group consists of the following conditions:

  • Plane contains a given point
  • Plane contains a given line
  • Plane does not contain a given point
  • Plane does not contain a given line
  • Plane contains no vertex of the selected figure
  • Plane contains no edge of the selected figure
  • Plane is parallel to a given line
  • Plane is parallel to a given plane
  • Plane is perpendicular to a given line
  • Plane is perpendicular to a given plane

Reference the given point, line or plane, as required, by comma-separated labels. To minimize the input, have the elements selected ahead of time in the selected figure.

The second group consists of the following conditions:

  • Plane cuts the volume of the selected figure in the given ratio

Enter valid expressions for the ratio's numerator and denominator.

The third group refers to the way the plane cuts the figure. The section may be shaped as:

  • Isoscelles triangle
  • Right triangle
  • Equilateral (regular) triangle
  • Parallelogram
  • Rhombus
  • Rectangle
  • Square

The changes to the answer can be undone.

3.1.2.6. Multiple choice

Enter an option. Click the Add button to add another option. Click on any option to highlight it. Click the Select button to select the highlighted option as the correct option. The correct option is displayed in bold face. Click the Remove button to remove the highlighted option.

Click OK to keep the changes to the answer. Click Cancel to discard the changes.

3.2. Solver's Answer

The Solver concludes the solution by providing a correct answer to the problem. Note, again, that the Solver's answer is considered correct if it matches the Editor's.

Select Answer from the Document menu. Enter a valid expression or comma-separated labels or make a selection, according to what the problem is asking for. If the problem is asking for a point, a point set, a segment set or a plane, have some appropriate elements, such as points, segments or a face selected ahead of time in the selected figure. Click OK.

An evaluation of the answer (corect or incorrect) will pop up.

If the answer is correct, it will conclude the solution log. No more actions will be possible unless the answer is undone. Save the solution so that it can be played back later on.

This action can be undone.

The problem cannot be answered during the solution playback. You have to stop the playback first.

See also:

4. Navigator

The Navigator allows you to manipulate problem, solution and figure files and organize them into folders. Both the files and the folders are stored on your computer in a single SQLite file.

The Navigator is initially displayed on the left side of the window. To hide the Navigator, select Toggle Navigator from the View menu or click the icon on the tool bar. To bring the Navigator back, click the same icon again.

You can tell the content of a file by its icon. Problem files have the  icon. Solution files have the  icon. Figure files have the  icon.

4.1. New Folder

To create a new folder, click the icon on the toolbar, then enter the folder name in the box at the bottom of the navigator. Folder names can only contain alphanumeric characters and spaces. Note that the subfolders and files in any given folder must have unique names.

The new folder will be added to the currently selected folder or, if no folder is selected, to the root folder.

New folders cannot be created during the solution playback. You have to stop the playback first.

4.2. Rename File

To rename the selected file or folder, click the button on the tool bar.

Folder and file names can only contain alphanumeric characters and spaces. Note that the subfoders and files in any given folder must have unique names.

Files and folders cannot be renamed during the solution playback. You have to stop the playback first.

4.3. Move File

To move a file or a folder to another folder, drag and drop it onto the desired folder.

Note that the subfoders and files in any given folder must have unique names.

Files and folders cannot be moved during the solution playback. You have to stop the playback first.

4.4. Delete File

To delete the selected file or folder, click the icon on the tool bar. Use caution as this action cannot be undone.

Files and folders cannot be deleted during the solution playback. You have to stop the playback first.

5. New Problem

The Editor creates a problem by either selecting New Problem from the Document menu or by clicking the button on the tool bar.

The current document, whether it is a problem or a solution, will be closed first. If there are unsaved modifications, a dialog will be displayed asking if the current document should be saved before it is closed.

Click Yes to save the current document.

Click No to close the current document without saving. Note that all unsaved modifications to the current document will be lost.

Click Cancel to dismiss the dialog with no action taken.

A new problem cannot be created during the solution playback. You have to stop the playback first.

See also:

6. Open Problem

To open a problem, double click it in the Navigator or select it and click the button on the tool bar. You can tell a problem file by its icon  .

The current document, whether it is a problem or a solution, will be closed first. If there are unsaved modifications, a dialog will be displayed asking if the current document should be saved before it is closed.

Click Yes to save the current document.

Click No to close the current document without saving. Note that all unsaved modifications to the current document will be lost.

Click Cancel to dismiss the dialog with no action taken.

A problem cannot be opened during the solution playback. You have to stop the playback first.

See also:

7. Save Problem

To save the changes to the current problem, the Editor either selects Save Document from the Document menu or clicks the button on the tool bar.

To save the problem under a different name, select Save Document As from the Document menu, then enter the file name in the box at the bottom of the navigator. File names can only contain alphanumeric characters and spaces. Note that the subfolders and files in any given folder must have unique names. The problem will be saved in the folder currently selected in the Navigator. If no folder is selected, the problem will be saved in the root folder.

If this is a final revision of the problem, the Editor may consider erasing all working calculations and measurements from the notepad so that solvers cannot see them.

Beware, though, that the action of clearing the notepad cannot be undone.

A final revision of the problem should also be provided with an answer.

The problem cannot be saved during the solution playback. You have to stop the playback first.

See also:

8. Lock Problem

To prevent any further editing of the problem, the Editor selects Lock Problem from the Document menu or clicks the button on the tool bar. When the Solver opens a locked problem file, Geometria will automatically start a new solution. The Solver will not be able to modify the problem itself.

Use caution and save an editable copy of the problem before locking it as this action cannot be undone.

9. Solve Problem

To solve the current problem or start a new solution, if the current problem is already being solved, select Solve Problem from the Document menu or click the on the tool bar.

If there are unsaved modifications, a dialog will be displayed asking if the current problem should be saved before it is closed.

Click Yes to save the current problem. Note that you will be solving the version of the problem you just saved.

Click No to close the current problem without saving. Note that all unsaved modifications to the current problem will be lost and you will be solving a problem different from the one you opened in the Navigator.

Click Cancel to dismiss the dialog with no action taken.

A new solution to the current problem cannot be started during the solution playback. You have to stop the playback first.

See also:

10. Open Solution

To open a solution, double click on it in the Navigator or select it and click the button on the tool bar. You can tell a solution file by its icon  .

Once the solution is opened, it can be played back.

The current document, whether it is a problem or a solution, will be closed first. If there are unsaved modifications, a dialog will be displayed asking if the current document should be saved before it is closed.

Click Yes to save the current document.

Click No to close the current document without saving. Note that all unsaved modifications to the current document will be lost.

Click Cancel to dismiss the dialog with no action taken.

A solution cannot be opened during another solution's playback. You have to stop the playback first.

See also:

11. Save Solution

To save the changes to the current solution, the Solver either selects Save Document from the Document menu or clicks the button on the tool bar.

To save the solution under a different name, select Save Document As from the Document menu, then enter the file name in the box at the bottom of the navigator. File names can only contain alphanumeric characters and spaces. Note that the subfolders and files in any given folder must have unique names. The solution will be saved in the folder currently selected in the Navigator. If no folder is selected, the solution will be saved in the root folder.

If this is a final revision of the solution, do not forget to conclude it with a correct answer.

The solution cannot be saved during its playback. You have to stop the playback first.

See also:

12. Solution Log

12.1. Playback

As the Solver progresses through the solution, his actions are recorded in the solution log. A saved solution can later be opened and played back.

To start the solution playback, click the button. To play back the next step, click the button again.

To jump forward or backward, select an arbitrary record in the log.

To erase all records following the selected record, click the button. Use caution since this action cannot be undone.

To rewind the solution playback so that you can start over, click the button.

To stop the solution playback and resume solving the problem, click the button.

12.2. Solution Comments

To comment on a solution step, double click on the small box on the right side of the record. A dialog will pop up. Type in some comments and click OK. Click Cancel to discard the changes. The comments support HTML markup.

A commented solution is marked with the  icon. To view the comments, pause the mouse over the corresponding record.

Comments cannot be edited during the solution playback. You have to stop the playback first.

See also:

13. Share Document

To share a document with another user, you need to export it first. Save the exported content to a regular file on your computer. Share the saved file. The other user will open the file and import its content.

13.1. Export Document

To export the current document, select Export Document from the Document menu. The JSON-formatted content of the document will pop up. Click in the text area, select all, copy, and paste into a text editor. Save the file in plain text format. The saved file can be shared with other users.

A solution cannot be exported during its playback. You have to stop the playback first.

13.2. Import Document

To import the document another user has shared with you, open the file in a text editor, select all and copy. Select Import Document from the Document menu. Paste into the text area and click OK.

The current document, whether it is a problem or a solution, will be closed first. If there are unsaved modifications, a dialog will be displayed asking if the current document should be saved before it is closed.

Click Yes to save the current document.

Click No to close the current document without saving. Note that all unsaved modifications to the current document will be lost.

Click Cancel to dismiss the dialog with no action taken.

A document cannot be imported during the solution playback. You have to stop the playback first.

Chapter 3. Figures

1. New Figure

Geometria offers a gallery of common figures, such as regular prisms, regular pyramids and the five platonic solids, namely, the tetrahedron, cube, octahedron, dodecahedron and icosahedron.

Select a figure from the Figure->New Figure menu to add it to the current document.

When selecting a prism or a pyramid, enter a number of sides between 3 and 100.

This action can be undone.

A new figure cannot be added during the solution playback. You have to stop the playback first.

See also:

2. Open Figure

To open a figure file, double click it in the Navigator or select it and click the button on the tool bar. You can tell a figure file by its icon  .

The figure will be added to the current document.

In addition, a gallery of common figures is available to choose from, including prisms, pyramids and platonic solids.

A new figure cannot be added during the solution playback. You have to stop the playback first.

This action can be undone.

A figure cannot be opened during the solution playback. You have to stop the playback first.

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3. Clone Figure

To clone a figure, select Clone Figure from the Figure menu or click the button on the tool bar.

A copy of the selected figure will be added to the current document.

This action can be undone.

Figures cannot be cloned during the solution playback. You have to stop the playback first.

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4. Save Figure

To save a figure, select it in the current document, then select Save Figure from the Figure menu or click the button on the tool bar. Enter the file name in the box at the bottom of the navigator. File names can only contain alphanumeric characters and spaces. Note that the subfolders and files in any given folder must have unique names.

The figure will be saved in the folder currently selected in the Navigator. If no folder is selected, the figure will be saved in the root folder.

A figure cannot be saved during the solution playback. You have to stop the playback first.

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5. Print Figure

To print a figure:

  • Select Print Figure from the Figure menu. An HTML-formatted text will pop up.
  • Click in the text area, select all and copy.
  • Paste into a text editor.
  • Save to a file in plain text format with extension html, for example, MyFigure.html.
  • Drag the file into your web browser.
  • Scale the figure if necessary and print it.

6. Rename Figure

To rename a figure in the current document, select it, then select Rename Figure from the Edit menu.

A figure name must consist of alphanumeric characters and begin with a letter. Special and accented characters are not allowed. A figure name cannot contain spaces.

This action can be undone.

The Solver cannot rename a figure that is part of the problem. However, any figures created in the process of solving the problem can be renamed.

A figure cannot be renamed during the solution playback. You have to stop the playback first.

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7. Rename Point

To rename a point in a figure, the Editor  selects the point, right-clicks and selects Rename Point from the popup menu. Alternatively, Rename Point can be selected in the Draw menu.

A dialog will pop up. Make sure the correct point is selected, enter a new label for the point and click OK. A valid label consists of a capital letter, which may be followed by one or more digits. Point labels in a figure must be unique.

Renaming a point may result in some of the notepad records being updated accordingly.

The Solver cannot rename a point in any figure that is part of the problem. However, points in figures that have been created in the process of solving the problem can be renamed.

This action can be undone.

Points cannot be renamed during the solution playback. You have to stop the playback first.

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8. Remove Figure

To remove a figure from the current document, select Remove Figure from the Edit menu or click button on the tool bar.

Removing a figure that has been measured will result in some of the variables being pinned.

This action can be undone.

The Solver cannot remove a figure that is part of the problem. However, any figures created in the process of solving the problem can be removed.

A figure cannot be removed during the solution playback. You have to stop the playback first.

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9. Erase Elements

9.1. Erase Segment

To erase a segment, select either the segment or the end points between which to erase, right-click and select Erase Segment. from the popup menu. Alternatively, select Erase Segment from the Edit menu. The segment cannot belong to an edge.

If a dialog pops up, fill in the missing point labels and click OK.

Erasing a segment that has been referenced in measurements will result in some of the variables being pinned.

This action can be undone.

A segment cannot be erased during the solution playback. You have to stop the playback first.

9.2. Erase Selection

To erase one or more elements, such as points and/or segments, select them, right-click and select Erase Selection from the popup menu. Alternatively, select Erase Selection from the Edit menu or click the button on the tool bar.

Vertices and segments that belong to edges will not be erased.

To erase all points and segments in a figure (with the exception of vertices and edges), use the Select All action first.

To erase only a part of a drawn segment, use the Erase Segment action.

Erasing elements that have been referenced in measurements will result in some of the variables being pinned.

This action can be undone.

The selection cannot be erased during the solution playback. You have to stop the playback first.

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10. View

10.1. Attitude

Geometria employs a fixed right-hand reference system. The Z axis of the reference system is parallel to the vertical edge of the screen. However, Editors and Solvers require no knowledge of the technical aspects of the reference system.

At any given time, the figure has a certain attitude. The attitude can be described in terms of angles the figure forms with the reference system. When a new figure is added to the document, it has the default attitude. The figure's attitude changes as the figure is rotated.

To revert a figure to the initial attitude, select Initial Attitude from the View menu or right-click and select Initial Attitude from the popup menu.

When a figure is saved, its current attitude becomes its initial attitude. To revert to the attitude the figure had when it was first added to the document, before any subsequent rotations, select Default Attitude from the View menu.

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10.2. Rotate Figure

Turn the selector off and stroke a figure with the mouse to rotate the figure. The rotation speed and angle will depend on the stroke.

To pin a rotating figure, click on it.

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10.3. Select Elements

10.3.1. Toggle Selector

Right-click on a figure and select Toggle Selector from the popup menu. Alternatively, select Toggle Selector from the View menu or click the button on the tool bar.

When the selector is on (its tool bar button highlighted), elements can be selected by clicking on them. When the selector is off, dragging the mouse will rotate the figure.

The selector is on by default.

10.3.2. Select and Deselect

Make sure the selector is on.

Click an element, such as a point, a segment or a face, to select (highlight) it. Any number of elements can be selected. To select a face, turn the wireframe off.

To select all points and segments in a figure (with the exception of vertices and edges), select Select All from the Edit menu. This action offers a convenient way to erase all elements at once.

To deselect a selected element, click on it again. To clear the selection completely, right-click and select Deselect All from the popup menu or select Deselect All from the Edit menu.

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10.4. View Modes

10.4.1. Wireframe

By default, figures are displayed as wireframes and the button on the tool bar is highlighted. To turn the wireframe off on the selected figure, select Toggle Wireframe from the View menu or click the button.

In a wireframe, the segments and points on the visible side of the figure are solid, whereas the ones on the back side are dashed. Any point or segment can be selected, whether it is visible or not. Faces can only be selected when the wireframe is off.

If the wireframe is off, the segments and points on the back side of the figure are not shown. Only points, segments and faces on the visible side of the figure can be selected.

10.4.2. Set Color

When the wireframe is off, the figure is painted in a certain color. To change the color, select Set Color from the View menu or click the button on the tool bar. Select a color in the popup dialog and click OK.

The color of a figure cannot be changed during the solution playback. You have to stop the playback first.

10.4.3. Labels

By default, figure points are labeled. To hide the labels, select Toggle Labels from the View menu or click the button on the tool bar. Labels will not show if the wireframe is off.

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10.5. Zoom Figure

To zoom in on a figure, select Zoom In from the View menu or click the button on the tool bar.

To zoom out, select Zoom Out from the View menu or click the button on the tool bar.

To resize a figure so it fits into the window, right-click and select Fit To View from the popup menu. Alternatively, select Fit To View from the View menu or click the button on the tool bar.

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11. Measure

11.1. Measure Distance

To measure the distance between two points, select both points or the segment that joins them, right-click and select Measure Distance from the popup menu. Alternatively, select Measure Distance from the Measure menu, or click the button on the tool bar.

A dialog will pop up. Fill in, if necessary, the missing point labels, assign a unique variable and click OK. A record will appear on the notepad, for example:

myDistance = |AB| : MyCube

where myDistance is the variable assigned to the distance between points A and B in figure MyCube.

This action can be undone.

Distances cannot be measured during the solution playback. You have to stop the playback first.

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11.2. Measure Angle

To measure an angle, select two segments that make it, right-click and select Measure Angle from the popup menu. Alternatively, select Measure Angle from the Measure menu, or click the button on the tool bar.

A dialog will pop up. Fill in, if necessary, the missing point labels, assign a unique variable and click OK. A record will appear on the notepad, for example:

myAngle = <ABC : MyPyramid

where myAngle is the variable assigned to the angle between AB and AC in figure MyPyramid.

All angles are expressed in radians.

This action can be undone.

Angles cannot be measured during the solution playback. You have to stop the playback first.

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11.3. Compute Area

To compute the area of a face, select some elements that identify uniquely that face, for example, two segments or three points, right-click and select Compute Area from the popup menu. Alternatively, select Compute Area from the Measure menu, or click the button on the tool bar.

A dialog will pop up. Make sure the face selection is correct, assign a unique variable and click OK. A record will appear on the notepad, for example:

myArea = area[ABC] : MyCube

where myArea is the variable assigned to the area of the face that contains points A, B, C in figure MyCube.

This action can be undone.

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Areas cannot be computed during the solution playback. You have to stop the playback first.

11.4. Compute Total Area

To compute the total area of a figure's surface, select Compute Total Area from the Measure menu.

A dialog will pop up. Assign a unique variable and click OK. A record will appear on the notepad, for example:

myArea = total area : MyCube

where myArea is the variable assigned to the total area of figure MyCube.

This action can be undone.

The total area cannot be computed during the solution playback. You have to stop the playback first.

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11.5. Compute Volume

To compute the volume of a figure, select Compute Volume from the Measure menu, or click the button on the tool bar.

A dialog will pop up. Assign a unique variable and click OK. A record will appear on the notepad, for example:

myVolume = volume : MyCube

where myVolume is the variable assigned to the volume of figure MyCube.

This action can be undone.

The volume cannot be computed during the solution playback. You have to stop the playback first.

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12. Draw

12.1. Join Points

To join two points with a segment, select both points, right-click and select Join Points from the popup menu. Alternatively, select Joint Points from the Draw menu, or click the button on the tool bar.

The two points must belong to the same face.

If a dialog pops up, fill in the missing point labels and click OK.

This action can be undone.

Points cannot be joined during the solution playback. You have to stop the playback first.

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12.2. Draw Perpendicular

To draw the perpendicular through a point to a segment, select the point and the segment, then select Draw Perpendicular from the popup menu. Alternatively, select Draw Perpendicular from the Draw menu, or click the button on the tool bar.

The point and the segment must belong to the same face. The point may or may not belong to the segment.

If a dialog pops up, fill in the missing point labels and click OK.

If both the point and the segment belong to two different faces at the same time, a perpendicular in each of those faces may be drawn.

This action can be undone.

Perpendiculars cannot be drawn during the solution playback. You have to stop the playback first.

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12.3. Divide Segment

To divide a segment in a given ratio, select the segment or its end points, right-click and select Divide Segment from the popup menu. Alternatively, select Divide Segment from the Draw menu, or click the button on the tool bar.

A dialog will pop up. Fill in, if necessary, the missing point labels and type in valid expressions for the numerator and denominator of the ratio. Click OK.

The ratio is defined as AX : XB, where X is the division point, AB the segment. For example, if the ratio is 1 : 2, X will be twice closer to A than to B. If the ratio is 3 : 1, X will be three times closer to B than to A.

The midpoint divides the segment in the ratio 1 : 1 and therefore can be drawn as shown above. However, a convenience Draw Midpoint action offers a shortcut.

This action can be undone.

A segment cannot be divided during the solution playback. You have to stop the playback first.

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12.4. Draw Midpoint

To draw the midpoint of a segment, select the segment or its end points, right-click and select Draw Midpoint from the popup menu. Alternatively, select Draw Midpoint from the Draw.

If a dialog pops up, fill in the missing point labels and click OK.

Note that drawing the midpoint is equivalent to dividing the segment in the ratio 1 : 1.

This action can be undone.

Midpoints cannot be drawn during the solution playback. You have to stop the playback first.

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12.5. Divide Angle

To divide an angle in a given ratio, select two segments that make the angle, right-click and select Divide Angle from the popup menu. Alternatively, select Divide Angle from the Draw menu, or click the button on the tool bar.

The segments that make the angle must belong to the same face.

A dialog will pop up. Fill in, if necessary, the missing point labels and type in valid expressions for the numerator and denominator of the ratio. Click OK.

The ratio is defined as <AOX : <XOB, where OX is the division segment, <AOB the angle. For example, if the ratio is 1 : 2, OX will make with OB a twice greater angle than with OA. If the ratio is 3 : 1, OX will make with OA three times greater an angle than with OB.

The bisector divides the angle in the ratio 1 : 1, therefore can be drawn as shown above. However, the convenience Draw Bisector action offers a shortcut.

This action can be undone.

An angle cannot be divided during the solution playback. You have to stop the playback first.

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12.6. Draw Bisector

To draw the bisector of an angle, select two segments that make the angle, right-click and select Draw Bisectory from the popup menu. Alternatively, select Draw Bisector from the Draw menu.

The segments that make the angle must belong to the same face.

A dialog may pop up. Fill in, if necessary, the missing point labels. Click OK.

Note that drawing the bisector is equivalent to dividing the angle in the ratio 1 : 1.

This action can be undone.

Bisectors cannot be drawn during the solution playback. You have to stop the playback first.

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12.7. Draw Intersection

To draw the intersection point of two lines, select two segments that belong to the lines, right-click and select Intersect Lines from the popup menu. Alternatively, select Intersect Lines from the Draw menu, or click the button on the tool bar.

Both segments, as well as the intersection point of their lines, must belong to the same face.

A dialog may pop up. Fill in, if necessary, the missing point labels. Click OK.

This action can be undone.

Intersections cannot be drawn during the solution playback. You have to stop the playback first.

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12.8. Lay Distance

To draw a point on a given segment at a given distance from another given point, select both the point and the segment, right-click and select Lay Distance from the popup menu. Alternatively, select Lay Distance from the Draw menu, or click the button on the tool bar.

The point and the segment must belong to the same face. The point may or may not belong to the segment.

A dialog will pop up. Fill in, if necessary, the missing point labels and type in a valid expression for the distance. Click OK.

This action may result in the construction of one or two distinct points.

This action can be undone.

Distances cannot be laid during the solution playback. You have to stop the playback first.

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12.9. Lay Angle

To draw a segment from the end point of another segment so that the two segments make a given angle, select the segment and its end point, right-click and select Lay Angle from the popup menu. Alternatively, select Lay Angle from the Draw menu, or click the button on the tool bar.

A dialog will pop up. Fill in, if necessary, the missing point labels and type in a valid expression for the angle. Click OK.

The angle must be greater than 0 and less than 180 degrees and be entered in radians, as opposed to degrees. For example, to lay an angle of 60 degrees, enter PI/3.

If the segment belongs to two different faces at the same time, this action may result in the angle being laid off in each of those faces.

This action can be undone.

Angles cannot be laid during the solution playback. You have to stop the playback first.

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13. Transform

13.1. Scale Figure

To scale a figure in the direction of a given segment by a given factor, select the segment or its end points, right-click and select Scale Figure from the popup menu. Alternatively, select Scale Figure from the Transform menu or click the button on the tool bar.

A dialog will pop up. Fill in, if necessary, the missing point labels and type in a valid expression for the scaling factor. Click OK.

As a result of scaling, the variables assigned to prior measurements performed upon the selected figure will become pinned.

This action can be undone.

The Solver cannot scale a figure that is part of the problem. However, the figures that have been created in the process of solving the problem can be scaled.

A figure cannot be scaled during the solution playback. You have to stop the playback first.

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13.2. Shear Figure

Shearing refers to an affine transformation where planes parallel to a fixed reference plane slide along a fixed line contained in the reference plane.

If XY is the fixed line in the reference plane, the shearing transformation is uniquely identified by points X, Y and the image of any point Z such that XZ is perpendicular to the reference plane.

To shear a figure relatively to segment XY and point Z, as described above, select segment XY, its point X and point Z, right-click and select Shear Figure from the popup menu. Alternatively, select Shear Figure from the Transform menu or click the button on the tool bar.

If a dialog pops up, fill in the missing point labels and click OK.

As a result of shearing, the variables assigned to prior measurements performed upon the selected figure will become pinned.

This action can be undone.

The Solver cannot shear a figure that is part of the problem. However, the figures that have been created in the process of solving the problem can be sheared.

A figure cannot be scaled during the solution playback. You have to stop the playback first.

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13.3. Cut Figure

To cut a figure with a plane, select some elements that uniquely identify that plane, such as three points, a point and a segment, or two segments, right-click and select Cut Figure from the popup menu. Alternatively, select Cut Figure from the Transform menu or click the button on the tool bar.

The plane must pass through the figure's interior.

If a dialog pops up, fill in the missing point labels and click OK.

The two resulting fragments of the figure will be displayed in separate views. The original figure will be kept intact in its own view.

This action can be undone.

A figure cannot be cut during the solution playback. You have to stop the playback first.

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13.4. Join Figures

To join two figures at their faces Face1 and Face2 respectively, select some elements, such as points and segments, on Face1, then select the corresponding elements on Face2. Select Join Figures from the popup menu or click the button on the tool bar.

A dialog will pop up. Select the names of the figures. A figure can be joined with a copy of itself. Therefore the same figure can be selected in both lists.

Select Join at selected elements or, if you would rather ignore the selection, Join arbitrarily and click OK.

A new view will appear with the combined figure in it. The two original figures will be kept intact in their own views.

The combined solid must be convex. Face1 and Face2 must have the same shape, that is have equal angles and proportional sides at corresponding vertices. Face1 and Face2 do not have to have the same size. If need be, one of the figures will be automatically scaled to fit the other.

This action can be undone.

Figures cannot be joined during the solution playback. You have to stop the playback first.

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