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Geometry
Custom Apps
The Geometry Custom Apps embed dynamic 2D and 3D shapes and animations. Click
on the name of a tool below for a description and related help topics:
Contour
Slicing Polyhedra
Slicing Cones, Cylinders, and Spheres
Surface of Revolution
Graphical Linear Programming

Contour
Represent threedimensional objects and surfaces with contour lines
or horizontal and vertical cross sections. 
The help topics below include information on: Getting
Started and Contour Settings, inserting
a moveable Relief Line whose topographic profile
is also displayed, and options for customizing the display of a 3D
Graph representing the data.
Getting Started
 Choose Contour Diagrams from the Quick Access  Advanced Apps
menu. Alternatively, access Spreadsheet and choose Contour / 3D
Plot within the Tools  Graph menu. There are options to model
a contour diagram from preset or usergenerated data.
Create a Contour / 3D Plot from preset data: Select
from the available data sets such as NASA Pioneer. Click OK. A Spreadsheet
window for the data and a Geometry window for the contour diagram will
appear.
Create a Contour / 3D Plot from usergenerated data: When prompted
with data set options, click Cancel (or access a blank Algebra Spreadsheet).
Type data values into the spreadsheet cells. Choose Tools  Graphs  Contour
/ 3D Plot. A separate window with the corresponding contour diagram will
appear.
Help Tip: Type values in a rectangular array of cells
in the upper left hand corner of the spreadsheet.
 Reading the tables and contour diagrams.
 Each numeric value in a spreadsheet cell is represented in its corresponding
rectangular grid position on the contour diagram. For example, the value
in cell A1 is represented in the upperleft hand corner of the contour
diagram grid. Each gridline intersection represents a spreadsheet cell
value.
 Contour lines represent parallel cross sections of an object or surface.
When contour lines form closed curves, they are surrounding either a
local maximum or local minimum. Most contour lines are labeled according
to their elevation.
Contour Settings
 Choose Contour  Controls to change the settings of the contour
diagram. Click the X button of the Contour Settings Window to simultaneously
close the controls and apply the new settings.
 Starting Contour Value: Type a number for the desired starting
contour value for the contour diagram. Any contour line values less than
the indicated number will not be shown on the diagram, however, the coloring
of the diagram will not change. That is, only the contour lines with
values greater than or equal to the specified starting value will be
shown on the Contour / 3D Plot.
 Contour Spacing: Type a number for the desired spacing between
contour lines displayed on the contour diagram. In general, the lower
the contour spacing value (e.g., 0.5) the closer the contour lines will
be spaced (i.e., a more finegrained version of the contour diagram will
be displayed).
Relief Line
 Choose Contour  Relief Line to display (or hide) a dynamic topographic
profile in the lower righthand corner of the window and the corresponding
movable relief line (shown in shades of green across the map). A topographic
profile is a vertical crosssection view along a line across a portion of
a map.
 Notice the color of the topographic profile corresponds to the coloring
of the contour map.
 Click and drag near the center of the relief line to change it's position
relative to the map. Click and drag an endpoint of the relief line to
make the horizontal line oblique or vertical.
Customizable 3D Views
Select entries of the 3D View menu to activate customizable features of the 3D
plot. Choose an entry to activate (a check mark will appear next to the entry),
select it again to deactivate (indicated by no check mark).
 Choose 3D View  On to turn on a 3D Plot of the corresponding
contour diagram.
 Select 3D View  Contour to hide (or show) the contour map.
When the contour diagram is hidden and the 3D Plot is On, the 3D graph will
appear larger in the viewing window with a xyz axis displayed.
 Select 3D View  Draw Grid /Mesh to show (or hide) the grid
on the 3D model. Note the 3D Plot must be On for this option to have an affect
on the display.
 The Light Source option of the 3D View menu affects the brightness or
dimness on the coloration on the 3D rendition of the map. Note the 3D Plot
must be On for this option to have an affect on the display.

Slicing Polyhedra
Visualize common polyhedra and manipulate 3D modeling options such
as cutting and unfolding. 
Instructions
 Set up the model by choosing a solid from the Polyhedron menu.
 Adjust the graph view.
 Click and drag the shape on the screen to rotate it in the 3D space.
 It might be helpful to adjust the transparency of the shape before
manipulating.
 Choose to View  Zoom In ,
Zoom Out ,
and Zoom Fit .
 The Vertices, Edges, and Faces of a solid can be shown or hidden by
selecting (or deselecting) them in the View menu.
 Model the slicing or unfolding of the selected polyhedra.
 Choose Enable in the Cut, Unfold, or Transform menus.
 Drag the available slider bars to adjust the model accordingly
 Use the Tool menu to save, print, or otherwise record your work.
See also Save & Print.

Slicing Cones, Cylinders, and Spheres
Dynamically manipulate a plane slicing a cone, cylinder, or sphere,
and view the corresponding cross section. 
This custom app allows for the manipulation of the position of the plane
(drag the tilt slider) and the view of the cone, cylinder, and sphere (choose
from the Surface menu then drag the solids themselves) in order to examine
the resultant intersection. The help topics below include information on: Slicing
Plane Attributes and the Tool Menu.
Slicing Plane Attributes
 Tilt (Angle from xyplane): The tilt angle is determined by the
angle formed between the xyplane and the shaded plane shown. Notice that
the 2D rendition of the angle and the conic section dynamically update as
the tilt is changed.
Note: To adjust the Settings, choose the red box to view the
Configuration Panel in order to set minimum and maximum values for the
tilt angle. Unit and block increments can also be adjusted. Click OK to
make these changes or Cancel to exit without change.
 Vertical Translation: Drag the horizontal slider bar to adjust
the vertical translation of the plane that cuts across the doublecone.
Note: To adjust the Settings, click the red box to set minimum
and maximum values for the vertical translation of the plane. Unit and
block increments can also be adjusted. Click OK to change these settings
or Cancel to exit without change.
 Choose Wire Frame from the Options menu to show the frame of the double
cone instead of the shaded surface.
 Choose Label Conic from the Options menu to show labels of the type of
conic section shown on both the 3D View and the Standard View.
 Choose Extend Surface or Shrink Surface from the Options menu to adjust
the size of the surface in the 3D View.
Tool menu
The Tool menu offers options to Copy, Save, Print, and Close the custom app.
 Choose Tool  Copy to Clipboard to store an image of the object
to be pasted into another document.
 Choose Tool  Save to store each of the 3D and 2D views as images
(with a GIF filename extension). Choose an appropriate name to allow for
searches of those images if needed.
 Choose Tool  Print to print the current view of the 3D solid
and the 2D slice.
See also Save & Print.
 Choose Tool  Close or the X button to close this custom app.

Surface of Revolution
Draw and manipulate a line or connected series of line segments
in twodimensional space to simultaneously generate a surface of revolution
in threedimensional space. 
A surface of revolution is a surface formed by rotating (revolving)
a curve about a line. The help topics below include information on: the Draw
Tools for defining lines and curves, the Revolution
Settings for setting the angle and axis of revolution, the Tool
menu, and Options menu.
Draw Tools
 Choose to
Clear All in the current drawing(s) and reset the axis to the original position.
Alternatively, choose Tool  Restart to obtain the original starting
view, settings, and graph.
 Choose the Draw tool to
define a new curve or line. Click and release the mouse over the 2D coordinate
grid to define points. Points will automatically be connected with a segment
to the point preceding it. Another drawing technique is to click and drag
as you hold the mouse button down.
Dynamic Drawing: Notice that as points are drawn on
the coordinate grid, the surface of revolution is correspondingly defined
in the threedimensional space.
 Use the Select tool to
click on and drag defined points to new locations on the coordinate grid.
Observe how the 3D surface of revolution automatically updates.
Revolution
Settings
 Angle of Rotation: Drag the slider bar as the angle of rotation
for the surface of revolution is dynamically updated.
Note: The default setting for the angle of rotation is 180° and
ranges from 0° to 360°. Click the red box to the right of this
slider bar to set different minimum and maximum values, and indicate the
unit and block increment. Click OK to update the settings or Cancel to
exit without changing the settings.
 Axis of Rotation: Choose the or buttons
to change the axis of rotation to the y or zaxis, respectively.
Tool menu
The Tool menu offers options to Restart, Print, and Close the custom app.
 Choose Tool  Restart ()
to obtain the original starting view, settings, and graph.
 Choose Tool  Print to print the currently selected frame.
See also Save & Print.
 Choose Tool  Close or the X button to close this custom ap.
Return to the main screen to make a new tool selection.
Options menu
The Options menu allows you to modify the display of the 3D surface of revolution.
 Choose Options  Show Edges to toggle the wireframe of the surface
of revolution.
 Select Options  Show Surface to toggle the quadrilateral faces
that comprise the surface of revolution.
 Choose Options  Show Meridian to toggle the 2D curve used to
sweep out the surface of revolution.
 Choose Options  Show 3D Axes to toggle the xyz coordinate
axes.
 Choose Options  Rotate about Horizontal Axis or the button
to perform the revolution about the yaxis rather than the zaxis.
 Choose Options  Rotate about Vertical Axis or the button
to perform the revolution about the zaxis rather than the yaxis.

Graphical Linear Programming
Analyze an optimization situation by using constraint inequalities
and an objective function to find the minimum/maximum. Twodimensional
(2D) and threedimensional (3D) linear programming problems can be
examined. 
The help topics below describe how to build a linear
programming situation, how to analyze that situation,
and the tools available for saving your work or converting
to the Spreadsheet environment.
Build a Linear Programming Situation
 Choose to
Clear All the current drawing(s) and remove any constraint inequalities and
objective function that may be present. Alternatively, choose Tools  New
to do the same.
 Specify whether you will be examining a 2D or 3D linear programming situation.
The default is 2D. Switch to a threedimensional linear programming problem
by selecting View  3D.
 Enter a constraint inequality in the box next to the Constraint Inequalities
checkbox. Use >= for greater than or equal to and <= for less than
or equal to. (For a 2D problem, the default variables used for function f are x and y.
For a 3D problem, the default variables are x, y, and z.)
Repeat as needed until all constraint inequalities have been entered by first
clicking the + button to add each new constraint inequality.
Change the Variables: If desired, you can change the
letters used to denote the function and/or any of the variables. Select
the Settings tab and rename the function/variables, making sure to hit
Enter or Tab after you change an entry.
 Enter the Objective Function as an expression (note that the lefthand
side of the equation is already provided).
Analyze the Situation
What is your purpose? Maybe you want to analyze the linear programming situation
one constraint inequality at a time, resulting in the determination of the
feasible region, and finalizing by using the objective function to find the
minimum/maximum. Or perhaps you want to skip all the details and get right
to analyzing the feasible region. Your intent determines the way you use this
tool.
 View the region covered by a constraint inequality by clicking the checkbox
next to that inequality. The window settings for the display automatically
adjust to accommodate the new graph. Continue to click checkboxes to show
all constraint inequalities graphed simultaneously (presumably showing a
region of overlap, the feasible region).
Show Only Positive Values: Typically for a linear programming
situation, you are only interested in considering positive values. Thus,
the default setting in the Constraints menu is to do just that, Show Only
Positive Values.
 View only the boundary values of the region covered by a constraint inequality
by selecting Constraints  Show Only Boundary. This may be particularly
helpful in 3D mode.
 View the feasible region by selecting Constraints  Show Feasible
Region or by clicking the button.
 With only the feasible region displayed, quickly find the solution(s)
to the problem by selecting Constraints  Show
Vertices.
Note: You may instead want to watch how the graph of the objective
function can be moved through the feasible region as you try to find the
solution(s). Select Objective  Show Obj. Fun. = Value to enable the graph
and a point on that graph that you can use to drag the graph through the
region. (This automatically turns on the Trace Point.) For a 3D situation,
you may want to also select Objective  Auto Rotate
View before moving the objective function plane through the feasible region.
This option reorients the display so that the objective
function plane is perpendicular to your computer screen; in essense, this
makes the plane into a line which you can more easily move through the
region towards the solution(s).
Tool menu
The Tool menu offers basic options for starting new problems or saving your
work. You can also export your work to Spreadsheet for further analysis.
 Choose Tools  New to give the original starting view, settings,
and graph.
 Choose Tools  Print to print the currently selected frame.
 To save your work there are two options.
 Choose Tool  Save to maintain the constraint inequalities
and objective function for use later. With this option use the filename
extension .txt or .tcms. Later choose Tool  Open to access
this saved work.
 Alternatively choose Tool  Save as Gif to store an image
of the graphical representation to your computer (with filename extension
gif). This cannot be opened within TCMSTools, but may be viewed
in an image viewer or pasted into a document. Likewise, the Copy to Clipboard
option stores an image of the graph that may be pasted into a document
for viewing.
See also Save & Print.
 Choose Tools  Convert to Spreadsheet to export your constraint
inequalities and objective function to the Solver tool within Spreadsheet
for further analysis.