Applications-of-GeoGebra/C3/3D-Geometry/English-timed
From Script | Spoken-Tutorial
| Time | Narration |
| 00:01 | Welcome to this tutorial on 3D Geometry. |
| 00:05 | In this tutorial, we will learn how to use GeoGebra to view:
And construct different structures in 3D space |
| 00:17 | Solids of rotation of polynomial functions |
| 00:21 | Trigonometric functions in 3D space |
| 00:25 | Here I am using:
Ubuntu Linux OS version 16.04 |
| 00:32 | GeoGebra 5.0.481.0 hyphen d |
| 00:39 | To follow this tutorial, you should be familiar with: |
| 00:43 | GeoGebra interface
Geometry |
| 00:48 | For relevant tutorials, please visit our website. |
| 00:53 | This image shows the rectangular coordinate system. |
| 00:58 | It is made up of mutually perpendicular axes and planes formed by them. |
| 01:04 | The axes are x (in red), y (in green) and z (in blue). |
| 01:11 | All points in 3D space are denoted by their x y z coordinates. |
| 01:18 | The point of intersection of the three axes is the origin O 0 comma 0 comma 0. |
| 01:27 | The gray rectangle in the image depicts the XY plane. |
| 01:33 | The planes divide space into 8 octants. |
| 01:38 | Point A is in the XOYZ octant and has the coordinates 4 comma 4 comma 2. |
| 01:48 | Let us draw a 3D pyramid in GeoGebra. |
| 01:53 | I have already opened a new window in GeoGebra. |
| 01:58 | This time, we work with Algebra, 2D Graphics and 3D Graphics views. |
| 02:05 | Under View, select 3D Graphics. |
| 02:09 | Click in 2D Graphics View to draw in 2D. |
| 02:14 | Drag the boundary to see 2D Graphics properly. |
| 02:19 | Click in 2D Graphics. |
| 02:22 | In 2D Graphics view, click on the Polygon tool and click on origin 0 comma 0. |
| 02:31 | This creates point A at the origin. |
| 02:35 | Then click on 2 comma 0 to create point B. |
| 02:40 | Click on 2 comma 2 for C and on 0 comma 2 to draw D. |
| 02:48 | Finally, click again on A. |
| 02:52 | Note that a quadrilateral q1 is seen in 2D and 3D Graphics views. |
| 03:00 | The length of each side is 2 units. |
| 03:04 | Click on the Move tool. |
| 03:07 | Click in 2D Graphics and drag the background. |
| 03:11 | Drag the boundary to see 3D Graphics properly. |
| 03:16 | Click in 3D Graphics and under Pyramid, on the Extrude to Pyramid or Cone tool. |
| 03:25 | In 3D Graphics view, click on the square. |
| 03:29 | In the Altitude text-box that opens, type 3 and click OK. |
| 03:36 | A pyramid e appears in 3D Graphics view. |
| 03:40 | Its base is the quadrilateral q1. |
| 03:44 | Its apex is E 1 comma 1 comma 3. |
| 03:49 | Its altitude or height is 3 units. |
| 03:54 | Rotation of a Polynomial |
| 03:57 | Let us rotate f of x equals minus 2 x raised to 4 minus x cubed plus 3 x squared. |
| 04:07 | We will rotate the part that lies in the second quadrant, in XY plane, about the x-axis. |
| 04:16 | I have already opened a new window in GeoGebra. |
| 04:21 | We will initially work with Algebra and 2D Graphics views and open 3D Graphics view later. |
| 04:29 | In the input bar, type the following line. |
| 04:33 | To type the caret symbol, hold Shift key down and press 6. |
| 04:36 | Spaces here denote multiplication.
Press Enter. |
| 04:46 | Under Perpendicular Line, click on Parallel line and on the y-axis. |
| 04:54 | Keep the cursor on the x-axis. |
| 04:58 | Drag it along until you see function f, x-axis at the intersection of f and x-axis. |
| 05:07 | Click on this intersection point. |
| 05:10 | Point A appears. |
| 05:13 | Click on Slider and in Graphics view. |
| 05:18 | A Slider dialog-box opens. |
| 05:21 | Leave a as the Name. |
| 05:24 | Change Min value to minus 1.5, Max value to 0 and Increment to 0.05. |
| 05:34 | Click OK. |
| 05:36 | This creates slider a, which changes the value of a from minus 1.5 to 0. |
| 05:45 | It will focus on the part of the graph in the second quadrant. |
| 05:51 | In the input bar, type the following in parentheses. |
| 05:55 | a comma f a in parentheses.
Press Enter. |
| 06:02 | This creates point B whose x coordinate is the value of a. |
| 06:09 | Its y-coordinate lies along the curve described by the function f between x equals 1.5 and 0. |
| 06:19 | Right-click on slider a and check Animation On. |
| 06:25 | Point B travels along function f as a changes. |
| 06:31 | Right-click on slider a and uncheck Animation On. |
| 06:37 | In the input bar, type a comma 0 in parentheses and press Enter. |
| 06:47 | This creates point C. |
| 06:50 | As its x co-ordinate a changes, C moves below point B along the x-axis. |
| 06:58 | Under Line, click on Segment and click on B and C to join them. |
| 07:07 | Click on Move Graphics View and drag the background to the left. |
| 07:13 | Click on View and check 3D Graphics to see the 3D Graphics view. |
| 07:20 | Note that what is drawn in 2D Graphics appears in the XY plane, in 3D Graphics. |
| 07:27 | Click in 3D Graphics view and on Rotate 3D Graphics View. |
| 07:34 | Rotate 3D Graphics to see the curve properly. |
| 07:41 | Place the cursor on the y-axis in green. |
| 07:46 | Click to see an arrow aligned with the y-axis. |
| 07:51 | Drag to pull the y-axis in or outwards to see the curve. |
| 07:58 | In the input bar, type the following line. |
| 08:02 | This creates circle c with center at point C. |
| 08:07 | Its radius is equal to f of a corresponding to the value of a on slider a. |
| 08:15 | Its rotation is around the x-axis.
Press Enter. |
| 08:21 | In Algebra view, right-click on circle c and check Trace On option. |
| 08:28 | Right click on slider a and select Animation On option. |
| 08:35 | Observe the solid traced as a changes. |
| 08:39 | Watch both 2D and 3D Graphics views. |
| 08:44 | Segment BC moves between the x-axis and function f. |
| 08:50 | The part of function f that is in the second quadrant in 2D, rotates around the x-axis. |
| 08:58 | Drag 3D Graphics to see it from another angle. |
| 09:03 | Finally, let us look at trigonometric functions in 3D. |
| 09:09 | I have already opened a new window in GeoGebra. |
| 09:14 | Under View, click on 3D Graphics. |
| 09:19 | Drag the boundary to see 2D Graphics properly. |
| 09:23 | Click in 2D Graphics, then on the Slider tool and in Graphics view. |
| 09:32 | A slider dialog-box opens. |
| 09:35 | By default, the Number radio-button is selected.
In the Name field, type t. |
| 09:43 | Set Min to minus 6, Max to 16 and increment of 0.1.
Click OK. |
| 09:54 | This creates a slider t which will change t from minus 6 to 16. |
| 10:01 | In the input bar, type f t in parentheses equals cos t in parentheses and press Enter. |
| 10:12 | Click in 2D Graphics. |
| 10:15 | Under Move Graphics View, click on Zoom Out and click in 2D Graphics. |
| 10:23 | Click on Move Graphics View and drag the background. |
| 10:28 | You can see the graph of the cosine function of f of t, in 2D and 3D Graphics views. |
| 10:37 | Similarly, in the input bar, type g t in parentheses equals sin t in parentheses.
Press Enter. |
| 10:49 | Sine function graph of g of t appears. |
| 10:53 | In the input bar, type h t in parentheses equals t divided by 4 and press Enter. |
| 11:05 | Line h of t is of the form y equals mx where slope m is 1 divided by 4. |
| 11:14 | Click in 3D Graphics view. |
| 11:17 | Click on the Point tool and click in the gray area in 3D Graphics view.
This creates point A. |
| 11:26 | Drag the boundary to see its co-ordinates properly. |
| 11:30 | In Algebra view, double-click on A. |
| 11:34 | Change the coordinates to the following. Press Enter. |
| 11:39 | The x- coordinate of A is cos t. |
| 11:44 | The y-coordinate is sin t and t divided by 4 is its z coordinate. |
| 11:53 | Right-click on slider t and click on Object Properties. |
| 11:58 | A Preferences dialog-box opens. |
| 12:02 | Click on Slider tab. |
| 12:05 | Under Animation, for Repeat, choose option “Increasing” from the dropdown menu. |
| 12:12 | Close the Preferences dialog box. |
| 12:15 | In Algebra view, right-click on A and select Trace On. |
| 12:22 | Right-click on slider t and check Animation On. |
| 12:27 | Point A traces a helix in 3D space with coordinates mentioned earlier. |
| 12:34 | Click in Rotate 3D Graphic View and rotate the background. |
| 12:39 | Rotate 3D Graphics view so you are looking down the z-axis at the XY plane. |
| 12:46 | Note that the traces of A are the circumference of a unit circle. |
| 12:52 | Point A moves along the circle as angle t changes. |
| 12:58 | In 2D, its coordinates are cos t comma sin t. |
| 13:05 | Let us summarize. |
| 13:07 | In this tutorial, we have learnt how to use GeoGebra to view: |
| 13:13 | And construct different structures in 3D space |
| 13:17 | Solids of rotation of polynomial functions |
| 13:21 | Trigonometric functions in 3D space |
| 13:25 | As an assignment:
Construct a prism and a cylinder anywhere in 3D space. |
| 13:33 | Draw lines to pierce the structures and find their intersection points. |
| 13:39 | Graph the given polynomial. |
| 13:42 | Show the solid formed due to rotation of the peak, in the first quadrant, in the XY plane. |
| 13:50 | As another assignment,
You tried to fly a kite off a cliff. The kite got dumped into the lake below. |
| 13:59 | You gave out 325 feet of string. |
| 14:03 | The angle of declination from where you stand at the cliff’s edge to the kite is 15 degrees.
How high is the cliff? |
| 14:13 | The video at the following link summarizes the Spoken Tutorial project.
Please download and watch it. |
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| 14:51 | This is Vidhya Iyer from IIT Bombay, signing off.
Thank you for joining. |
