Geometer’s Sketchpad Tour 5&6: Transformations
Adapted by Geoff Hagopian from Key Curriculum Press’ The Geometer’s Sketchpad – Learning Guide.
Translation, rotation, compression/dilation and reflection are the building blocks of transformational geometry. Geometer’s Sketchpad is great for exploring this stuff. You start by investigating a basic problem involving reflections and rotations and then, as a bonus, you get a kaleidoscope. While you’re at it, work some more Measure menu features and build new custom tools.
The applet below gives you some idea of where you're headed here. Click Animate to see what happens when one of the papa pentagons gets going!
Ideas for this Section:
- Quick polygon interiors.
- Reflections across mirror lines.
- The Sketchpad calculator.
- Rotations about a point by an angle.
- How to creat a custom tool that defines a multi-step transformation.
- How to dilate objects about a marked center.
- How to merge independent points to paths.
The following basic transformations are available in Sketchpad:
It turns out we can eliminate rotation from this list and still achieve all the transformations we could with it by combining reflections through various lines. Seeing how this works is one of the objectives of this lab.
Constructing Mirrors and Polygonal Interior
1. Draw to segments that share a vertex. This is in the shape of a “V.”
2. Use the Point tool construct a point inside the “V” and then, holding down the shift key, construct 4 more points. With these points selected, choose Pentagon Interior from the Construct menu
To test our hypothesis (we can achieve rotation by reflection), reflect the interior over one of the V segments and then reflect the reflection over the other segment.
3. Deselect all, select, say, the upper segment and then choose Mark Mirror fro the Transform menu.
4. Select the pentagon interior by clicking on it and then choose Reflect from the Transform menu.
5. Right click on the reflected interior and choose the color option to change its color. Leave that interior selected.
6. Double click on the other line segment to change the line of reflection choose Reflect from the Transform menu again.
Try dragging around various points and observe the reflective properties of the lines with respect to the interiors. Notice that the third interior could be a rotation of the first. Is it? We’ll put the conjecture to some empirical tests.
7. With the Segment tool, draw a third segment just outside the V and sharing the vertex of the V, as shown.
8. Right click on the new segment and choose “dashed” so that you can distinguish this segment from the others.
9. Use the Text tool to label the endpoints in (alphabetical) order as shown.
Now the goal is to rotate the original interior about this new segment to see if the third interior can be reproduced.
10. Deselect all objects and then select the three points defining the new larger angle in the following order: point A, then point B and thirdly point C.With these three point selected (in that order) choose Mark Angle form the Transform menu.
11. Select the common vertex (point B) and choose
from the Transform menu Mark Center
12. Select the original interior and choose Rotate from the Transform menu. Look to see that the “By Marked Angle” option is chosen, and then click Rotate.
13. Deselect all and then select A, B & C, being sure to select B second. Now choose Angle from the Measure menu. The angle measurement will appear on the page.
14. Repeat the previous step, except to measure angle ABD.
Rotating by a Marked Calculation
Notice that by dragging D you can make rotated interior and the doubly reflected interior coincide and that when this happens the angle ABD is twice angle ABC (or approximately.) To be sure, try using the Sketchpad’s calculator as outlined in the following steps.
15. Select point D and remove it and its children by pressing the delete or backspace key.
16. Calculate twice the angle ABC by choosing Calculate from the Measure menu. The Calculation window opens. Type “2*” in the box and then click on the measure of angle ABC in the Sketchpad page. You’ll see something like the picture at right. Click OK.
17. Now Rotate the original interior again, this time by twice angle ABC. This is done by selecting 2.mRABC and choosing Mark Angle from the Transform menu, then selecting the original interior and choosing Rotate from the Transform menu.
18. The result is interesting in that depending on how many half-rotations of angle ABC are completed. You may get a picture like the one at right, where the rotated interior overlaps the doubly reflected interior, but you may also get a figure where they are adjacent, but not overlapping.
19. Before proceeding on, delete the calculation of twice angle ABC (and its children) by selecting and deleting it. Now might be a good time to save the sketch.
What happens if you continue the process of reflecting the final image through the two lines and repeating this over and over to each final result? To explore how to answer this question you’ll build a custom tool.
The tool you’ll build should reflect any given object across two straight lines, resulting in two reflected images. You’ll first need to make the construction (start with what you have after completing step 19.) Select those objects the tool’s construction is based on and the results the tool will produce. These results will be the two new reflected interiors.
20. Adjust the angle of the V so it is approximately 45 degrees, as shown at right.
21. Select the desired objects for the tool to operate on by first clicking on the original interior, the first line AB and then the second line AC, in that order.
22. Continue by selecting the desired outcomes: the two reflected interiors. You’ll have five objects selected at this stage.
23. Choose Create New Tool from the Custom Tool menu on the Toolbox and name it, say, Double Reflection. Click OK. You’ll see this tool’s name now listed in the Custom Tool menu.
24. Choose Double Reflection from the Custom Tool menu. Click on these objects in order: The doubly reflected (rotated) interior, the first segment AB and the second segment AC. Two more interiors appear as each segment is selected.
25. Now, with the Double Reflection tool still active, click on the reflected interior (the original interior reflected in AB) and then on each of the two line segments, producing two more interiors, for a total of seven, as shown in the diagram above.
26. To review these last few actions, choose Undo from the Edit menu (ctrl+Z) twice and then Redo twice to get back to the seven interiors picture.
27. Experiment by dragging vertices of the original pentagon (which now has many children.) Also, try dragging points of the V around.
28. Adjust the V angle to something like 42° or 93° and apply the Double Reflection tool repeatedly—always to the lastly generated interiors. If you get lost, try doing Undo/Redo to get oriented as to which interior is most recent. Pay attention to the messages at the bottom of the screen that say “Match pre-image…” and “…to segment.” When you have 25-30 reflected interiors you should have a picture something like shown at right..
29. To control the angle, it helps to move the point A (or C) far from the center. You can use the arrow key on the keyboard to move the selected point one pixel at a time.
30. Look at what happens for special mirror angles such as 30°, 36°, 45°, 60° and 72°. What conjecture is suggested as to how many interiors are concurrent for a given angle? Can you prove the conjecture?
31. Keep pressing ctrl+Z (Undo) until you’re back to just seven interiors. Adjust the mirror angle to as near to 60° as you can.
32. Select only the original interior and use the Hide command from the Display menu (or ctrl+H) to hide it.
The Dilate command shrinks objects toward or stretches them away from a marked center by a specifies scale factor. A negative scale factor will dilate the object right through the center point and out the other side!
33. Double click point B to mark it as a center for subsequent Dilate transformations. You’ll see that funny radiating animation to verify your action.
34. Select three of six alternating reflected interiors and change their color to one you like using the Color option from the Display menu. With these objects still chosen (or choose them again) execute the Dilate command from the Transform menu.
35. Put -1 in the upper field and 2 in the lower one to enter -1/2 for the scale factor and click Dilate. The dilated images appear half the size of the originals and half the distance from the point, but on the opposite side (and rotated 180°.)
36. With the three new interiors selected (this may already be true) choose a new color from the Display menu and then Dilate again with the parameter -1/2 (now default.) Give these three newly dilated interiors yet another color to your liking.
37. Repeat steps 35-37 with the other three of the six original interiors. This should 18 interiors more or less like those shown in the picture here. Experiment with dragging various points, interiors.
Animating and Merging
38. Select an interior in the sketch and choose Animate from the Display menu. After fiddling with the thing for a bit, I produce the following image. I find it engaging!
The Adam and Eves of this diagram are the five points with define the original pentagon interior. Never mind the Biblical inaccuracies…the point is that animating any interior then goes back an independently animates each vertex in a pseudo-random walk (never mind why it’s pseudo.)
You can get more control by selecting a point or combination of points that define the original pentagon. You can also merge points and combinations of point follow combinations of paths. Wow, that’s control!
37. If the thing is still animating, stop it with the stop buttom on the Motion Controller (or choose Stop Animation from the Display menu.)
38. Select one of the vertex points (of the original pentagon) and one the segments of the V. With these (and only these) selected, click Merge Point to Segment from the Edit menu. The point will move to and merge with the segment so that it can only move along its length.
39. Repeat the last step for a different vertex and the other segment of the V.
40. Use the Compass tool (circle tool) to build three circles in blank spaces of the sketch—the circles can overlap existing objects, but should not be attached to any.
41. You will have a sketch looking something like the sketch to the right.
42. Animating any interior now will now animate each of the five vertices around its constrained path: either a segment or a circle.
43. You can use the Motion Controller to adjust the animation speed and direction of individual points. Mash buttons and wiggle away!
44. If you have a kaleidoscope you’d like to save, you might select only the interiors by doing the following:
· Deselect all objects.
· Choose the Point tool and then choose Select All Points from the Edit menu.
· Chose Hide Points from the Display menu.
· Repeat the last two sub-steps, first with the Compass tool, them with the Segment tool.
45. You could end up with something much more cool than this, but at least this is missing the distraction of rigging: