get acq transformations

Isometries with Sketchpad

Translations and Vectors

Transformations in geometry are the tools for moving figures around the plane. Sketchpad has a powerful set of transformation tools that we explore here.

Visualization using a point

One tool we will use to visualize transformations is to start with a point P and then transform it to P'. Then we will drag P around. We may connect P to P' with a segment or trace P' as P moves. Also, we may transform P' to P'' and see what added insight that brings.

Visualization using a "blob" (an irregular figure)

Another approach to visualization is to create an irregular shape as a Sketchpad Polygon Interior, made from a set of several vertex points. We will call such a shape a "blob". When the blob is transformed, it gives a visual idea of what the transformation does. Since we are working with Sketchpad, we can reshape the blob by dragging vertices. Here is an example, but your blob will be different.

Translating a "blob" (an irregular figure) by a fixed vector

The simplest transformations are the translations. Intuitively speaking, we translate a figure when we slide it without rotating it.

To translate the blob, select the polygon interior (and the vertex points too) and choose Translate from the Transform menu. You will be given a choice to check of By Polar Vector or By Rectangular Vector. Check polar vector for example; then type in the distance and the angle of the transformation. Here is our blob translated by 3 cm in the angle direction 0. If you move the shape, notice how the image moves. If you drag a vertex, notice how the shapes stay the same.

This answers the common question of how to construct a horizontal or vertical line in Sketchpad that stays horizontal or vertical. Take a point A and translate A horizontally to get a point B. Then construct line AB.

Try this exercise again, but this time choose Rectangular Vector and type in some horizontal and vertical displacements.

Translating a "blob" by dynamic vector AB

This is a kind of translation with is powerful in Sketchpad and unavailable when drawing on paper.

We begin with a sketch consisting of a segment AB and a polygon interior "blob". It does not matter where the segment AB is located.

• Select points A and B (in this order) and choose Mark Vector AB from the Transform menu.

• Next, select the blob (including the interior). It is easiest to drag a selection rectangle around the shape rather than shift clicking). Choose Translate from the Transform menu. You will be presented with a dialog box with some choices. Choose By Marked Vector; then click OK.

• Drag A or B around and see how the translated blob moves.

Piano tracks -- visualizing with points. Imagine that the blob is a piano and you and some friends have pushed the piano across the room. Unfortunately, the feet of the piano, located at the vertices of the blob, leave deep scratches in your floor. The scratches are parallel to each other, of the same length and in the same direction.

• Tracks in our sketch. To see these tracks in our Sketchpad figure, we can

Do a couple of visualization exercises. First, just drag the segment AB across the figure so that A is on top of one of the vertices of the original blob. See how the segment shows the scratch from the old position to the new one, no matter which vertex we choose!

Alternatively, you can construct the segments connecting the corresponding vertices in the two blobs and see that these segments are in fact all parallel, with the same length and direction.

Visualization of repeated translation by vector AB (A Row of Blobs)

Continue with the figure from the last exploration. The vector AB should still be the marked vector (if you have marked another vector since then, mark AB again). Now select the translated blob and again choose Translate (by Marked Vector). You now have a third blob, which is selected. If you immediately go back to the menu (without clicking in the sketch) and choose Translate again, the new blob is translated. Do this until you have 5 or 6 blobs.

• Observing the row of blobs. Observe how the blobs are arranged. Move A and B around and see what happens when AB gets longer and shorter. If you pick a point on the first blob and the corresponding translated points on the other blobs, notice that all the points lie on a line parallel to AB (check this by constructing this line).

A row of points. Draw a new point P in your sketch and translate P by vector AB to get P', then translate P' by the same vector to get P'' and continue to get several points. In other words, construct the first part of the orbit of P by F. How are the points related? What is the distance between successive points?

Answer.

A row of lines. Draw a line m and translate the line by AB several times to construct the first part of the orbit of the line. How are the lines related in general? How are the lines related if m is parallel to line AB? What can be said about the distance between the lines?

Answer.

Translating by a second vector CD

Now add a vector CD to the figure and Mark the vector CD. Returning to the points P and P', translate both of them by CD.

• A grid of blobs. Select all the blobs in the figure and choose Translate several times repeatedly from the Transform menu (without clicking in the sketch, so the translated blobs stay selected at the end of each operation). Now you have an arrangement of blobs marching off in two directions! Move D around and observe what happens when CD is parallel to AB and when it is perpendicular to AB.

A lattice of points. Return to the point P and the "row of points". Select the whole row and translate by CD. This gives a lattice of points forming parallelograms.

Problems and Questions about Translations

Midpoint Triangle. In a new sketch, draw a triangle ABC. Using translations without using the Construct menu, construct a triangle DEF so that ABC is the midpoint triangle of DEF.

Explain your method.

Paths in the lattice. In the sketch with the lattice, make some paths in this way. Let F denotes translation by AB and let G denote translation by CD. Pick a point M and find the point N which is FFGFG(M) doing this step by step and connecting each new point to the previous point by a segment, so that there is a path of segments from M to N. Then repeat the exercise (using segments of different colors) to construct paths from M to GGFFF(M) and FGFGF(M). What relationship do you find among these points? What general property of translations is illustrated?

Answer.

Rotations

How to rotate a shape with Sketchpad using a Marked Angle

Place a point O in a figure. Then also draw an angle EFG.

Create an irregular polygonal shape S as before.

Select the point O and choose Mark Center "O" from the Transform menu (or just double-click on O for the same result).

Then select the 3 points E, F, G in order and choose Mark Angle "EFG" " from the Transform menu.

Then select the blob and choose Rotate from the Transform menu. A dialog box will appear to ask you for the angle of rotation. There will be a box already checked By Marked Angle. Leave this as it is and click on OK.

The rotated image S’ of S will appear.

Rotate S’ in the same way to get S’’. Then rotate S’’ to get S’’’ and continue to get several shapes.

Observe the changes

If you change the shape of S or another one of the shapes, the other shapes changes also.

Watch how S’ and the other shapes change position as you drag the center O. What happens to the pattern if you drag O to a position at a vertex of S or inside S?

Drag a point such as G to change the angle and see how the pattern moves as the angle changes. For some angles, the shapes will begin to overlap and repeat, so only a finite number of shapes are visible.

Drag S. Does this change the number of shapes?

Measure the angle EFG to see better the relationship between angle size and the number of shapes in the figure.

How to rotate a shape with Sketchpad using a Fixed Angle

First draw a polygon to be rotated, let's say a free-hand quadrilateral. Select the points that are the vertices of the polygon and then choose Polygon Interior from the Construction menu to fill in the shape.

Now create a point P. (You can show the label by clicking the point with the hand tool and you can change the name by double-clicking the label with the hand tool.)

Next, select the point P and choose Mark Center "P" on the Transform menu. Then select the shape by dragging and/or shift clicking and choose Rotate from the Transform menu. A "Dialog Box" appears with the number of degrees highlighted. Type 60; then click OK.

The new shape will be highlighted. If you immediately choose Rotate again (the 60 will still be there too) you can rotate the new shape. Continue doing this until you have 6 shapes.

Once you have your 6 shapes, move the center P to new locations and observe the result.

Also, you can move the points of the shapes and get new shapes.

Each of the whole figures we get is said to have 6-fold rotational symmetry or a 60-degree rotational symmetry. This means that if you rotate the whole figure by 60 degrees with center P, then the figure is moved onto itself.

Here are some suggested activities. You may want to check with your neighbor and choose different angles of rotation to increase the number of shapes you get to look at.

Reflections
How to reflect a shape with Sketchpad

Make a shape S as before and also draw a line AB. Now select the line by clicking on it and then choose Mark Mirror in the Transform menu. To reflect the shape, select the shape and choose Reflect in the Transform menu.

Move the shapes around and see what reflection looks like when the shape crosses the line. What happens if you reflect the new shape over the same line again?

Reflect a point P to get the image point P' and connect the points with a segment. Do this for several points. What is the relationship between these segments and the mirror line?

Double Parallel Mirrors

In a new sketch draw a line m and construct a line n parallel to m. Make an irregular "blob" S as before.

Reflect S across m to get S’. Then reflect S’ across n to get S’’. Reflect S’’ across m to get S’’’. Continue until you have several shapes. These shapes are of two kinds, some were obtained by an odd number of reflections and some were obtained by an even number of reflections. Let’s call these the even images and the odd images.

Change the shade and/or color of the odd image shapes so they are same shade and color as each other but different from the even images.

What kind of transformation is needed to transform S directly into S’’? What kind would be needed to transform any even image to another?
What transformations would carry any odd image to another odd image? What would carry an odd image to an even image?

Conclusions

If M is reflection in line m and N is reflection in line n, what isometry is the composition MN? What is NM? What is (MN)(NM)?

What is MNM?

Given two other lines u and v which are parallel to each other, If line reflection in u is U and line reflection in v is V, exactly what must be true about u and v to have UV = MN? Can you illustrate this with a sketch?

Double Intersecting mirrors.

Start a new sketch. Draw two intersecting lines m = OM and n = ON.

Repeat the same exercise as before. Start with a shape S and reflect in m to get S’, reflect S’ in n to get S’’, etc.

Shade the even and odd sets of shapes as before.

By what isometries are the even shapes related? By what isometries are the odd shapes related? Be as precise as you can about exactly what isometries you observe. It may help to measure some angles.

Add some points and reflect them twice as well. For clarity, connect them with segments.

O is the point of intersection of the two reflecting lines. You can measure the angle at O formed by a point P and its double reflection. For example, measure the angle POP" by shift-clicking to select the 3 points P, O, P" and then choosing Angle from the Measure menu. You don't have to draw line segments or rays to define the angle. Also measure the angle between the lines, MON.

• How do the measurements compare among various double reflections such as POP", QOQ", and FOF" for some point F which is a vertex of S? How do they change when the lines change? What is the angle between the lines?

What happens if you move the lines so that they are (nearly) parallel?

Analysis: Rotational symmetry from line symmetry.

Move the mirror lines to see the effect on the even and odd sets of shapes.

See if you can move the lines so that you get a symmetric shape, as in the second figure. The whole figure has mirror symmetries, but if you look only at the dark shapes, you see a figure with only rotational symmetries. What is the angle of the rotation? How is the angle of rotational symmetry related to the angle between the lines?

Conclusions

Let M be reflection in line m and N be reflection in line n (not to be confused with the points M and N). What isometry is the composition MN? What is NM? What is (MN)(NM)?

What is MNM?

Given two other lines u and v which are intersecting to each other. If line reflection in u is U and line reflection in v is V, for what lines u and v will UV = MN? Exactly what must be true about u and v? Can you illustrate this with a sketch?

Glide Reflections: How to glide reflect a shape with Sketchpad

In sketch draw a line AB. Mark the line as a mirror and mark vector AB as a vector as before.

Make a shape S.

Reflect S across the line to get S’.
Translate S by the vector to get S’’.

Hide S’.

The transformation that takes S to S’’ is the glide reflection g_AB.

You can add this transformation to the Transform menu by selecting S and S’’ and choosing Define Transform from the Transform menu. Name it something like G_AB, or Glide Ref AB.

Now select S’’ and apply this G_AB from the Transform menu. Continue applying the transformation to the new images.

Once again, you can group the shapes according to even and odd numbers of applications of G_AB and color or shade them differently.

What isometries will map an even shape to other even shapes? What isometries will map an odd shape to other odd shapes?

Conclusions

If G is the glide reflection defined by A and B as above, what isometry is GG?

For what other points C and D will the glide translation by C and D be the same as the G defined by A and B?

Given two glide reflections G1 and G2, do a Sketchpad experiment to study the composition G2 G1. What kind of isometry do you find this composition to be?