Dynamic Geometry workshop: 2

Sketchpad Continued — 2nd session Sebastian Lisken
lisken@Mathematik.Uni-Bielefeld.DE

Transformations

We have experienced many items in the "Construct" menu. Let's now see what the "Transform" menu can do. Sketchpad knows four transformations:

	• Translations,
	• rotations,
	• dilations and
	• reflections.

They can be applied to most objects (excluding loci and external objects).

How to use transformations

Working with transformations in Sketchpad requires an extra step before the usual sequence of

selecting objects and then applying a menu function can be performed. First, you

mark the objects that define the translation. For example, if you want to rotate, you must define a rotation center by marking a previously constructed point. You may also define the rotation angle by marking an angle between three points or a measured or calculated angle.

To mark objects that define transformations, you select them in the usual way and then choose the appropriate "Mark" item in the "Transform" menu. If what you mark is a single point (e.g. a center for rotations or dilations) or line (a mirror for reflections), you can also double-click the object. A short "flashing" animation confirms that you have marked the object.

After marking, you select the objects that you actually want to transform, and choose the appropriate transformation from the "Transform" menu.

Get some first experience with transformations by working through the instructions in this sketch.

Get some practice

Now try a few practical examples. Can you do the following?

Construct a parallelogram, a square or an equilateral triangle without auxiliary objects like circles or parallel lines. If you do, drag the points and observe how they seem to move more freely, even those that are actually dependent (constructed) objects.
Construct a slider that carries over a linear movement from one line to another. For an interesting application, see Bezier.
Try to think of other mechanical devices where movement, either linear (along a line) or angular (along a circle), is transferred from one place to another. For example, invent the cog-wheel: make a circle roll along the outline of another circle - construct a point on the wheel that shows the rolling movement. Or try your hand at a rope going round a corner. You will need more specific help if you want to translate a linear movement into an angular one, for example have a wheel roll along a straight line. (So, in other words, the railway is harder to invent than the cog-wheel.)

Glossary

The terms select, mark, and choose are confusingly similar in their English meaning, but are used in a very specific way here.

select To click objects with the mouse. If you need to select several objects, use the Shift key while clicking. You can also use Shift and click an already selected object to deselect it.

mark To tell Sketchpad which objects should be used as parameters for transformations. For example, you need to tell Sketchpad which center it should use for a rotation or a dilation. You mark a center point by selecting it and then choosing "Transform | Mark Center".

choose To execute some function in some menu. The phrase choose "Transform | Mark Center", for example, means to execute the "Mark Center" item in the "Transform" menu. You can do this by clicking the left mouse button in the menus, sometimes by clicking the right mouse button within a sketch, or by using the keyboard.

drag Another important procedure: To drag an object means to click it with the mouse, then to keep the mouse key pressed and move the mouse so that the object moves across the screen accordingly.

select	To click objects with the mouse. If you need to select several objects, use the Shift key while clicking. You can also use Shift and click an already selected object to deselect it.
mark	To tell Sketchpad which objects should be used as parameters for transformations. For example, you need to tell Sketchpad which center it should use for a rotation or a dilation. You mark a center point by selecting it and then choosing "Transform \| Mark Center".
choose	To execute some function in some menu. The phrase choose "Transform \| Mark Center", for example, means to execute the "Mark Center" item in the "Transform" menu. You can do this by clicking the left mouse button in the menus, sometimes by clicking the right mouse button within a sketch, or by using the keyboard.
drag	Another important procedure: To drag an object means to click it with the mouse, then to keep the mouse key pressed and move the mouse so that the object moves across the screen accordingly.