In the figure below we have created a rectangle ABCD. The new figure EFGH is the image of ABCD under a shear transformation in the x direction.
In general an affine transformation is an invertible transformation of points (x,y) in the Euclidean plane with the form
The restriction on ad-bc not equaling zero guarantees that an affine transformation is invertible, i.e. that the effects of the transformation can be reversed. We require this condition for a general Euclidean transformation because we think of such transformations as movements or re-configurations of Euclidean objects that do not destroy any essential geometric properties of those objects. Thus, any Euclidean transformation should be reversible, or invertible.
From the definition above we see that any affine transformation is uniquely defined by a choice of the numbers a,b,c,d,e,and f (with ad-bc non-zero). Let us look at how we can use affine transformations in Geometry Explorer to carry out the shear transformation discussed above.
To begin with construct a rectangle ABCD as shown in the figure above. To define an affine transform, we need to go to the ``Custom'' menu in the Transform Panel and select ``Affine...''. A dialog box titled ``Build an Affine Mapping'' will pop up as shown below. We can put in values for a,b,c,d,e, and f in the appropriate text fields in this dialog box. (Note that we can quickly go from one text field to another by hitting the Tab key on the keyboard.) To define a shear transformation put in the following values: a=1.0, b=1.0, c=0.0, d=1.0, e=0.0, and f=0.0. Then, name the transformation ``shear'' and hit the Okay button in the dialog box to finish the definition of the shear transformation.
Note the checkbox labeled "Check Inverse."
This can be used to define a non-invertible
transformation. By clicking the checkbox off, the program will
allow non-invertible transformations to be carried out. Such
transformations are useful, for example, in constructing iterated
function systems (IFS). For more information on the topic of IFS,
consult the pdf manual for Geometry Explorer.
At this point, our new transformation can be used to transform objects in the Canvas. To transform rectangle ABCD, first select the rectangle by dragging a selection box around the rectangle. Then, click on the ``Custom'' menu. You will see the transformation ``shear'' listed near the bottom of the pull-down menu. Drag down to ``shear'' and select this menu item. The rectangle will be transformed as shown below.
Affine transformations include all of the standard Geometry Explorer transformations of rotations, reflections, translations, and dilations. An interesting exercise would be to determine the values of a,b,c,d,e,and f that give each of the four types of standard transformations.
Affine transformations are strictly Euclidean transformations and are not available in hyperbolic geometry. However, there is a general class of hyperbolic transformations called Moebius Transformations that act in a manner very similar to the class of affine transformations. Look here for an example using Moebius Transformations.
Important Note: Circles do not
transform
as one would expect under an affine transformation such as a
shear.
When a circle is transformed in Geometry Explorer, the center and
radius
points are transformed and a new circle is constructed on these two
transformed
points. Thus, circles always transform to circles.
Under
a shear transformation, if all of the points of the circle were to be
transformed,
then the circle could transform to an ellipse. However,
for
performance reasons, transformations in Geometry Explorer of objects
such
as circles, arcs, lines, filled areas, etc, only transform defining
points
of these objects. Ellipses are planned for a future release of
Geometry
Explorer, at which time shears will transform circles in the standard
way.