Introduction to Isometries: The Holy Grail
Let's review our story. We began by thinking about creating
computer animations. To do an animation of a teapot, for example, we
first constructed a model of the teapot out of polygons, since
polygons are relatively easy to draw. To animate the teapot, then,
all we had to do was figure out how to move around a polygon.
For a computer to effectively move around a polygon, we needed a
formula for computing new locations for the coordinates of the
vertices of the polygons. However, coming up with the proper
transformations is difficult, since in general, transformations will
distort the shapes of the objects they move. So we began looking for
transformation that introduced no distortion, often called isometries.
A bit of thinking revealed four different kinds of isometries. If
you think of picking up a paper square from a table top, and setting
it back down, the result will either be a reflection, translation,
rotation or glide reflection. We also concluded that these
transformations are all fundamentally different by considering whether
or not they have fixed points and how many, and whether or not they are
orientation reversing, i.e. producing mirror images.
Finally, we convinced ourselves that these are the only isometries
of the plane by asking how they combine. Doing one transformation
followed by another always produced a net effect which was again
a transformation already on
the list. For eaxmple, two reflections produce the net effect of a
rotation or a translation, depending on whether their mirror lines
cross.
So, all that remains is to produce the formulas for each kind of
isometry, the holy grail of our quest. A bit of an anticlimax,
perhaps, since typically, the actual formulas for isometries are
buried deep in the software we use, and of relatively little general
interest. However, understanding about isometries and how
they work pays you back, whether you are just looking in the mirror,
or sitting down to watch the latest special effect extravaganza from
Hollywood.
And just in case you are a programmer or a mathematician, a compendium of useful
isometry formulas is available in the
Science U library.
Complete Seminar Series available in the Science U library.
