Math 5270
Transformational Geometry
Day 3
Summer 13
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Isometries
Transformation
A transformation of the plane is simply a function f : R2 → R2 . In
other words, a function that sends points to points.
Isometry
A transformation f is called an isometry if it sends any two points,
P1 and P2 , to points f (P1 ) and f (P2 ) the same distance apart. In
other words, an isometry is a function f with the property
|f (P1 )f (P2 )| = |P1 P2 |
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Translations
Definition
using vectors explicitly.
using vectors implicitly.
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How do we show that translations preserve length?
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Is ”preserving collinearity” a consequence of ”preserving
length”?
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gebraic way to show that straight lines are sent to
straight lines
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What else do translations preserve?
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Fixed points, fixed sets
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What happens when you compose two translations?
Three? More than that?
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Identity? Inverses?
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Group of translations T
Group
A group is a set G together with a mapping G × G → G which is
associative, has an identity element and every element has an
inverse.
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Students often think of translating first in the
left/right direction, then up/down direction. Why might
this be a problem, and why isn’t it?
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The idea was to translate one of the circles in
the direction and by magnitutde a. If the
image of the circle does not intersect the
other circle the task is not possible. If it does,
then the point of intersection is a away from
a point on the original circle (since it's the
image of a point on the original circle) and it
lies on the second circle. So, the image and its
preimage are the endpoints of the segment
we were ;ooking for.