Math 5270
Transformational Geometry
Day 1
Summer 13
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Introductions to the course and each other
Fill out the survey
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Introductions to the course and each other
Fill out the survey
Share the responses to questions #1, 3, 6 and 10 with your
table-mates.
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Symmetries
of quadrilaterals
Use geogebra to answer the questions on the first page.
Work in pairs.
Class discussion in about half an hour.
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Groups of transformations determine geometries
The fewer symmetries ... the more general space.
Different groups of transformations... different geometries.
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Groups
of transformations
determine
geometries
The fewer symmetries ... the more general space.
Different groups of transformations... different geometries.
Euclidean, but others, too.
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Euclid and Hilbert’s approach vs. coordinate approach
Descartes: geometry was as Euclid described it, and numbers
assist in studying geometric figures.
Discovery of things non-Euclidean led to the idea of defining
points, lines, lengths...
Arithmetization of geometry
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Describe
how
a coordinate
plane - www.PDFAnno
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Coordinate plane
Courtesy of Parallel Postulate:
The problem here seems to be that this plane is unlike Euclid’s
plane! We have a favored point and favored lines...
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Annotator
List
three transformations
of of
thePDF
plane
in your- www.PDFAnno
notes
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Produced with a Trial Version of PDF Annotator - www.PDFAnno
Produced with a Trial Version of PDF Annotator - www.PDFAnno
Transformations
For each of the given transformations, describe their effect on
points of the plane, coordinate axes, and characteristics they leave
invariant.
f (x, y ) = (x 3 , y 3 )
g (x, y ) = (2x, 3y )
√
h(x, y ) = ( 3 x, e y )
i(x, y ) = (cos x, sin x)
j(x, y ) = (−x, x + 3)
l(x, y ) = (x 3 − x, y )
m(x, y ) = (3y , x + 2)
n(x, y ) = (x + 2, y − 3)
p(x, y ) = (0.6x − 0.8y , 0.8x + 0.6y )
q(x, y ) = (−0.6x + 0.8y , 0.8x + 0.6y )
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