3.7.3Suppose thatf(A) = (1.8,1.6),f(B) = (1,1), andf(C) = (0.4,1.8). Howdo I know that this is a glide reflection?3.7.4State a simple test for telling whetherfis a translation, rotation, or glidereflection from the positions off(A),f(B), andf(C).3.8DiscussionThe discovery of coordinates is rightly considered a turning point in thedevelopment of mathematics because it reveals a vast new panorama ofgeometry, open to exploration in at least three different directions.•Description of curves by equations, and their analysis by algebra.This direction is calledalgebraic geometry, and the curves describedby polynomial equations are calledalgebraic curves. Straight lines,described by the linear equationsax+by+c=0, are called curves ofdegree 1. Circles, described by the equations(x−a)2+(y−b)2=r2,are curves ofdegree 2, and so on.One can see that there are curves of arbitrarily high degree, so mostof algebraic geometry is beyond the scope of this book. Even thecurves of degree 3 are worth a book of their own, so for them, andother algebraic curves, we refer readers elsewhere. Two excellentbooks, which show how algebraic geometry relates to other parts ofmathematics, areElliptic Curvesby H. P. McKean and V. Moll andPlane Algebraic Curvesby E. Brieskorn and H. Kn¨orrer.
643Coordinates•Algebraic study of objects described by linear equations (such aslines and planes). Even this is a big subject, calledlinear algebra.Although it is technically part of algebraic geometry, it has a specialflavor, very close to that of Euclidean geometry. We explore planegeometry from the viewpoint of linear algebra in Chapter 4, and laterwe make some brief excursions into three and four dimensions.The real strength of linear algebra is its ability to describe spacesof any number of dimensions in geometric language.Again, thisinvestigation is beyond our scope, but we will recommend additionalreading at the appropriate places.•The study of transformations, which draws on the special branch ofalgebra known asgroup theory. Because many geometric transfor-mations are described by linear equations, this study overlaps withlinear algebra. The role of transformations was first emphasized bythe German mathematician Felix Klein, in an address he delivered atthe University of Erlangen in 1872. His address, known by its Ger-man name theErlanger Programm, characterizes geometry as thestudy oftransformation groupsand theirinvariants.So far, we have seen only one transformation group and a handful ofinvariants—the group of isometriesofR2and what it leaves invariant(length, angle, straightness)—so the importance of Klein’s idea can hardlybe clear yet. However, in Chapter 4 we introduce a very different group oftransformations and a very different invariant—theprojective transforma-tionsand thecross-ratio—so readers are asked to bear with us. In Chapters7 and 8, we develop Klein’s idea in some generality and give another sig-nificant example, the geometry of the “non-Euclidean” plane.
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- Spring '08
- Geometry, triangle, Euclidean geometry, Euclid