A Course in Modern GeometriesSpringer Science & Business Media, 9. mars 2013 - 441 sider A Course in Modern Geometries is designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary school. Chapter 1 presents several finite geometries in an axiomatic framework. Chapter 2 continues the synthetic approach as it introduces Euclid's geometry and ideas of non-Euclidean geometry. In Chapter 3, a new introduction to symmetry and hands-on explorations of isometries precedes the extensive analytic treatment of isometries, similarities and affinities. A new concluding section explores isometries of space. Chapter 4 presents plane projective geometry both synthetically and analytically. The extensive use of matrix representations of groups of transformations in Chapters 3-4 reinforces ideas from linear algebra and serves as excellent preparation for a course in abstract algebra. The new Chapter 5 uses a descriptive and exploratory approach to introduce chaos theory and fractal geometry, stressing the self-similarity of fractals and their generation by transformations from Chapter 3. Each chapter includes a list of suggested resources for applications or related topics in areas such as art and history. The second edition also includes pointers to the web location of author-developed guides for dynamic software explorations of the Poincaré model, isometries, projectivities, conics and fractals. Parallel versions of these explorations are available for "Cabri Geometry" and "Geometer's Sketchpad". Judith N. Cederberg is an associate professor of mathematics at St. Olaf College in Minnesota. |
Inni boken
Resultat 6-10 av 91
Side 14
... proof , figures are included as part of the proofs whenever appropriate ; but the narrative portions of the proofs are constructed so as to be completely independent of the figures . In models P.1 , P.2 , and P.3 , the number of points ...
... proof , figures are included as part of the proofs whenever appropriate ; but the narrative portions of the proofs are constructed so as to be completely independent of the figures . In models P.1 , P.2 , and P.3 , the number of points ...
Side 15
... proof of the previous theorem , it can be shown that these lines are distinct and there are no other lines through P. So in this case there are exactly n + 1 lines through P. Case 2 ( P is on 1 ) : Assume P = P1 . Axiom P.1 guarantees ...
... proof of the previous theorem , it can be shown that these lines are distinct and there are no other lines through P. So in this case there are exactly n + 1 lines through P. Case 2 ( P is on 1 ) : Assume P = P1 . Axiom P.1 guarantees ...
Side 18
... proof of Theorem P.5 . ] 13. Prove : In an affine plane of order n , each line contains exactly n points . 14. Prove : In an affine plane of order n , each line 1 has exactly n - 1 lines that do not intersect 1 . 15. Prove : In an ...
... proof of Theorem P.5 . ] 13. Prove : In an affine plane of order n , each line contains exactly n points . 14. Prove : In an affine plane of order n , each line 1 has exactly n - 1 lines that do not intersect 1 . 15. Prove : In an ...
Side 27
... 1 is illustrated in the proofs of the following two theorems , which verify that the correspondence between poles and polars is one - to - one . P R S P FIGURE 1.9 Proof DC.1 . Theorem 1.5 . Desargues ' Configurations 27.
... 1 is illustrated in the proofs of the following two theorems , which verify that the correspondence between poles and polars is one - to - one . P R S P FIGURE 1.9 Proof DC.1 . Theorem 1.5 . Desargues ' Configurations 27.
Side 28
Judith N. Cederberg. P R S P FIGURE 1.9 Proof DC.1 . Theorem DC.2 Each point has exactly one polar . Proof Let P be an arbitrary point . By Axiom DC.2 , P has at least one polar p . Assume P has a second polar p ' . By Axioms DC.4 and DC ...
Judith N. Cederberg. P R S P FIGURE 1.9 Proof DC.1 . Theorem DC.2 Each point has exactly one polar . Proof Let P be an arbitrary point . By Axiom DC.2 , P has at least one polar p . Assume P has a second polar p ' . By Axioms DC.4 and DC ...
Innhold
1 | |
5 | |
17 | |
Geometric Transformations of the Euclidean Plane | 99 |
4 | 116 |
6 | 128 |
7 | 135 |
13 | 175 |
Projective Geometry | 213 |
10 | 269 |
Appendices | 389 |
Geometry | 399 |
Andre utgaver - Vis alle
Vanlige uttrykk og setninger
AABC affine transformation algebra analytic angle sum APQR assume asymptotic triangles axiomatic system axis collineation congruent Construct contains Corollary corresponding Definition determined dimension direct isometry distance distinct points elements elliptic geometry equation equilateral triangle Euclid's Euclidean geometry Euclidean plane exactly Exercise fifth postulate FIGURE Find the matrix fractal frieze group frieze pattern glide reflection H(AB homogeneous coordinates homogeneous parameters hyperbolic geometry ideal points incident invariant points label maps Mathematics matrix representation midpoint non-Euclidean geometry Note P₁ pair parallel lines pencil of points pencils of lines perpendicular perspective plane of order Playfair's axiom point conic point set points and lines polar projective geometry Proof Let proof of Theorem properties prototile Prove Theorem real numbers result rotation Saccheri quadrilateral segment self-similarity sensed parallel set of points sides Sierpinski triangle similar straight lines symmetry groups tiling translation ultraparallel unique vector verify vertices