**Pure and Applied Undergraduate Texts**

Volume: 34;
2019;
142 pp;
Hardcover

MSC: Primary 51;

**Print ISBN: 978-1-4704-4760-1
Product Code: AMSTEXT/34**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

**Electronic ISBN: 978-1-4704-5147-9
Product Code: AMSTEXT/34.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

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#### Supplemental Materials

# Two-Dimensional Geometries: A Problem-Solving Approach

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*C. Herbert Clemens*

This book on two-dimensional geometry uses a
problem-solving approach to actively engage students in the learning
process. The aim is to guide readers through the story of the subject,
while giving them room to discover and partially construct the story
themselves. The book bridges the study of plane geometry and the study
of curves and surfaces of non-constant curvature in three-dimensional
Euclidean space. One useful feature is that the book can be adapted to
suit different audiences.

The first half of the text covers
plane geometry without and with Euclid's Fifth Postulate, followed by
a brief synthetic treatment of spherical geometry through the excess
angle formula. This part only requires a background in high school
geometry and basic trigonometry and is suitable for a quarter course
for future high school geometry teachers. A brief foray into the
second half could complete a semester course.

The second half of the text gives a uniform treatment of all the
complete, simply connected, two-dimensional geometries of constant
curvature, one geometry for each real number (its curvature),
including their groups of isometries, geodesics, measures of lengths
and areas, as well as formulas for areas of regions bounded by
polygons in terms of the curvature of the geometry and the sum of the
interior angles of the polygon. A basic knowledge of real linear
algebra and calculus of several (real) variables is useful background
for this portion of the text.

Exercises are freely available electronically: click
here.

#### Readership

Undergraduates interested in secondary school mathematics teaching; also some engineering and physics majors.

#### Reviews & Endorsements

This delightful text tells the story of two-dimensional geometries in seven parts...The material presented here would give future teachers a depth and breadth of geometric understanding that would allow them to teach for understanding...Clemens succeeds in telling a story of plane geometries so that various topics flow naturally, one topic building toward the next.

-- Sr. Barbara Reynolds, MAA Reviews

#### Table of Contents

# Table of Contents

## Two-Dimensional Geometries: A Problem-Solving Approach

- Cover Cover11
- Title page iii4
- Chapter 1. Introduction 110
- Part I . Neutral geometry 716
- Part II . Euclidean (plane) geometry 1726
- Part III . Spherical geometry 3948
- Part IV . Usual dot-product for three-dimensional Euclidean space 4958
- Part V . 𝐾-geometry 6978
- Part VI . Return to spherical geometry 97106
- Part VII . Hyperbolic geometry 111120
- Back Cover Back Cover1153