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Hardcover ISBN:  9781470447601 
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Product Code:  AMSTEXT/34.B 
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MAA Member Price:  $156.60 $118.35 
AMS Member Price:  $139.20 $105.20 
Hardcover ISBN:  9781470447601 
Product Code:  AMSTEXT/34 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470451479 
Product Code:  AMSTEXT/34.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9781470447601 
eBook ISBN:  9781470451479 
Product Code:  AMSTEXT/34.B 
List Price:  $174.00 $131.50 
MAA Member Price:  $156.60 $118.35 
AMS Member Price:  $139.20 $105.20 

Book DetailsPure and Applied Undergraduate TextsVolume: 34; 2019; 142 ppMSC: Primary 51
This book on twodimensional geometry uses a problemsolving 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 nonconstant curvature in threedimensional 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, twodimensional 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.
Ancillaries:
ReadershipUndergraduates interested in secondary school mathematics teaching; also some engineering and physics majors.

Table of Contents

Cover

Title page

Chapter 1. Introduction

1.1. Design of this book

1.2. Parts V–VII: How many twodimensional geometries are there?

1.3. Parts IV–VII: Some needed multivariable calculus and linear algebra facts

1.4. References and notation

Part I . Neutral geometry

Chapter 2. Euclid’s postulates for plane geometry

2.1. Neutral geometry

2.2. Sum of angles in a triangle in NG

2.3. Are there rectangles in NG?

Part II . Euclidean (plane) geometry

Chapter 3. Rectangles and cartesian coordinates

3.1. Euclid’s Fifth Postulate, the Parallel Postulate

3.2. The distance formula in EG

3.3. Law of Sines and Law of Cosines

3.4. Dilations in EG

3.5. Similarity in EG

Chapter 4. Concurrence and circles in Euclidean geometry

4.1. Concurrence theorems in EG, Ceva’s theorem

4.2. Properties of circles in EG

4.3. Circles and sines and cosines

4.4. Crossratio of points on a circle

4.5. Ptolemy’s theorem

Part III . Spherical geometry

Chapter 5. Surface area and volume of the 𝑅sphere in Euclidean threespace

5.1. Volumes of pyramids

5.2. Magnification principle

5.3. Relation between volume and surface area of a sphere

5.4. Surface area

5.5. Areas on spheres in Euclidean threespace

Part IV . Usual dotproduct for threedimensional Euclidean space

Chapter 6. Euclidean threespace as a metric space

6.1. Points and vectors in Euclidean threespace

6.2. Curves in Euclidean threespace and vectors tangent to them

6.3. Surfaces in Euclidean threespace and vectors tangent to them

Chapter 7. Transformations

7.1. Rigid motions of Euclidean threespace

7.2. Orthogonal matrices

7.3. Linear fractional transformations

Part V . 𝐾geometry

Chapter 8. Changing coordinates

8.1. Bringing the North Pole of the 𝑅sphere to (0,0,1)

8.2. 𝐾geometry: Euclidean lengths and angles in (𝑥,𝑦,𝑧)coordinates

8.3. Congruences, that is, rigid motions

Chapter 9. Uniform coordinates for the twodimensional geometries

9.1. The twodimensional geometries in (𝑥,𝑦,𝑧)coordinates

9.2. Central projection

9.3. Stereographic projection

9.4. Relationship between central and stereographic projection coordinates

Part VI . Return to spherical geometry

Chapter 10. Spherical geometry from an advanced viewpoint

10.1. Rigid motions in spherical geometry

10.2. Spherical geometry is homogeneous

10.3. Lines in spherical geometry

10.4. Central projection in SG

10.5. Stereographic projection in SG

Part VII . Hyperbolic geometry

Chapter 11. The curvature 𝐾 becomes negative

11.1. The world sheet and the light cone

11.2. Hyperbolic geometry is homogeneous

11.3. Lines in hyperbolic geometry

11.4. Central projection in HG

11.5. Stereographic projection in HG

Definitions

Bibliography

Index

Back Cover


Additional Material

Reviews

This delightful text tells the story of twodimensional 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


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This book on twodimensional geometry uses a problemsolving 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 nonconstant curvature in threedimensional 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, twodimensional 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.
Ancillaries:
Undergraduates interested in secondary school mathematics teaching; also some engineering and physics majors.

Cover

Title page

Chapter 1. Introduction

1.1. Design of this book

1.2. Parts V–VII: How many twodimensional geometries are there?

1.3. Parts IV–VII: Some needed multivariable calculus and linear algebra facts

1.4. References and notation

Part I . Neutral geometry

Chapter 2. Euclid’s postulates for plane geometry

2.1. Neutral geometry

2.2. Sum of angles in a triangle in NG

2.3. Are there rectangles in NG?

Part II . Euclidean (plane) geometry

Chapter 3. Rectangles and cartesian coordinates

3.1. Euclid’s Fifth Postulate, the Parallel Postulate

3.2. The distance formula in EG

3.3. Law of Sines and Law of Cosines

3.4. Dilations in EG

3.5. Similarity in EG

Chapter 4. Concurrence and circles in Euclidean geometry

4.1. Concurrence theorems in EG, Ceva’s theorem

4.2. Properties of circles in EG

4.3. Circles and sines and cosines

4.4. Crossratio of points on a circle

4.5. Ptolemy’s theorem

Part III . Spherical geometry

Chapter 5. Surface area and volume of the 𝑅sphere in Euclidean threespace

5.1. Volumes of pyramids

5.2. Magnification principle

5.3. Relation between volume and surface area of a sphere

5.4. Surface area

5.5. Areas on spheres in Euclidean threespace

Part IV . Usual dotproduct for threedimensional Euclidean space

Chapter 6. Euclidean threespace as a metric space

6.1. Points and vectors in Euclidean threespace

6.2. Curves in Euclidean threespace and vectors tangent to them

6.3. Surfaces in Euclidean threespace and vectors tangent to them

Chapter 7. Transformations

7.1. Rigid motions of Euclidean threespace

7.2. Orthogonal matrices

7.3. Linear fractional transformations

Part V . 𝐾geometry

Chapter 8. Changing coordinates

8.1. Bringing the North Pole of the 𝑅sphere to (0,0,1)

8.2. 𝐾geometry: Euclidean lengths and angles in (𝑥,𝑦,𝑧)coordinates

8.3. Congruences, that is, rigid motions

Chapter 9. Uniform coordinates for the twodimensional geometries

9.1. The twodimensional geometries in (𝑥,𝑦,𝑧)coordinates

9.2. Central projection

9.3. Stereographic projection

9.4. Relationship between central and stereographic projection coordinates

Part VI . Return to spherical geometry

Chapter 10. Spherical geometry from an advanced viewpoint

10.1. Rigid motions in spherical geometry

10.2. Spherical geometry is homogeneous

10.3. Lines in spherical geometry

10.4. Central projection in SG

10.5. Stereographic projection in SG

Part VII . Hyperbolic geometry

Chapter 11. The curvature 𝐾 becomes negative

11.1. The world sheet and the light cone

11.2. Hyperbolic geometry is homogeneous

11.3. Lines in hyperbolic geometry

11.4. Central projection in HG

11.5. Stereographic projection in HG

Definitions

Bibliography

Index

Back Cover

This delightful text tells the story of twodimensional 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