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Two-Dimensional Geometries: A Problem-Solving Approach
 
C. Herbert Clemens Ohio State University, Columbus, OH
Two-Dimensional Geometries
Hardcover ISBN:  978-1-4704-4760-1
Product Code:  AMSTEXT/34
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-5147-9
Product Code:  AMSTEXT/34.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-1-4704-4760-1
eBook: ISBN:  978-1-4704-5147-9
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
Two-Dimensional Geometries
Click above image for expanded view
Two-Dimensional Geometries: A Problem-Solving Approach
C. Herbert Clemens Ohio State University, Columbus, OH
Hardcover ISBN:  978-1-4704-4760-1
Product Code:  AMSTEXT/34
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-5147-9
Product Code:  AMSTEXT/34.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-1-4704-4760-1
eBook ISBN:  978-1-4704-5147-9
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 Details
     
     
    Pure and Applied Undergraduate Texts
    Volume: 342019; 142 pp
    MSC: Primary 51

    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.

    Ancillaries:

    Readership

    Undergraduates 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 two-dimensional 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. Cross-ratio of points on a circle
    • 4.5. Ptolemy’s theorem
    • Part III . Spherical geometry
    • Chapter 5. Surface area and volume of the 𝑅-sphere in Euclidean three-space
    • 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 three-space
    • Part IV . Usual dot-product for three-dimensional Euclidean space
    • Chapter 6. Euclidean three-space as a metric space
    • 6.1. Points and vectors in Euclidean three-space
    • 6.2. Curves in Euclidean three-space and vectors tangent to them
    • 6.3. Surfaces in Euclidean three-space and vectors tangent to them
    • Chapter 7. Transformations
    • 7.1. Rigid motions of Euclidean three-space
    • 7.2. Orthogonal matrices
    • 7.3. Linear fractional transformations
    • Part V . 𝐾-geometry
    • Chapter 8. Changing coordinates
    • 8.1. Bringing the North Pole of the 𝑅-sphere to (0,0,1)
    • 8.2. 𝐾-geometry: Euclidean lengths and angles in (𝑥,𝑦,𝑧)-coordinates
    • 8.3. Congruences, that is, rigid motions
    • Chapter 9. Uniform coordinates for the two-dimensional geometries
    • 9.1. The two-dimensional 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
  • Reviews
     
     
    • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 342019; 142 pp
MSC: Primary 51

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.

Ancillaries:

Readership

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 two-dimensional 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. Cross-ratio of points on a circle
  • 4.5. Ptolemy’s theorem
  • Part III . Spherical geometry
  • Chapter 5. Surface area and volume of the 𝑅-sphere in Euclidean three-space
  • 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 three-space
  • Part IV . Usual dot-product for three-dimensional Euclidean space
  • Chapter 6. Euclidean three-space as a metric space
  • 6.1. Points and vectors in Euclidean three-space
  • 6.2. Curves in Euclidean three-space and vectors tangent to them
  • 6.3. Surfaces in Euclidean three-space and vectors tangent to them
  • Chapter 7. Transformations
  • 7.1. Rigid motions of Euclidean three-space
  • 7.2. Orthogonal matrices
  • 7.3. Linear fractional transformations
  • Part V . 𝐾-geometry
  • Chapter 8. Changing coordinates
  • 8.1. Bringing the North Pole of the 𝑅-sphere to (0,0,1)
  • 8.2. 𝐾-geometry: Euclidean lengths and angles in (𝑥,𝑦,𝑧)-coordinates
  • 8.3. Congruences, that is, rigid motions
  • Chapter 9. Uniform coordinates for the two-dimensional geometries
  • 9.1. The two-dimensional 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 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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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