An Elementary Course in Synthetic Projective Geometry

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Ngôn Ngữ Nội Dung Sách
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Năm xuất bản
2005
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            CHAPTER I - ONE-TO-ONE CORRESPONDENCE

            1. Definition of one-to-one correspondence

            2. Consequences of one-to-one correspondence

            3. Applications in mathematics

            4. One-to-one correspondence and enumeration

            5. Correspondence between a part and the whole

            6. Infinitely distant point

            7. Axial pencil; fundamental forms

            8. Perspective position

            9. Projective relation

            10. Infinity-to-one correspondence

            11. Infinitudes of different orders

            12. Points in a plane

            13. Lines through a point

            14. Planes through a point

            15. Lines in a plane

            16. Plane system and point system

            17. Planes in space

            18. Points of space

            19. Space system

            20. Lines in space

            21. Correspondence between points and numbers

            22. Elements at infinity

            PROBLEMS

            CHAPTER II - RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE-TO-ONE CORRESPONDENCE WITH EACH OTHER

            23. Seven fundamental forms

            24. Projective properties

            25. Desargues's theorem

            26. Fundamental theorem concerning two complete quadrangles

            27. Importance of the theorem

            28. Restatement of the theorem

            29. Four harmonic points

            30. Harmonic conjugates

            31. Importance of the notion of four harmonic points

            32. Projective invariance of four harmonic points

            33. Four harmonic lines

            34. Four harmonic planes

            35. Summary of results

            36. Definition of projectivity

            37. Correspondence between harmonic conjugates

            38. Separation of harmonic conjugates

            39. Harmonic conjugate of the point at infinity

            40. Projective theorems and metrical theorems. Linear construction

            41. Parallels and mid-points

            42. Division of segment into equal parts

            43. Numerical relations

            44. Algebraic formula connecting four harmonic points

            45. Further formulae

            46. Anharmonic ratio

            PROBLEMS

            CHAPTER III - COMBINATION OF TWO PROJECTIVELY RELATED FUNDAMENTAL FORMS

            47. Superposed fundamental forms. Self-corresponding elements

            48. Special case

            49. Fundamental theorem. Postulate of continuity

            50. Extension of theorem to pencils of rays and planes

            51. Projective point-rows having a self-corresponding point in common

            52. Point-rows in perspective position

            53. Pencils in perspective position

            54. Axial pencils in perspective position

            55. Point-row of the second order

            56. Degeneration of locus

            57. Pencils of rays of the second order

            58. Degenerate case

            59. Cone of the second order

            PROBLEMS

            CHAPTER IV - POINT-ROWS OF THE SECOND ORDER

            60. Point-row of the second order defined

            61. Tangent line

            62. Determination of the locus

            63. Restatement of the problem

            64. Solution of the fundamental problem

            65. Different constructions for the figure

            66. Lines joining four points of the locus to a fifth

            67. Restatement of the theorem

            68. Further important theorem

            69. Pascal's theorem

            70. Permutation of points in Pascal's theorem

            71. Harmonic points on a point-row of the second order

            72. Determination of the locus

            73. Circles and conics as point-rows of the second order

            74. Conic through five points

            75. Tangent to a conic

            76. Inscribed quadrangle

            77. Inscribed triangle

            78. Degenerate conic

            PROBLEMS

            CHAPTER V - PENCILS OF RAYS OF THE SECOND ORDER

            79. Pencil of rays of the second order defined

            80. Tangents to a circle

            81. Tangents to a conic

            82. Generating point-rows lines of the system

            83. Determination of the pencil

            84. Brianchon's theorem

            85. Permutations of lines in Brianchon's theorem

            86. Construction of the penvil by Brianchon's theorem

            87. Point of contact of a tangent to a conic

            88. Circumscribed quadrilateral

            89. Circumscribed triangle

            90. Use of Brianchon's theorem

            91. Harmonic tangents

            92. Projectivity and perspectivity

            93. Degenerate case

            94. Law of duality

            PROBLEMS

            CHAPTER VI - POLES AND POLARS

            95. Inscribed and circumscribed quadrilaterals

            96. Definition of the polar line of a point

            97. Further defining properties

            98. Definition of the pole of a line

            99. Fundamental theorem of poles and polars

            100. Conjugate points and lines

            101. Construction of the polar line of a given point

            102. Self-polar triangle

            103. Pole and polar projectively related

            104. Duality

            105. Self-dual theorems

            106. Other correspondences

            PROBLEMS

            CHAPTER VII - METRICAL PROPERTIES OF THE CONIC SECTIONS

            107. Diameters. Center

            108. Various theorems

            109. Conjugate diameters

            110. Classification of conics

            111. Asymptotes

            112. Various theorems

            113. Theorems concerning asymptotes

            114. Asymptotes and conjugate diameters

            115. Segments cut off on a chord by hyperbola and its asymptotes

            116. Application of the theorem

            117. Triangle formed by the two asymptotes and a tangent

            118. Equation of hyperbola referred to the asymptotes

            119. Equation of parabola

            120. Equation of central conics referred to conjugate diameters

            PROBLEMS

            CHAPTER VIII - INVOLUTION

            121. Fundamental theorem

            122. Linear construction

            123. Definition of involution of points on a line

            124. Double-points in an involution

            125. Desargues's theorem concerning conics through four points

            126. Degenerate conics of the system

            127. Conics through four points touching a given line

            128. Double correspondence

            129. Steiner's construction

            130. Application of Steiner's construction to double correspondence

            131. Involution of points on a point-row of the second order.

            132. Involution of rays

            133. Double rays

            134. Conic through a fixed point touching four lines

            135. Double correspondence

            136. Pencils of rays of the second order in involution

            137. Theorem concerning pencils of the second order in involution

            138. Involution of rays determined by a conic

            139. Statement of theorem

            140. Dual of the theorem

            PROBLEMS

            CHAPTER IX - METRICAL PROPERTIES OF INVOLUTIONS

            141. Introduction of infinite point; center of involution

            142. Fundamental metrical theorem

            143. Existence of double points

            144. Existence of double rays

            145. Construction of an involution by means of circles

            146. Circular points

            147. Pairs in an involution of rays which are at right angles. Circular involution

            148. Axes of conics

            149. Points at which the involution determined by a conic is circular

            150. Properties of such a point

            151. Position of such a point

            152. Discovery of the foci of the conic

            153. The circle and the parabola

            154. Focal properties of conics

            155. Case of the parabola

            156. Parabolic reflector

            157. Directrix. Principal axis. Vertex

            158. Another definition of a conic

            159. Eccentricity

            160. Sum or difference of focal distances

            PROBLEMS

            CHAPTER X - ON THE HISTORY OF SYNTHETIC PROJECTIVE GEOMETRY

            161. Ancient results

            162. Unifying principles

            163. Desargues

            164. Poles and polars

            165. Desargues's theorem concerning conics through four points

            166. Extension of the theory of poles and polars to space

            167. Desargues's method of describing a conic

            168. Reception of Desargues's work

            169. Conservatism in Desargues's time

            170. Desargues's style of writing

            171. Lack of appreciation of Desargues

            172. Pascal and his theorem

            173. Pascal's essay

            174. Pascal's originality

            175. De la Hire and his work

            176. Descartes and his influence

            177. Newton and Maclaurin

            178. Maclaurin's construction

            179. Descriptive geometry and the second revival

            180. Duality, homology, continuity, contingent relations

            181. Poncelet and Cauchy

            182. The work of Poncelet

            183. The debt which analytic geometry owes to synthetic geometry

            184. Steiner and his work

            185. Von Staudt and his work

            186. Recent developments

            INDEX

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