front cover -- copyright page -- title page -- CONTENTS -- PREFACE -- INTRODUCTION -- FOREWORD -- Chapter 1 MAMIKON'S SWEEPING-TANGENT THEOREM -- 1.1 INTRODUCTION -- 1.2 EVOLUTION OF MAMIKON'S THEOREM -- 1.3 APPLICATION TO SLICES OF SPHERICAL SHELLS -- 1.4 CONSTANT-LENGTH TANGENT SWEEP AND TANGENT CLUSTER OF A PLANE CURVE -- 1.5 VARIABLE-LENGTH TANGENT SWEEP. SPACE CURVES -- 1.6 APPLICATION TO THE TRACTRIX -- 1.7 SUBTANGENTS USED TO DRAW TANGENT LINES TO PLANE CURVES -- 1.8 EXPONENTIAL CURVES -- 1.9 AREA OF A HYPERBOLIC SEGMENT -- 1.10 AREA OF A PARABOLIC SEGMENT -- 1.11 REAL POSITIVE POWERS -- 1.12 GENERAL NEGATIVE POWERS -- 1.13 AN ALTERNATIVE APPROACH FOR NEGATIVE POWERS -- 1.14 A REVERSE TYPE OF APPLICATION -- 1.15 APPLICATION TO THE LIMAC¸ON OF PASCAL -- 1.16 APPLICATION TO PHYSICS -- NOTES ON CHAPTER 1 -- Chapter 2 CYCLOIDS AND TROCHOIDS -- 2.1 INTRODUCTION -- 2.2 AREA OF CYCLOIDAL CAP (PROOF OF LEMMA 2.1) -- 2.3 AREA OF CYCLOIDAL SECTOR (PROOF OF THEOREM 2.1) -- 2.4 EPICYCLOIDAL AND HYPOCYCLOIDAL CAP AND SECTOR -- 2.5 AREAS OF CYCLOIDAL RADIAL AND ORDINATE SETS -- 2.6 AREA OF A GENERAL TROCHOIDAL CAP AND SECTOR -- 2.7 NEW APPLICATIONS OF THEOREM 2.8 -- 2.8 SPECIAL RESULTS ON CYCLOIDAL AREA -- NOTES ON CHAPTER 2 -- Chapter 3 CYCLOGONS AND TROCHOGONS -- 3.1 INTRODUCTION -- 3.2 CYCLOGONS -- 3.3 AREA OF A CYCLOGONAL ARCH GENERATED BY AREGULAR POLYGON -- 3.4 TROCHOGONS: GENERALIZED CYCLOGONS -- 3.5 SPECIAL TROCHOGONS -- 3.6 ARCLENGTH OF CYCLOGONAL ARCHES -- 3.7 ARCLENGTH OF EPICYCLOGONS AND HYPOCYCLOGONS -- 3.8 SOME SPECIAL TROCHOGONS -- 3.9 INCOMPLETE TROCHOGONS -- 3.10 ARCLENGTH AND AREA OF INVOLUTOGONS -- 3.11 AREA AND ARCLENGTH OF AUTOGONS -- 3.12 ELLIPTIC, HYPERBOLIC, AND PARABOLIC CATENARIES -- 3.13 PEDAL CURVES AND STEINER'S THEOREMS -- 3.14 REDUCTION FORMULAS FOR ARCLENGTHS AND AREAS -- NOTES ON CHAPTER 3.

Chapter 4 CIRCUMGONS AND CIRCUMSOLIDS -- 4.1 INTRODUCTION -- 4.2 CIRCUMGONS -- 4.3 CIRCUMGONAL RINGS -- 4.4 CENTROIDS OF CIRCUMGONAL REGIONS -- 4.5 CENTROIDS OF CIRCUMGONAL RINGS -- 4.6 EXTENSIONS TO 3-SPACE -- 4.7 FAMILIAR CIRCUMSOLIDS -- 4.8 BUILDING BLOCKS OF A CIRCUMSOLID -- 4.9 APPLICATIONS OF THEOREM 4.13 -- 4.10 OPTIMAL CIRCUMGONS AND CIRCUMSOLIDS -- 4.11 INTERSECTION OF A CONE AND A CYLINDER HAVING THE SAME INSPHERE -- 4.12 CENTROIDS OF CIRCUMSOLIDS -- 4.13 CIRCUMSOLID SHELLS -- 4.14 CENTROIDS OF CIRCUMSOLID SHELLS -- NOTES ON CHAPTER 4 -- Chapter 5 THE METHOD OF PUNCTURED CONTAINERS -- PART 1: ARCHIMEDEAN GLOBES -- 5.1 INTRODUCTION -- 5.2 VOLUME OF A SPHERE -- 5.3 VOLUME OF A SPHERICAL SHELL -- 5.4 VOLUME OF AN ARCHIMEDEAN GLOBE -- 5.5 VOLUME OF AN ARCHIMEDEAN SHELL -- 5.6 SURFACE AREA OF AN ARCHIMEDEAN DOME -- 5.7 INCONGRUENT SOLIDS WITH EQUAL VOLUMES AND EQUAL SURFACE AREAS -- 5.8 QUADRATURE OF THE SINE CURVE -- 5.9 APPLICATION TO CENTROIDS -- PART 2: GENERALIZED ARCHIMEDEAN DOMES -- 5.10 REDUCIBLE SOLIDS -- 5.11 POLYGONAL ELLIPTIC DOMES AND SHELLS -- 5.12 GENERAL ELLIPTIC DOMES -- 5.13 NONUNIFORM ELLIPTIC DOMES -- 5.14 FORMULAS FOR VOLUME AND CENTROID -- 5.15 THE NECESSITY OF ELLIPTIC PROFILES -- NOTES ON CHAPTER 5 -- Chapter 6 UNWRAPPING CURVES FROM CYLINDERS AND CONES -- PART 1: UNWRAPPING FROM CYLINDERS -- 6.1 INTRODUCTION -- 6.2 UNWRAPPING AN ELLIPSE FROM A CIRCULAR CYLINDER -- 6.3 CURVE OF INTERSECTION OF TWO CYLINDERS -- 6.4 UNWRAPPING A CURVE FROM ANY CYLINDER -- 6.5 UNWRAPPING A CURVE FROM A CIRCULAR CYLINDER -- 6.6 ROTATING THE MAIN CYLINDER -- 6.7 CYLINDER TO CYLINDER -- 6.8 DRILLED CYLINDER -- 6.9 TILTED CUTTING CYLINDER -- 6.10 UNWRAPPING CURVES FROM A GENERAL CYLINDER -- 6.11 APPLICATIONS TO GRAPHICS -- PART 2: UNWRAPPING FROM CONES -- 6.12 UNWRAPPING CURVES FROM A RIGHT CIRCULAR CONE.

6.13 UNWRAPPED BASE AND PRESERVATION OF ARCLENGTH -- 6.14 REFORMULATED PROBLEM IN TERMS OF CEILING PROJECTION -- 6.15 CEILING PROJECTION AND UMBRELLA TRANSFORMATION -- 6.16 CONE TO CONE -- 6.17 UNWRAPPING A CONIC SECTION FROM A CONE TO A PLANE -- 6.18 EXAMPLES OF GENERALIZED CONICS -- 6.19 LIMITING CASES -- 6.20 OTHER CURVES ON A CONE -- 6.21 VERTICAL WALL PROJECTION -- 6.22 CEILING AND WALL PROJECTIONS OF A ROTATED CURVE -- 6.23 TILTED WALL PROJECTION -- 6.24 ARCLENGTH AND AREA -- NOTES ON CHAPTER 6 -- Chapter 7 NEW DESCRIPTIONS OF CONICS VIA TWISTED CYLINDERS, FOCAL DISKS, AND DIRECTORS -- 7.1 INTRODUCTION -- PART 1: FOCAL DISK-DIRECTOR DESCRIPTION OF NONCIRCULAR CONICS -- 7.2 DISK-DIRECTOR RATIO. FOCAL DISK-DIRECTOR PROPERTY -- 7.3 DISK-DIRECTOR RATIO RELATED TO ECCENTRICITY -- 7.4 FOCAL DISK-DIRECTOR THEOREM AND ITS CONVERSE -- PART 2: BIFOCAL DISK DESCRIPTION OF CONICS -- 7.5 BIFOCAL DISK PROPERTY -- 7.6 BIFOCAL DISK THEOREM AND ITS CONVERSE -- 7.7 TWO NEW CHARACTERIZATIONS OF THE CONICS -- PART 3: SUPPLEMENTARY RESULTS -- 7.8 MORE ON DIRECTORS AND THE BIFOCAL DISK PROPERTY -- 7.9 LOCATING A FOCAL DISK AND ITS DIRECTOR FOR A CONIC -- 7.10 EXAMPLES OF CONICS WITH FIXED FOCAL DISKS -- 7.11 APPLICATIONS OF THE BIFOCAL DISK PROPERTY TO TRACING CONICS -- 7.12 FOCAL DISKS AND DIRECTORS FOR THE ELLIPSE AS A SECTION OF A CIRCULAR CYLINDER -- 7.13 SURPRISING PROPERTY OF HYPERBOLAS -- NOTES ON CHAPTER 7 -- Chapter 8 ELLIPSE TO HYPERBOLA: "WITH THIS STRING I THEE WED" -- 8.1 STRING CONSTRUCTION FOR BOTH ELLIPSE AND HYPERBOLA -- 8.2 FOCAL CIRCLES FOR ELLIPSE AND HYPERBOLA -- 8.3 TWO LOCUS PROPERTIES RELATING THE ELLIPSE AND HYPERBOLA -- 8.4 EXTENDED BIFOCAL PROPERTY: ELLIPSE AND HYPERBOLA -- 8.5 BIFOCAL PROPERTIES TRANSFERRED TO THE PARABOLA -- 8.6 CIRCULAR DIRECTRICES AND WAVE MOTION -- 8.7 EXTENDED ECCENTRICITY PROPERTIES OF CONICS -- NOTES ON CHAPTER 8.

Chapter 9 TRAMMELS -- 9.1 INTRODUCTION. STANDARD TRAMMEL -- 9.2 ELLIPSE TRACED BY A POINT ON A STANDARD TRAMMEL -- 9.3 ASTROID AS THE ENVELOPE OF A STANDARD TRAMMEL -- 9.4 EQUATIONS FOR ELLIPSE, TRAMMEL, AND ASTROID -- 9.5 COMMON TANGENCY OF TRAMMEL, ELLIPSE, AND ASTROID -- 9.6 AREA OF AN ELLIPTICAL SECTOR -- 9.7 AREA OF AN ASTROIDAL SECTOR -- 9.8 ZIGZAG TRAMMEL -- 9.9 FLEXIBLE TRAMMEL -- 9.10 TANGENCY OF A FLEXIBLE TRAMMEL AND ITS TRACE -- 9.11 ENVELOPE OF A TRAMMEL AND OF ITS FAMILY OF TRACES -- 9.12 APPLICATION: GRAPHIC CONSTRUCTION OF ENVELOPES AND GOVERNORS AS A CLASSROOM ACTIVITY -- 9.13 REMARKS CONCERNING HOLDITCH'S THEOREM -- NOTES ON CHAPTER 9 -- Chapter 10 ISOPERIMETRIC AND ISOPARAMETRIC PROBLEMS -- PART 1: ISOPARAMETRIC REGIONS -- 10.1 INTRODUCTION -- 10.2 CONTOUR RATIOS -- 10.3 ISOPARAMETRIC CONTOURS OF DIFFERENT SHAPES -- 10.4 RING RATIOS -- 10.5 ISOPARAMETRIC INEQUALITY FOR RINGS -- 10.6 ISOPARAMETRIC RINGS -- 10.7 ISOPARAMETRIC RINGS WITH EQUAL INNER PERIMETERS AND EQUAL OUTER PERIMETERS -- 10.8 INCONGRUENT SOLIDS WITH PROPERTIES (a) TO (f) -- PART 2: DISSECTIONS OF ISOPARAMETRIC REGIONS -- 10.9 DISSECTIONS INVOLVING BOUNDARIES -- 10.10 COMPLETE DISSECTION OF POLYGONAL REGIONS -- 10.11 COMPLETE DISSECTION OF POLYGONAL FRAMES -- 10.12 COMPLETE DISSECTIONS WITHOUT FLIPPING -- 10.13 COMPLETE DISSECTIONS USED TO APPROXIMATE CURVILINEAR REGIONS -- 10.14 ISOPERIMETRIC PROPERTIES OF FRAMES -- 10.15 DESIGNATED COMPLETE DISSECTIONS -- 10.16 CONCLUDING REMARKS -- NOTES ON CHAPTER 10 -- Chapter 11 ARCLENGTH AND TANVOLUTES -- PART 1: ARCLENGTH -- 11.1 INTRODUCTION -- 11.2 ARCLENGTH OF TANGENCY CURVE: TANGENT IN THE FORWARD DIRECTION -- 11.3 ARCLENGTH OF TANGENCY CURVE: TANGENT IN THE BACKWARD DIRECTION -- 11.4 EXAMPLES: TANGENTS OF CONSTANT LENGTH -- 11.5 TANGENT SEGMENTS OF VARIABLE LENGTH -- 11.6 CLASSICAL INVOLUTE AND EVOLUTE.

PART 2: TANVOLUTES -- 11.7 TANVOLUTES -- 11.8 BASIC FUNCTIONS AND THREE BASIC PROBLEMS -- 11.9 BASIC DIFFERENTIAL EQUATIONS -- 11.10 CONSTANT ANGLE OF ATTACK: β-TANVOLUTES. -- 11.11 PROBLEM 1: FINDING β-TANVOLUTES OF A GIVEN CURVE -- 11.12 EXAMPLES ILLUSTRATING PROBLEM 1 -- 11.13 TANVOLUTES APPLIED TO PURSUIT PROBLEMS -- 11.14 PROBLEM 2. FINDING TANGENCY CURVE WITH KNOWN β-TANVOLUTE -- 11.15 PROBLEM 3. FINDING β-TANVOLUTES WHEN t IS KNOWN -- 11.16 GEOMETRIC BEHAVIOR OF β-TANVOLUTES -- 11.17 FURTHER EXAMPLES ILLUSTRATING PROBLEM 1 -- 11.18 CUSPS OF CYCLOIDAL SPECIAL TANVOLUTES -- 11.19 VARIABLE ANGLE OF ATTACK β -- NOTES ON CHAPTER 11 -- Chapter 12 CENTROIDS -- PART 1: CENTROIDS OF PLANE FIGURES -- 12.1 INTRODUCTION -- 12.2 THE ARCHIMEDES LEMMA -- 12.3 FUNDAMENTAL PROPERTIES OF CENTROIDS OF PLANE LAMINAS -- 12.4 APPLICATIONS -- PART 2: CENTROID OF n POINTS -- 12.5 CENTROIDS CONSTRUCTED GRAPHICALLY -- 12.6 ALTERNATIVE BISECTION INDUCTIVE METHOD -- 12.7 GENERALIZATION OF A PUTNAM PROBLEM -- NOTES ON CHAPTER 12 -- Chapter 13 NEW BALANCING PRINCIPLES WITH APPLICATIONS -- 13.1 INTRODUCTION -- 13.2 BALANCING REGULAR CIRCUMGONS IN A PLANE -- 13.3 BALANCE-REVOLUTION PRINCIPLE AND CIRCUMSOLIDS -- 13.4 MOMENT-WEDGE PRINCIPLE AND CYLINDRICAL WEDGES -- 13.5 BALANCING PORTIONS OF A SPHERE AND OF A CYLINDER -- 13.6 HIGHER-DIMENSIONAL BALANCING PRINCIPLES -- 13.7 ON THE SPHERE AND CYLINDROID IN n-SPACE -- 13.8 FURTHER EXTENSIONS TO n-SPACE, AND APPLICATIONS -- 13.9 FORMULAS FOR CENTROIDS -- 13.10 ON THE SPHERE AND ITS CIRCUMSOLIDS IN n-SPACE -- NOTES ON CHAPTER 13 -- Chapter 14 SUMS OF SQUARES -- 14.1 A LOCUS PROBLEM IN THE PLANE -- 14.2 FIRST BASIC THEOREM ON SUMS OF SQUARES OF DISTANCES IN m-SPACE -- 14.3 SECOND BASIC THEOREM -- 14.4 APPLICATIONS TO GEOMETRY -- 14.5 COMPOSITE SYSTEMS -- 14.6 EQUAL WEIGHTS: APPLICATIONS TO GEOMETRY.

14.7 SUMS OF SQUARES OF INTEGERS IN ARITHMETIC PROGRESSION.