THE THEORY OF PLANE CURVES

By MARK HERNANDEZ

CONTENTS I INTERSECTIONS OF CURVES INTRODUCTION AND HISTORY §1 INTERSECTIONS AT THE ORIGIN §2 HOMOGENEOUS COORDINATES §3 INTERSECTIONS IN HOMOGENEOUS COORDINATES §4 LINES AND TANGENTS II CONICS §5 CONICS AND INTERSECTIONS §6 PASCAL’S THEOREM §7 ENVELOPES OF CONICS III CUBICS §8 FLEXES AND SINGULAR POINTS §9 ADDITION ON CUBICS §10 COMPLEX NUMBERS §11 BEZOUT’S THEOREM §12 HESSIANS §13 DETERMINING CUBICS IV PARAMETRIZING CURVES §14 PARAMETRIZATIONS AT THE ORIGIN §15 PARAMETRIZATIONS OF GENERAL FORM §16 ENVELOPES OF CURVES REFERENCES