On Projective And Differential Aspects Of Conodis

The thesis consists of five chapters <The first chapter: IntroductionThe straight line is the generator of conoids. We start by giving afurther study of the straight line in the three dimensional projective spacep3. Considering the four degrees of freedom of the straight line and usingthe LINE GEOMETRY we define.(i) The line complex if one of the four degrees of freedom is replaced byone constraint.(ii) the line congruence if two of the four degrees of freedom are replacedby two constraints.(iii) the ruled surface if three of the four degrees of freedom are replacedby three geometrical constraints.The third case is the required case 111 which the TIlled surfacemay be a conoid. In the case of conoids, the three geometrical constraintsare three [;eometrical directrices. We distinguish between two subcases:(1) If there are not common points of the three directrices, for example:right and oblique circular conoids, right helicoids, right and obliquespherical conoids, hyperbolic paraboloid (as a special case).(2) IF there are common points of the directrices, for example, Pluckerconoid which is a cubic ruled surface. We held an analogy between p3and ps* : p3 ~ psSome results on linear complex, linear congruence and ruledsurfaces were obtained. The degree of a ruled surface given by threegeometric directrices is discussed also in this introduction.