Shade Lines of Curved Surfaces - Rotational and Helical Circular Surfaces

Uobičajeno je da se tangenta t na rastavnicu e oble plohe Φ konstruira na temelju činjnice da su t i zraka svjetlosti l par konjugiranih dijametara tzv. DUPINOVE indikatrise u promatranoj točki P. Ovaj rad opisuje drukčiji pristup konstrukciji takve tangente: rastavnica e definira se kao prodorna krivulja plohe Φ i specijalne pravčaste plohe Ψ koja ovisi o Φ i snopu zraka svjetlosti. Ψ se uvodi kao pridružena pravčasta ploha duž krivulje e. Taj pristup omogućuje jednostavnu, linearno i globalno primjenjivu konstrukciju tangente t za rotacijske i zavojne plohe, na način nacrtne geometrije. Metoda je također prikladna za klizne plohe isto kao i za centralnu rasvjetu. U nekim je slučajevima ploha Ψ pravčasta kvartika.
Typically a tangent t to the shade line e of a curved surface Φ is constructed by making use of the fact that t and the light ray l form a pair of conjugate diameters of the so-called DUPIN-indicatix of Φ at an investigated point P. This article describes a very different approach to developing such a tangent: The shade line e is defined as the intersection of Φ and a special ruled surface Ψ, which depends both on Φ and on the bundle of light rays. Ψ is introduced as accompanying ruled surface along e. This approach allows a simple, linear and globally applicable construction of t for rotational and helical surfaces by means of descriptive geometry. The method is also suitable for translation surfaces as well as for central illumination [4]. In a few cases Ψ is a ruled quartic.