engleski

Circular Surfaces CS(\alpha, p)

Sonja Gorjanc Faculty of Civil Engineering, University of Zagreb, Zagreb, Croatia e-mail: sgorjanc@grad.hr This lecture introduces a new concept of surface-construction: We consider a congruence of circles C(P1, P2) = C(p) in the Euclidean space, i.e. a two-parametric set of circles which pass through the points P1, P2 given by the coordinates (0, 0, ±p), where p = pq, q 2 R. It is a normal curve congruence with singular points on the z axes, [3]. C(p) is a hyperbolic, parabolic or elliptic if q is greater, equal or less then 0, respectively. For a piecewise-differentiable curve : I ! R3, I  R, we define a circular surface CS( , p) as the system of circles of C(p) which cut the curve . For the surfaces CS( , p) we derive parametric equations (which enable their visualizations in the program Mathematica) and investigate their properties if is an algebraic curve. In the general case, if is an algebraic curve of the order n, CS( , p) is an algebraic surface of the order 3n passing n times through the absolute conic and containing the n-fold straight line P1P2. But the order of CS( , p) is reduced if passes through the absolute points or if it cuts the line P1P2. The first examples of algebraic CS( , p) are parabolic cyclides (if is a line), Dupin’s cyclides (if is a circle) and rose-surfaces (if is a rose) [1], [4]. Furthermore, we consider cyclic-harmonic curves R(a, n, d) lying in the plane z = k, k 2 R, which are given by the polar equation  = cos(n d')+a, where n d is a positive rational number in lowest terms and a 2 R+ [ {; ; ; 0}; ; ; . A generalized rose-surface R(p, k, n, d, a) is the surface CS( , p) where the directing curve is the cyclic harmonic curve R(n, d, a) in the plane z = k. These surfaces have various attractive shapes, a small number of high singularities, and they are convenient for algebraic treatment and visualization in the program Mathematica. Some examples are shown in Fig. 1.

circular surfaces; congruence of circles; cyclic-harmonic curves; generalized rose-surfaces

nije evidentirano