DESIGN OF THE THIRD-ORDER CURVES OF THE TYPE THE HYPERBOLISM OF THE HYPERBOLA

Аннотация

The solution of many engineering problems requires the construction of curves with given characteristics. One way to create the curves is to obtain a new curve by some transformations of an already known curve. This method is especially important because the line of the family of lines belonging to the prototype, is definitely displayed in the relevant lines, of the family of lines of the constructed surface. And knowing the properties of the transformation and the preimage, it is easy to establish the properties of the image.
In this regard, this article proposes an algorithm (method) for constructing a third-order curve, where the initial image is a parabola. The choice of the parabola is based on the fact that among the set of curves of the third order when modeling the technical curves in CAD, the most widely used is a cubic parabola due to its’ uniqueness. The essence of the method is as follows. Each focal straight line of parabola crosses the parabola at two points, through which two straight lines (parabolic beams) run parallel to the focal axis, and a set of pairs of these lines form a bundle with an improper center. The set of points of intersection of the bundle of lines with the center at the point P with a projective to thatbundle of straight lines (parabolic beams) forms a curve of the third order, namely "hyperbolism of hyperbola" according to Newton's classification.