A method for adjusting the shape of semi-oval arches using Hügelschäffer’s construction

Abstract

There are numerous ways to construct arches as curved structural elements that span openings in architectural buildings. Arches are present in many historical styles, having the role of an identification mark. Contemporary architecture also uses a resource of inherited styles, combining them into an eclectic blend of modern and classic. However, regardless of the aspirations that investors, architects and future users may have towards some of the complex stylistic forms, the question is whether every contractor will be able to meet the set requirements. Although modern construction technology is apt to perform much more diverse forms than in earlier epochs, there are still situations when some problems, especially geometric ones, need to be solved, since the technology itself has not yet reached the point of its own thinking. One such problem was posed to the authors by a contractor who came up with the question: how to easily and accurately make a template for a semi-oval window arch, with a predefined point (M) through which the arch should pass. Hence, this is not about an elliptical arc for which there are a number of known constructions. The request was to offer a construction of a higher order curve, simple enough to enable quick and easy design and fabrication of templates on the construction site. In semi-oval arcs, the curve deviates from the elliptical one, giving greater curvature at the vertices of the major axis, and lesser at the vertex of the minor semi-axis. To solve this problem, we use the generalization of the Hügelschäffer’s egg curve construction. With input data: the major and minor semiaxes a and b, together with the given position of the point M which defines the deviation of the oval curve from the ellipse, we first determine the displacement of the minor circle of the Hügelschäffer’s construction along the y axis. Then, by applying the transformation of hyperbolism, we obtain the points of the semi-oval. The offered construction gives quick and accurate positions of points on the semi-oval arc, moreover, it allows the adjustment of the shape of the semi-oval arch according to needs (aesthetic, stylistic, constructive, functional, etc.). In the digital age, it is no problem to draw a higher order curve, but in order to simplify the construction for the purpose of making templates on the construction site, we can turn to the classic method - the approximation of curves with circles, as with semi-elliptical arcs. We find the centers of circles and use their successively connected arcs to accomplish the task with synthetic tools: compass and ruler.