The Design of Architectural Forms Based on Irregular Curves

Non-rectilinear shapes and smooth nature-like curves have been widely used in modern architecture. Such curve-based architectural forms are designed using the latest computational technologies and parametric digital modeling methods borrowed from the automotive and aviation industries. Graphically defined irregular curves are an essential component of architectural design. For the practical use of a graphically defined curve there are certain requirements: it should be approximated by a regular curve; the approximation error should not exceed 2%; the curve should pass through nodes and touch predefined direction vectors at these points; and the angle of the slope and curvature of the composite curve should change continuously. This article shows that these requirements can be fulfilled using a composite cubic Bézier curve. We have developed a graphic-analytical algorithm for the formation of the desired curve. A distinctive feature of the algorithm is that it takes into account the direction of the tangent vectors and the radii of the curvature at the nodes. Direction vectors are considered as a control for the constructed curve. We ensured second-order geometric smoothness at the nodes from the continuity of the slope and curvature. An experiment on the approximation of a nature-like curve (physical spline) of a composite cubic Bézier curve showed the approximation error was less than 1.5%.