In recent years, freeform shapes are gaining more and more popularity in architecture. Such shapes are often challenging to manufacture, and have motivated an active research field called architectural geometry. In this thesis, we investigate patterns of special curves on surfaces, which find applications in design and realization of freeform architectural shapes. We first consider families of geodesic curves or piecewise geodesic curves on a surface, which are important for panelization of the surface and for interior design. We propose a method to propagate a series of such curves across a surface, starting from a given source curve, so that the distance functions between neighboring curves are close to given target distance functions. We use Jacobi fields as first order approximation of the distance functions from a curve to its neighboring curves, and select a Jacobi field which is closest to the target distance function. A neighboring curve is then computed according to the selected Jacobi field by solving an optimization problem. Using different target distance functions, we can generate different patterns of geodesic/piecewise geodesic curves. Our method provides an intuitive and controllable way to design geodesic patterns on freeform surfaces. We then present a method to compute functional webs, which are three families of curves with regular connectivity, where the curves have given special properties. We consider planar, circular and geodesic properties of the curves, which facilitate the fabrication of curve elements. We discretize a web as a regular triangle mesh, where the curves are represented by edge polylines of the mesh. The shape of the web is determined by optimizing a target functional which penalizes the deviation of the curves from their target properties. Furthermore, for webs where all curves are planar, we also show they can be computed in an exact way using three families of planes. By enabling the design of webs composed of curve elements which are easily manufacturable, our method addresses the challenge in realization of webs which have emerged in recent architectural designs.