d) toroidal, e) double curved, three edges aligned, f) conical, g) anticlastic, h) synclastic. Panel types a, b c, f are developable, and therefore ruled surfaces, panels d, e, g and h are double curved surfaces.
There are a few advantages of conical glass elements over double curved elements. They are more cost efficiency and easier installation. This depends primarily on the geometry of the edges and further the sealing of the joints of watertight structures.
Cylindrical panels and elements of annealed glass that are based on ruled surfaces can be produced with the same technique (Fig. 1).
Fig. 4 Annealed Glass Bending with gravity force method is set up as depicted left and produces ruled surfaces, of which the types on the right are special cases: rotational cylindrical, rotational conical, general cylindrical and general conical (from top left to bottom right).
3 Geometrical Prerequisites
Fig. 5 (Left) for a conical surface, all rulings intersect in a common apex; (right) a general ruled surface does not fulfill this property. For an arbitrary point on either of the colored curves, the directions of its tangent and the intersecting ruling are called conjugate.
The tangent surfaces of space curves, general cylinders and general cones are subsumed in the class of developable surfaces. They are a subclass of ruled surfaces i.e. surfaces that carry a family of lines, with the special property that the tangent plane along a ruling does not change its direction, see (Do Carmo 1976).
3.1 Conjugate Directions
The tangent planes of a curve c(u) on a surface envelope a developable surface G, called the tangent developable. For a point x on c(u), there is exactly one tangent to the curve c(u) and exactly one ruling of G that touches at x; the directions of these two lines are called conjugate, see Fig. 5. If two families of curves on a surface are such that through every point, there is one member of each family and their directions are conjugate, we speak of a conjugate network of curves. For equivalent definitions of conjugate directions see (Zadravec et al 2010), who give different examples of conjugate networks, the most important of which are principal curvature lines.
(Liu et al 2011) show that conjugate networks are a good choice for the initialization of a planarization algorithm, which can be derived from the analytical definition of conjugate directions (Do Carmo 1976). We will use conjugate directions for the choice of a ruling direction of developable strips in sec. 4.3.
3.2 Developable Surfaces and Planarization
If one wants to approximate a given surface with developable surfaces or planar quads via an optimization, principal curvature networks are a good starting point, see (Liu et al 2011) and (Pottmann et al 2008).
To approximate a given surface with almost rectangular panels, see (Wallner et al. 2010), which relies on finding geodesic lines of almost constant distance, see (Kahlert et al 2011).
A completely different approach is to start with two curves and construct a developable surface between them, see (Pottmann et al 2007), by connecting points with coplanar tangents with a line l, i.e. the tangents and l lie in the same plane E. These planes E envelope a surface, the connecting developable surface, and the lines l are its rulings. Note that this surface is not unique for given input curves. The angle between the ruling l and the curves’ tangents varies. This approach has been followed by (Subag and Elber 2006) for the approximation with NURBS surfaces.
If one curve is given and the other one is to be approximated, this approach leads to nonlinear equations and has been done for B-spline curves and a developable B-spline by (Aumann 2004) and (Chu and Séquin 2002).
3.3 Conical and Cylindrical Panels
Take a general space curve c(u), connect every point on it and an arbitrary point a (the apex) through a line segment to get a surface s(u,v) = v • c(u) + (1-v) • a. In this parameterization, the u-isolines are the straight lines (rulings of the developable surface) and the v-isolines are curves similar to c(u). If you take any curve segment on a v-isoline and the corresponding, similar curve segment on another v-isoline, you have a conical panel i.e. a patch that lies on a general cone and two of the border curves lie on intersecting lines. We will model developable strips with these in section 4. The four - if you chose the apex instead of one the v-isolines, there are only three - corner points lie on the plane spanned by the bordering rulings, so planarization is obvious. Cylindrical panels are even easier, v-isolines are congruent to c(u) and u-isolines are parallel line segments, planarize as above.
Note that in both cases the u- and v-isolines form a conjugate curve network of sec. 3.1, as the cone (or cylinder) is c(u)’s tangent developable.
4 Modelling with Conical Panels
In this section we will present the main contribution of our work, after relating it to known methods in sec. 4.1. For two given curves we will construct a strip of conical panels i.e. a single curved surface by a direct algorithm in sec. 4.2. In sec. 4.3 we will explain how to choose a ruling direction that relates to the theory of conjugate directions, see sec. 3.3. Examples that have been constructed using this algorithm and suggestions for the predefined thresholds can be found in sec. 5.
We assume a given family of curves that outline a surface. Alternatively, take a modeling surface and chose a family of curves on them, such as the evenly spaced geodesics of (Wallner et al 2010) or (Kahlert et al 2011). If a network of (conjugate) directions is already given e.g. through methods of (Alliez et al 2003) or (Bommes et al 2009), one can also choose one of the families as input curves.
4.1 Connecting Developable and Tangent Developable
Given a family of curves, we will construct a strip between two consecutive curves that consists of conical panels i.e. panels that lie on general cones, see sec. 3.3.
As outlined in sec. 3.2, the construction of a connecting developable surface between two curves is solved, but the angle between the ruling and the curves’ tangents can behave wildly, especially if curves have small perturbations.
On the other extreme, one could choose one of the two curves – call it the base curve b – and, if an underlying surface S is given, construct the tangent developable surface G as in sec. 3.1. The rulings of G are by definition the conjugate directions for the points of b, so this is the method of (Wallner et al 2010). In general, the second or target curve will not lie on G.
4.2 Outline of the Algorithm
Fig. 6 Illustration explaining the approximation algorithm: a base curve b (green) is followed smoothly, while a second curve c (blue) serves as a target. Points xi-1 and xi are on b, ŷi is on c; the lines li-1 and li are coplanar and meet in apex ai – see the algorithm in the main text for further explanations.
Our algorithm is a combination of the most important characteristics of the constructions outlined in sec. 3.2. We input a base curve and a target curve, which lie in parallel planes. If they were general space curves, step 2 would have to be adapted to accommodate for tangent planes. The output is a strip of developable surfaces precisely through the base curve and approximately following the target curve.
1 Connect the start point x0 of the base curve b and the start point y0 of the target curve c with a line l0.
2 Choose a point xi on b and a point zi on c, e.g. by subdividing the curves