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An approach for designing a surface pencil through a given geodesic curve

Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve have attracted much interest. In the present paper, we propose a new method to construct a surface interpolating a given curve as the geodesic curve of it. Also, we analyze the conditions when the resulting surface is a ruled surface. In addition, developablity along the common geodesic of the members of surface family are discussed. Finally, we illustrate this method by presenting some examples.


Introduction
A rotation minimizing adapted frame (RMF) {T,U,V} of a space curve contains the curve tangent T and the normal plane vectors U, V which show no instantaneous rotation about T. Because of their minimum twist RMFs are very interesting in computer graphics, including free-form deformation with curve constraints [1 -6], sweep surface modeling [7 -10], modeling of generalized cylinders and tree branches [11 -15], visualization of streamlines and tubes [15 -17], simulation of ropes and strings [18], and motion design and control [19].
There are infinitely many adapted frames on a given space curve [20]. One can produce other adapted frames from an existing one by controlling the orientation of the frame vectors U and V in the normal plane of the curve. In differential geometry the most familiar adapted frame is Frenet frame {T, N, B}, where T is the curve tangent, N is the principal normal vector and B T N  is the binormal vector (see [21] for details). Beside of its fame, the Frenet frame is not a RMF and it is unsuitable for specifying the orientation of a rigid body along a given curve in applications such as motion planning, animation, geometric design, and robotics, since it incurs "unnecessary" rotation of the body [22]. Furthermore, Frenet frame is undefined if the curvature vanishes.
One of most significant curve on a surface is geodesic curve. Geodesics are important in the relativistic description of gravity. Einstein's principle of equivalence tells us that geodesics represent the paths of freely falling particles in a given space. (Freely falling in this context means moving only under the influence of gravity, with no other forces involved). The geodesics principle states that the free trajectories are the geodesics of space. It plays a very important role in a geometric-relativity theory, since it means that the fundamental equation of dynamics is completely determined by the geometry of space, and therefore has not to be set as an independent equation. Moreover, in such a theory the action identifies (up to a constant) with the fundamental length invariant, so that the stationary action principle and the geodesics principle become identical. The concept of geodesic also finds its place in various industrial applications, such as tent manufacturing, cutting and painting path, fiberglass tape windings in pipe manufacturing, textile manufacturing [23][24][25][26][27][28]. In architecture, some special curves have nice properties in terms of structural functionality and manufacturing cost. One example is planar curves in vertical planes, whichcan be used as support elements. Another example is geodesic curves, [29]. Deng, B. , described methods to create patterns of special curves on surfaces, which find applications in design and realization of freeform architecture. He presented an evolution approach to generate a series of curves which are either geodesic or piecewise geodesic, starting from a given source curve on a surface. In [29], he investigated families of special curves (such as geodesics) on freeform surfaces, and propose computational tools to create such families. Also, he investigated patterns of special curves on surfaces, which find applications in design and realization of freeform architectural shapes (for details, see [29] ). Most people have heard the phrase; a straight line is the shortest distance between two points. But in differential geometry, they say this same thing in a different language. They say instead geodesics for the Euclidean metric are straight lines. A geodesic is a curve that represents the extremal value of a distance function in some space. In the Euclidean space, extremal means 'minimal',so geodesics are paths of minimal arc length. In general relativity, geodesics generalize the notion of "straight lines" to curved space time. This concept is based on the mathematical concept of a geodesic. Importantly, the world line of a particle free from all external force is a particular type of geodesic. In other words, a freely moving particle always moves along a geodesic. Geodesics are curves along which geodesic curvature vanishes. This is of course where the geodesic curvature has its name from.
In recent years, fundamental research has focused on the reverse problem or backward analysis: given a 3D curve, how can we characterize those surfaces that possess this curve as a special curve, rather than finding and classifying curves on analytical curved surfaces. The concept of family of surfaces having a given characteristic curve was first introduced by Wang et.al. [28] in Euclidean 3-space. Kasap et.al. [30] generalized the work of Wang by introducing new types of marching-scale functions, coefficients of the Frenet frame appearing in the parametric representation of surfaces. Also, surfaces with common geodesic in Minkowski 3-space have been the subject of many studies. In [31] Kasap and Akyıldız defined surfaces with a common geodesic in Minkowski 3-space and gave the sufficient conditions on marching-scale functions so that the given curve is a common geodesic on that surfaces. Şaffak and Kasap [32] studied family of surfaces with a common null geodesic. Lie et al. [33] derived the necessary and sufficient condition for a given curve to be the line of curvature on a surface. Bayram et al. [34] studied parametric surfaces which possess a given curve as a common asymptotic. However, they solved the problem using Frenet frame of the given curve.
In this paper, we obtain the necessary and sufficient condition for a given curve to be both isoparametric and geodesic on a parametric surface depending on the RMF. Furthermore, we show that there exists ruled surfaces possessing a given curve as a common geodesic curve and present a criteria for these ruled surfaces to be developable ones. We only study curves with an arc length parameter because such a study is easy to follow; if necessary, one can obtain similar results for arbitrarily parameterised regular curves.

Backgrounds
has a constant s or t-parameter value. In this paper,    rsdenotes the derivative of r with respect to arc length parameter s and we assume that   rs is a regular curve, i.e.   0 where " " denotes the standard inner product in 3 [36]. Observe that such a pair U and V is not unique; there exist a one parameter family of RMF's corresponding to different sets of initial values of U and V. According to Bishop [20], a frame is an RMF if and only if each of is the necessary and sufficient condition for the frame to be rotation minimizing [37]. There is a relation between the Frenet frame (if the Frenet frame is defined) and RMF, that is, is the angle between the vectors N and U (see Fig. 2), [38].

s t s t T s s t U s s t V s
. . .

Examples of generating surfaces with a common geodesic curve
st  (Fig. 5).