Abstract

We handle the problem of finding a hypersurface family from a given asymptotic curve in . Using the Frenet frame of the given asymptotic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be asymptotic. We illustrate this method by presenting some examples.

1. Introduction

Asymptotic curves are encountered in differential geometry frequently. A surface curve is called asymptotic if its tangent vectors always point in an asymptotic direction, that is, the direction in which the normal curvature is zero. In an asymptotic direction, the surface is not bending away from its tangent plane.

Asymptotic curves on a surface can be seen in many differential geometry books [1–5]. Rastogi [2] obtained the differential equation of hyperasymptotic curves by a new method and showed some properties of these curves. Aminov [6] established more general expressions for the curvature of asymptotic curves of submanifolds in the Riemannian space. Romero-Fuster et al. [3] studied asymptotic curves on generally immersed surfaces in . Both the general and rational developable surface pencils through an arbitrary parametric curve as its common asymptotic curve were analyzed by Liu and Wang [7]. Bayram et al. [8] tackled the problem of finding a surface pencil from a given asymptotic curve.

However, while differential geometry of a parametric surface in can be found in textbooks such as those by Struik [5], Willmore [9], Stoker [4], and do Carmo [10], differential geometry of a parametric surface in can be found in textbooks such as the contemporary literature on geometric modeling [11, 12]. Also, there is little literature on differential geometry of parametric surface family in [8, 13–16] but not in . Besides, there is an ascending interest in fourth dimension [13, 14, 17].

Furthermore, various visualization techniques about objects in Euclidean -space are presented [18–20]. The fundamental step to visualize a 4D object is projecting first into the 3-space and then into the plane. In many real world applications, the problem of visualizing three-dimensional data, commonly referred to as scalar fields, arises. The graph of a function , where is open, is a special type of parametric hypersurface with the parametrization in 4-space. There exists a method for rendering such a 3-surface based on known methods for visualizing functions of two variables [21].

In this paper, we consider the four-dimensional analogue problem of constructing a parametric representation of a surface family from a given asymptotic as in Bayram et al. [8], who derived the necessary and sufficient conditions on the marching-scale functions for which the curve is an asymptotic curve on a given surface. We express the hypersurface pencil parametrically with the help of the Frenet frame of the given curve. We find the necessary and sufficient constraints on the marching-scale functions, namely, coefficients of Frenet vectors, so that both the asymptotic and parametric requirements are met.

2. Preliminaries

Let us first introduce some notations and definitions. Bold letters such as , will be used for vectors and vector functions. We assume that they are smooth enough so that all the (partial) derivatives given in the paper are meaningful. Let be an arc-length curve. If is the moving Frenet frame along , then the Frenet formulas are given by where , , , and denote the tangent, principal normal, first binormal, and second binormal vector fields, respectively, and denote the th curvature functions of the curve [20].

From elementary differential geometry we have

Using Frenet formulas one can obtain the following:

The unit vectors and are given by where is the vector product of vectors in .

Since the vectors , , , and are orthonormal, the second curvature and the third curvature can be obtained from (3) as where “” denotes the standard inner product.

Let be the standard basis for four-dimensional Euclidean space . The vector product of the vectors , , and is defined by (see [22, 23]).

If , , and are linearly independent, then is orthogonal to each of these vectors.

3. Hypersurface Family with a Common Isoasymptotic

A curve on a hypersurface is called an isoparametric curve if it is a parameter curve; that is, there exists a pair of parameters and such that . Given a parametric curve , it is called an isoasymptotic of a hypersurface if it is both an asymptotic and an isoparametric curve on .

Let ,  , be a curve, where is the arc-length. To have a well-defined principal normal, assume that ,  .

Let , , , and be the tangent, principal normal, first binormal, and second binormal, respectively; and let , , and be the first, the second, and the third curvature, respectively. Since is an orthogonal coordinate frame on , the parametric hypersurface passing through can be defined as follows: where , , , and are all functions. These functions are called the marching-scale functions.

We try to find out the necessary and sufficient conditions for which a hypersurface has the curve as an isoasymptotic.

First, to satisfy the isoparametricity condition there should exist and such that ,  ; that is,

Secondly, the curve is an asymptotic curve on the hypersurface if and only if the normal curvature along the curve, where is the shape operator and is the tangent vector to the curve. The normal of the hypersurface can be obtained by calculating the vector product of the partial derivatives and using the Frenet formula as follows:

Remark 1. Because along the curve , by the definition of partial differentiation, we have

Using (8) we have where

So

Thus, any hypersurface defined by (7) has the curve as an isoasymptotic if and only if is satisfied. We call the set of hypersurfaces defined by (7) and satisfying (15) an isoasymptotic hypersurface family.

4. Examples

Example 1. Let ,  , be a curve parametrized by arc-length. For this curve, Let us choose the marching-scale functions as So, we have the hypersurface where ,  ,  ,  , and , which is a member of the isoasymptotic hypersurface family, since it satisfies (15).
By changing the parameters and we can adjust the position of the curve on the hypersurface. Let us choose and . Now the curve is again isoasymptotic on the hypersurface and the equation of the hypersurface is The projection of a hypersurface into 3-space generally yields a three-dimensional volume. If we fix each of the three parameters, one at a time, we obtain three distinct families of 2-spaces in 4-space. The projections of these 2-surfaces into 3-space are surfaces in 3-space. Thus, they can be displayed by 3D rendering methods.
So, if we (parallel) project the hypersurface into the subspace and fix , we obtain the surface where ,   in 3-space illustrated in Figure 1.