Abstract
The relatively parallel adapted frame or Bishop frame is an alternative approach to define a moving frame that is well defined even when the curve has vanished second derivative, and it has been widely used in the areas of biology, engineering, and computer graphics. The main result of this paper is using the relatively parallel adapted frame for classification of singularity type of Bishop spherical Darboux image and Bishop dual which are deeply related with a space curve and making them visualized by computer.
1. Introduction
In 1975, Bishop [1] introduced a new beautiful frame called the relatively parallel adapted frame or Bishop frame, which could provide the desired means to ride along a space curve with and . After that, many research papers related to the Bishop frame have been treated in the Euclidean space [2–7], Minkowski space [8, 9], and dual space [10]. This special frame is also extended to study canal and tubular surfaces [9]. It has applications in the areas of biology and computer graphics [11, 12]. For example, it may be possible to compute information about the shape of sequences of DNA using a curve defined by the Bishop frame. The Bishop frame may also provide a new way to control virtual cameras in computer animations. Typical computer graphics applications of the parallel transport frame include the generation of ribbons and tubes from 3D space curves and the generation of forward-facing camera orientations given an appropriate initial camera path. If the curve is coarsely refined, but is smooth enough to generate appropriate frame control points from the parallel transport frame algorithm, the resulting frames can be used as control points for any desired degree of smooth spline interpolations using the methods of Shoeemake [13]. Rotating camera orientations relative to a stable forward-facing frame can be added by various techniques such as that of Hanson and Ma [14]. In many (though not all) applications of camera animation, it is desirable to have the camera gaze direction pointing forward along a space curve throughout the motion. Typical examples would include a flight looking through the front window of an airplane cockpit, riding on a roller coaster, or sliding down a bannister. Other applications require the camera gaze direction to remain in some fixed skew orientation relative to the camera path, for example, a passenger looking out an airplane window. All such applications are easily accommodated using the parallel transport mechanism applied to an initial camera orientation.
Computer vision is the automatic analysis of sequences of images for the purpose of recovering three-dimensional surface shape. In recent years, several branches of mathematics, both ancient and modern, have been applied to computer vision. Projective geometry, which in its mathematical form dates back at least two centuries, is used to describe the relationship between points and lines in different images of the same object. Differential geometry, which is even older, though it received its definitive modern look in the first half of the nineteenth century, is used to describe the shape of curves and surfaces in engineering. A minor revolution in mathematical thought and technique occurred during the 1960s, largely through the inventive genius of the French mathematician René Thom. His ideas partly inspired by Whitney gave birth to what is called singularity theory, a term which includes catastrophes and bifurcations. Today's singularity being a direct descendant of differential calculus is certain to have a great deal of interest to say about geometry and therefore about all the branches of mathematics, physics, engineering, and other disciplines where the geometrical spirit is a guiding light [15]. More recently, developments in singularity theory have enriched the field of geometry by making possible a wealth of detail only dreamed of fifty years ago. Likewise, developments in the speed and power of computers over the last decade have turned other dreams into reality and made possible real-world applications of mathematical theory.
There are several articles on singularities of Frenet curve in Euclidean space and Minkowski space [15–19]. The curves in these articles are three times differentiable curves such that and must be linearly independent and . The main tools in these articles are Serret-Frenet formulas and some related functions on curves such as the distance-squared functions and the height functions. With the help of Serret-Frenet formulas, some classical invariants of extrinsic differential geometry can be treated as singularities of these two functions. However, the Frenet-Serret frame is not defined for all points along every curve. A new frame is needed for the kind of mathematical analysis that is typically done with computer graphics. Bishop introduced the relatively parallel adapted frame or Bishop frame, which could provide the desired means to ride along a space curve such that and . However, to the best of the authors' knowledge, no literature exists regarding the singularities of space curves according to the relatively parallel adapted frame. Thus, the current study hopes to serve such a need, and it is inspired by the works of Bishop [1], Izumiya et al. [17], and Wang and Pei [18]. In this paper, we introduce the notions of Bishop spherical Darboux image, Bishop dual, Bishop height functions, and extended height functions on a space curve embedded in Euclidean -space. These definitions are similar to those carried out by Bruce and Izumiya in Euclidean -space [15, 17]. Adopting the relatively parallel adapted frame [1] as the basic tool and using the same methods as in [15, 17], we find that the relationships between the singularities of the discriminant of extended height functions and the sets of bifurcations of the height function correspond to Bishop dual and Bishop spherical Darboux image separately and geometric invariants of curves in Euclidean -space. We also get some meaningful properties of Bishop helix. The main result of this paper is in Theorem 1 using the relatively parallel adapted frame for classification of singularity type of some objects deeply related with space curves.
The rest of this paper is organized as follows. Firstly, we introduce some basic concepts and the main results in the next two sections. Then, we introduce two different families of functions on that will be useful to the study of geometric invariants of regular curve. Afterwards, some general results on the singularity theory are used for families of function germs, and the main result is proved. Finally, we give two examples to illustrate the main results, and the conclusion of the work is drawn.
2. Preliminaries and Notations
Let be a regular unit speed Frenet curve in . We know that there exist an accompanying three-frames called Frenet frame for Frenet curve. Denote by the moving Frenet frame along the unit speed Frenet curve . Then, the Frenet formulas are given by Here, and are called curvature and torsion, respectively [15].
We can parallel transport an orthogonal frame along a curve simply by parallel transporting each component of the frame. The parallel transport frame is based on the observation that while for a given curve model is unique, we may choose any convenient arbitrary basis for the remainder of the frame, so long as it is in the normal plane perpendicular to at each point. If the derivatives of depend only on and not each other, we can make and vary smoothly throughout the path regardless of the curvature. Therefore, we have the alternative frame equations Here, we will call the set as Bishop frame and and as Bishop curvatures. The relation matrix can be expressed as One can show that so that and effectively correspond to a Cartesian coordinate system for the polar coordinates and with . Here, Bishop curvatures are also defined by
The orientation of the parallel transport frame includes the arbitrary choice of integration constant , which disappears from (and, hence, from the Frenet frame) due to the differentiation [1]. The unit sphere with center in the origin in the space is defined by For any regular unit speed curve , is called a Bishop Darboux vector. We define a vector and call it a modified Bishop Darboux vector along . We define the Bishop spherical Darboux image of the curve as In fact, The Bishop rectifying developable of is defined by The Bishop dual of is defined by Denote the tangent Bishop spherical indicatrix, the first Bishop spherical indicatrix, and the second Bishop spherical indicatrix in [5] by , and separately. A regular unit speed curve is called a Bishop slant helix according to Bishop frame provided that the unit vector of has constant angle with some fixed unit vector ; that is, . We define a new invariant of a regular curve in , and we can describe -slant curve by the invariant .
3. Singularity Classification and Its Visualization
The main results of this paper are in the following theorem.
Theorem 1. Let be a regular unit speed curve with . Then one has the following.(1)(a)The Bishop spherical Darboux image is locally diffeomorphic to a line at if and only if .(b)The Bishop spherical Darboux image is locally diffeomorphic to the cusp at if and only if and . (2) (a)The Bishop dual is locally diffeomorphic to the plane at if and only if , where and are real numbers such that .(b)The Bishop dual is locally diffeomorphic to the cuspidal edge at if and only if
(c)The Bishop dual is locally diffeomorphic to the at if and only if
where the ordinary cusp is .
The cuspidal edge is .
The swallowtail is .
Note that when we consider the case , we can get analogous results to the previous theorem; so, we omit them. The pictures of cuspidal edge and swallowtail will be seen in Figure 1.
(a)
(b)
4. Geometric Invariant of Space Curve and Bishop Height Functions
In this section, we will introduce two different families of functions on that will be useful to the study of geometric invariants of regular curve. Let be a regular unit speed curve. Now, we define a family of smooth functions on as follows: We call it Bishop height function. For any , we denote that . We also define a family of functions on as follows: We call it extended Bishop height function of . We denote that . Then, we have the following proposition.
Proposition 2. Let be a regular unit speed curve with . Then, one has the following.(A) (1) if and only if there are real numbers and such that and .(2) if and only if .(3) if and only if and .(4) if and only if and .(5) if and only if and .(B) (1) if and only if .(2) if and only if there are real numbers and such that and .(3) if and only if and .(4) if and only if , and .(5) if and only if .(6) if and only if , , and .
Proof. As some equations are rather long, we omit the variable .(1)Since . We have that there are real numbers and such that . By the condition that , we get that . The converse direction also holds.(2)Since , we have that . It follows from the fact that . Therefore, we have that
(3)Since , by the conditions of , we have that
This is equivalent to the condition .(4)Since , by the conditions of , we have that
This is equivalent to the condition .(5)Since + − , by the conditions of and a rather long computation, we get that
This is equivalent to the condition .
(B) The proof of B follows from the proof of A; so, we omit it.
Proposition 3. Let be a regular unit speed curve with ; then, if and only if each of is a constant vector.
Proof. Suppose that . By straightforward calculations, we have that Thus, if and only if .
Corollary 4. Let be a regular unit speed curve with ; then, if and only if the Bishop spherical Darboux image is a constant map.
By the previous propositions, we can recognize that the function has special meanings.
Theorem 5. Let be a unit speed curve with nonzero natural curvatures; then, is a B-slant helix if and only if . (cf., [5]).
We also can slightly extend the main results in [5] to the following proposition.
Proposition 6. Let be a regular unit speed curve with and . Then, one has the following claims.(1) is a B-slant helix.(2)The bishop spherical Darboux image is a constant map.(3)The tangent spherical indicatrix is a circle in osculating plane.(4)The Bishop rectifying developable of is a cylindrical surface given by , where .(5)The first Bishop spherical indicatrix is a circle on the unit spheres , and the direction of the circle is given by the constant vector .(6)The second Bishop spherical indicatrix is a circle on the unit spheres , and the direction of the circle is also given by the constant vector .
Proof. (1) Suppose that . By straightforward calculations, we have
This means that . By the definition of B-slant helix, we know that is a B-slant helix.
(2) By Corollary 4, we can claim easily.
(3) By Corollary 3.2 in [5], we also can claim .
(4) The claim in is clear by definition.
(5) Suppose that . Since
we get that
So, is constant. This means that the first Bishop spherical indicatrix is a circle on the unit spheres , and the direction of the circle is given by the constant vector .
Suppose that . By the analogous computation as , we get that
is constant. This means that the second Bishop spherical indicatrix is a circle on the unit spheres , and the direction of the circle is also given by the constant vector .
5. Versal Unfoldings and Proof of the Main Result
In this section, we use some general results on the singularity theory for families of function germs. Detailed descriptions can be found in the book [15]. Let be a function germ. We call an -parameter unfolding of , where . We say that has -singularity at if for all , and . We also say that has -singularity at if for all . Let be an unfolding of , and let has -singularity at . We denote the -jet of the partial derivative at by for . Then, is called a versal unfolding if the matrix of coefficients has rank , , where . We now introduce an important set concerning the unfolding. The discriminant set of is the set
The bifurcation set of is the set Then, we have the following well-known result (cf., [15]).
Theorem 7. Let be an -parameter unfolding of which has the -singularity at .
Suppose that is a (p) versal unfolding. Then, one has the following.(a)If , then is locally diffeomorphic to .(b)If , then is locally diffeomorphic to .(c)If , then is locally diffeomorphic to .
Suppose that is a (p) versal unfolding. Then, one has the following.(a)If , then is locally diffeomorphic to .(b)If , then is locally diffeomorphic to .
We consider a unit speed regular curve with and the height function of . By Proposition 2, the discriminant set of is given as follows:
The bifurcation set of is
Then, we have the following proposition.
Proposition 8. (1) If has -singularity () at , then is a versal unfolding of .
(2) If has -singularity at , then is a versal unfolding of .
Proof. (1) We denote that
Under this notation, we have that
Thus, we have that
We also have that
Therefore, the -jet of at is given by
It is enough to show that the rank of the matrix is 2, where
Denote that . Then, we have that
Note that is a singular point, where
(2) Under the same notations as (1), we have that
It is enough to show that the rank of the matrix is 2, where
This follows from the proof of . This completes the proof.
Proof of Theorem 1. (1) The bifurcation set of is
The assertion (1) of Theorem 1 follows from Proposition 2, Proposition 8, and Theorem 7.
(2) The discriminant set of is
The assertion of Theorem 1 follows from Proposition 2, Proposition 8, and Theorem 7.
6. Examples
As applications and illustration of the main results (Theorem 1), we give two examples in this section.
Example 9. Let be a unit speed curve of defined by
with respect to an arclength parameter (Figure 2).
The curvature and torsion of this curve are, respectively, as follows:
Using the Bishop curvature equation (5), we obtain
We can calculate the geometric invariant
Using (3), we obtain the Bishop frame as follows:
Using (10) and (44), we obtain the Bishop spherical Darboux image (Figure 3)
We define Bishop Gauss surface by
This surface is deeply related with Bishop spherical Darboux image and Bishop dual. We make both Bishop spherical Darboux image and Bishop Gauss surface visualized by Maple in Figure 4. We can see that Bishop spherical Darboux image lies in Bishop Gauss surface, and it has a cusp point.
For the Bishop dual, we should define a diffeomorphic mapping and make its diffeomorphic image visualized in Figure 5. We can see the locus of singular points which is the red curve and some other cuspidal edge points in this surface. The diffeomorphic mapping is defined by
where and . Thus, the diffeomorphic image of Bishop dual is
We can see the diffeomorphic image of Bishop dual in Figure 5.
Example 10. Let be a unit speed curve of defined by with respect to an arclength parameter (Figure 6). The curvature and torsion of this curve are, respectively, as follows: Using the Bishop curvature equation (5), we obtain the following: We can calculate the geometric invariant . Using the same method as Example 9, we can get Bishop spherical Darboux image, Bishop Gauss surface, and Bishop dual of this curve. The pictures of Bishop spherical Darboux image, Bishop Gauss surface, and diffeomorphic image of Bishop dual are visualized in Figures 7 and 8. We can see that Bishop spherical Darboux image lies in Bishop Gauss surface, and it is locally diffeomorphic to a line. The diffeomorphic image of Bishop dual has some cuspidal edge points.
7. Conclusions
In this paper, we introduce the notions of Bishop spherical Darboux image, Bishop dual, Bishop height functions, and extended height functions on a space curve embedded in Euclidean -space. We use the Bishop-Serret-Frenet formulas and Bishop height functions to study these objects from the singularity viewpoint. We establish the relationships between the singularities of the discriminant of extended height functions and the sets of bifurcations of the height function, which are Bishop dual and Bishop spherical Darboux image separately, and geometric invariants of a space curve in Euclidean -space. We also get some meaningful properties of Bishop helix. Note that the Bishop dual and the Bishop spherical Darboux image according to Bishop frame are very different from those in [15, 17]; however, we found that they have some analogous singularity properties under some different conditions. The main result of this paper is Theorem 1 using the relatively parallel adapted frame for classification of singularity type of some objects deeply related with a space curve and making them visualized by computer. As applications of our main results, we give two examples.
Acknowledgments
The authors would like to thank the referee for his/her valuable suggestions which improved the first version of the paper. The second author (corresponding author) was partially supported by NSF of China, no. 11271063. The first author was partially supported by the Fundamental Research Funds for the Central Universities (Graduate Innovation Fund of Northeast Normal University, no. 12SSXT140), Preparatory Studies of Provincial Innovation Project, no. SY201225, and Young Teachers Project in Science and Technology of Mudanjiang Normal University, no. QY201102.