Studies of curves in 3D-space have been developed by many geometers and it is known that any regular curve in 3D space is completely determined by its curvature and torsion, up to position. Many results have been found to characterize various types of space curves in terms of conditions on the ratio of torsion to curvature. Under an extracondition on the constant curvature, Y.L. Seo and Y. M. Oh found the series solution when the ratio of torsion to curvature is a linear function. Furthermore, this solution is known to be a rectifying curve by B. Y. Chen’s work. This project, uses a different approach to characterize these rectifying curves

Logan, Julie

Oh, Yun Myung

English

Andrews University

Andrews University Digital Commons @ Andrews University Faculty Publications 2017 Characterization of Rectifying and Sphere Curves in R^3 Yun Myung Oh Andrews University, ohy@andrews.edu Julie Logan Andrews University, julie@andrews.edu Follow this and additional works at: https://digitalcommons.andrews.edu/pubs  Part of the Algebraic Geometry Commons, Applied Mathematics Commons, and the Geometry and Topology Commons Recommended Citation Oh, Yun Myung and Logan, Julie, "Characterization of Rectifying and Sphere Curves in R^3" (2017). Faculty Publications. 708. https://digitalcommons.andrews.edu/pubs/708 This Article is brought to you for free and open access by Digital Commons @ Andrews University. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Digital Commons @ Andrews University. For more information, please contact repository@andrews.edu. American Journal of Undergraduate Research  Volume 14 | Issue 2 | June 2017  91Characterization of Rectifying and Sphere Curves in   Julie Logan and Yun Myung Oh*    Department of Mathematics, Andrews University, Berrien Springs, MI  Student: julie@andrews.edu Mentor: ohy@andrews.edu  ABSTRACT Studies of curves in 3D-space have been developed by many geometers and it is known that any regular curve in 3D space is completely determined by its curvature and torsion, up to position. Many results have been found to characterize various types of space curves in terms of conditions on the ratio of torsion to curvature. Under an extra condition on the constant curvature, Y. L. Seo and Y. M. Oh found the series solution when the ratio of torsion to curvature is a linear function. Furthermore, this solution is known to be a rectifying curve by B. Y. Chen’s work. This project, uses a different approach to characterize these rectifying curves.  This paper investigates two problems. The first problem relates to figuring out what we can say about a unit speed curve with nonzero curvature if every rectifying plane of the curve passes through a fixed point  in . Secondly, some formulas of curvature and torsion for sphere curves are identified. KEYWORDS Space Curve; Rectifying Curve; Curvature; Torsion; Rectifying Plane; Tangent Vector; Normal Vector; Binormal Vector  INTRODUCTION Consider a unit speed curve , where  is an interval on the real number line. Since it is a unit speed curve (i.e.  has magnitude 1), the unit tangent vector is   Definition 1. The curvature, , of a unit speed curve, , is defined as  .             Equation 1. The principal normal vector, , is defined by dividing  by its magnitude:  .         Equation 2. We then have . The binormal vector which is perpendicular to both  and  is defined as .  Figure 1. The Frenet-Serre frame. These three vectors  form an orthonormal basis in  along the curve which is called the Frenet-Serre frame (Figure 1).  So,                               Equation 3.  for functions , , and . It is easy to see that c is zero from the fact that B is a unit vector field. The fact that  is zero follows from the perpendicularity of  and  together with the fact that , which follows from . We are ready to define the torsion of the curve. Definition 2. The torsion, , of the curve, , is defined by the equation . The curvature measures the deviation of a curve from being a line and torsion measures the deviation of a curve from being contained in a single plane. Since we know that , , and  are all mutually perpendicular to each other, we have . Using the facts that  and  we derive the following: American Journal of Undergraduate Research  Volume 14 | Issue 2 | June 2017  92                         .              Equation 4. Thus, we have the following Frenet-Serret formula:                           Equation 5. According to the Fundamental Theorem of Curves, any regular curve in 3D space is completely determined by its curvature and torsion, up to position. Several characterization facts have been found over the years. We know that any curve with constant curvature and zero torsion is a circle and the curve with a constant ratio of torsion to curvature is known to be a general helix.  Now, we need to introduce three types of planes along the curve. The osculating plane to a unit speed curve  is the plane perpendicular to , the normal plane of  is the plane perpendicular to  and the rectifying plane of  is the plane perpendicular to  (Figure 2).   Figure 2. The normal, osculating and rectifying planes. It has been shown that if every normal plane to the curve  passes through a fixed point  in , then the curve lies on a sphere, and if every osculating plane to the curve  passes through a fixed point  in , then the curve lies on a plane.1 Thus, it is natural to investigate the case where every rectifying plane goes through a given point .  Definition 3. A rectifying curve is a space curve whose position vector lies in its rectifying plane.3   The idea of rectifying curves was introduced by B. Y. Chen and he provided many fundamental properties of the curves together with classification results.3 For the second problem investigated in this paper, we will need the following fact about the curvature and torsion of “sphere curves”, i.e. curves in  lying on a sphere. Proposition: If  is a unit speed curve with , , then  lies on a sphere if and only if  .1               Equation 6.  RESULTS The first problem we worked on was finding a necessary condition for a unit speed curve if every rectifying plane contains the point  in .  Since the rectifying plane is orthogonal to , we have . Thus, taking the derivative of both sides we find .      Equation 7. Then by substituting from the Frenet-Serret formula we have                                                                                                       Equation 8. since . We know .           Equation 9. Thus,                                                                                         Equation 10. Then, taking another derivative of Equation 8, we obtain                        Equation 11. American Journal of Undergraduate Research  Volume 14 | Issue 2 | June 2017  93So by Equation 10, we have  and by simplification we obtain       Equation 12. Therefore,                                                                            Equation 13.                                                Equation 14. Then working with the denominator of Equation 13,           Equation 15. Equation 13 becomes                          Equation 16. In a similar manner, since we know that ,                  Equation 17. Therefore                                                               Equation 18.  We then let . Equation 18 becomes  .      Equation 19. We take another derivative of Equation 18, to find          Equation 20. and since ,         Equation 21. We then obtain                      Equation 22. and we now note that ,  and  are linearly independent. Thus,                                                                                                                        Equation 23.                                                                                  .                                                                   Equation 24.  Then, working with Equation 24, we have ,  and  for some constants ,  and arc length . This solution satisfies Equation 23 as well.  Here is the summary of the first result: Theorem A: Suppose  is a unit speed curve with nonzero curvature.  If every rectifying plane contains the point  in , i.e., if  is a rectifying curve, then its ratio  is a linear function.  Note: This result was obtained by B. Y. Chen in Theorem 2,3 but it was derived here by a different method.  Our next task is to get the formula for the curvature and torsion for a sphere curve. From the Proposition,   is a sphere curve in  if and only if . Let . Then . Thus,  .                  Equation 25.American Journal of Undergraduate Research  Volume 14 | Issue 2 | June 2017  94Substituting ,              Equation 26. which yields             Equation 27. By rearranging the terms .           Equation 28. Taking the integral of both sides to solve the separable differential equation we obtain  for a constant and then  for a constant . By solving for , we find ,       Equation 29. and since we know that , we use this formula for  to get  ,       Equation 30. for a differentiable function  and a constant .  The above result can be summarized as follows: Theorem B: For a sphere curve in , its curvature  and  are given by   and                    Equation 31. for a constant  and a differentiable function   DISCUSSION  Centrodes are a particular kind of rectifying curve and they are useful in mechanics and joint kinematics.4 Another example of application appeared in the Journal of Hand Surgery.5  Using the formulas for curvature and torsion of sphere curves we have found, future work would include finding a formula for the original sphere curve satisfying given curvature and torsion with specific  and differentiable function . This is a continuation of work by Ye Lim Seo and Yun Myung Oh published in American Journal of Undergraduate Research in Jan, 2015.6    REFERENCES  1. Millman, R. and Parker, G. (1977) Elements of differential geometry, Prentice Hall, Englewood Cliffs, N. J. , 33–37 2. Curvature, Torsion and the Frenet Frame. blogimages.bloggen.be/gnomon/attach/203774.pdf. 3. Chen, B. Y. (2003) When Does the Position Vector of a Space Curve Always Lie in Its Rectifying Plane?, Amer. Math. Monthly 110, 147–152. 4. Chen, B. Y. and Dillen, F. (2005) Rectifying Curves as Centrodes and Extremal Curves, Bulletin of the Institute of Mathematics Academia Sinica 33 No. 2, 77–90.  5. Weiler, P.J. and Bogoch, R. E. (1995) Kinematics of the distal radioulnar joint in rheumatoid-arthritis-an in-vivo study using centrode analysis, J. Hand Surgery 20A, 937–943. 6. Oh, Y. M. and Seo, Y. L. (2015) A Curve Satisfying  with constant , American Journal of Undergraduate Research 12, 57–62.  ABOUT THE STUDENT AUTHOR Julie Logan is an undergraduate student at Andrews University and this paper is a requirement of the honors program at Andrews University. She has interest in mathematics and has earned several awards for excellence in mathematics classes. She is planning to apply for graduate work in mathematics.  PRESS SUMMARY It is known that for every rectifying curve, the ratio  is a linear function. In this paper, we give a different proof of the result. Besides, we also derive formulas for the curvature and the torsion of sphere curves as well.     ACKNOWLEDGEMENTS The authors thank the Andrews University Research Office and J. N. Andrews Honors program for assistant through the 2015–2016 Undergraduate Research Assistant Scholarship. The authors are very grateful to the referees for several valuable suggestions.  

Characterization of Rectifying and Sphere Curves in R^3

https://digitalcommons.andrews.edu/cgi/viewcontent.cgi?article=1709&amp;context=pubs

Characterization of Rectifying and Sphere Curves in R^3

Abstract

Similar works

Full text

Available Versions

Andrews University