1. INTRODUCTION:

Differential geometry is that part of geometry which is treated with the help of differential calculus. Differential geometry provides the language and theory to understand geometry. It provides the study of properties of arbitrary small pieces of curves and surfaces. Its importance in pure and applied mathematics can be viewed with the help of following example. A plane which is flying from Tokyo to Los Angeles flies from north of either city. Although the two cities have same latitude yet the plane do not fly straight east from Tokyo to Los Angeles i.e. the path do not follow a circle of latitude. Actually the airlines minimize its distance to be covered in order to minimize travel time and fuel cost. So it follows a minimum curved path on curved surface of earth. Differential geometry provides the language and theory to understand this problems.¹

2. Historical background of developments in Differential Geometry:

Although the Greeks were able to differentiate between various types of curves like a straight line, a circle, conic section etc. yet they did not have any precise theory regarding curves. It was Euclid who first gave the definition of a tangent to the circle. The Greeks also know various kinds of surfaces such as sphere, cylinder, spheroid etc. but they did not have any surface theory. In 17^th century analytical geometry was developed which uses analytic functions to describe the curves. The foundation stone of differential geometry was laid down when Leibnitz and Newton described fundamental principles of calculus.²for general surfaces, Euler gave the concept of normal sections of surfaces. In 1776, Meusnier gave the concept of skew sections to supplement Euler formula. Monge (1776-1818) together with K.F. Gauss (1777-1855) is considered as father of differential geometry. Monge was the first author of a book on differential geometry. Gaspard Monge made several contributions in differential geometry. He gave the concept of family of surfaces, enveloping surfaces and their characteristics. Meusnier and Dupin were his students. They further carried the study of their mentor. Gauss fundamental work on surface theory introduced the intrinsic geometry of surfaces in which curvilinear coordinates were used. Through Monge’s influence, geometry began to flourish at the Polytechnic school and through Monge’s methods, Dupin was led to find asymptotic lines and conjugate lines. In 1826, A. L. Cauchy published his book on differential geometry to promote differential geometry. In 1730, Clairaut and Euler considered space curves like helix. They analyzed how much the curves twisted out of the plane the measure of which they called as torsion.

3. Formation of a helix:

A helix is the form assumed by a straight line drawn in any plane, when that plane is wrapped around the curved surface of a right circular cylinder. Conversely a helix is a straight line when the cylinder on which it is drawn is converted into a plane.³ for finding the mathematical equation of a helix, we consider a straight line y = mx in a plane. Let us wrap this plane on a right circular cylinder so that the axis of x may coincide with the circular base, the line will become the helix. Also the ordinate y of any point on the curve is proportional to the value of x measured along the circle. So the coordinates of the projection of this point on the base of the cylinder are x = a cost and y = a sint where a is radius of the circular base. But the height z of the point is proportional to arc t. So the equation of the helix is

x = a cos z/h , y = a sin z/h , x²+ y² = a²

Further dx = - a/h ( sin z/h ) dz = - y/h dz and dy = a/h ( cos z/h) dz = x/h dz

So ds²= ( a²+h²)/h²dz² which implies that dz/ds is constant.

In other words the angle made by the tangent at any point on the helix with z axis is constant. This angle is same which the tangent at any point on the helix makes with the generators of the cylinder.

3.1 Definition:

A helix is a smooth curve in three-dimensional space such that the tangent line at any point on the curve makes a constant angle with a fixed line. The fixed line is called the axis of helix. A circular helix is a curve drawn on surface of a circular cylinder, cutting the generators at a constant angle.^[4]

3.2 Parametric equations of a cylindrical helix:

A cylindrical helix has the following parametric equations:

x = r cost, y = r sint ,z = pt where r is radius of helix .

The pitch of a helix is defined as the displacement of one complete helix turn, measured along the axis of helix. The above curve lies on the right circular cylinder of radius r.

The equation z = pt moves the points of curve uniformly in z – direction. When t increases or decreases by 2π, x and y return to their original values but z increases or decreases by 2πp, the pitch of the helix.^[5]

Here r > 0 , if p > 0 the helix is right handed if p < 0 the helix is left handed.^[6]

First we list the important properties related to circular helix in the form of following theorems:

Theorem 1:

The necessary and sufficient condition for a curve to be helix is that its curvature and torsion are in constant ratio.^[7]. Also the only curve for which both curvature and torsion are both constant is a circular helix.^[8]

Theorem 2:

The circular helix is a curve of constant curvature.^[9]

3.3 Osculating circle at a point on the helix:

The word "osculate" means "to kiss." The osculating circle of a curve at a given point on the curve is defined as the circle passing through that point and two additional points on the curve infinitesimally close to that point. This circle is the one among all tangent circles at the given point that approaches the curve most tightly. Leibniz gave this kissing circle the name “circulus osculans” (Latin for "kissing circle") .The center and radius of the osculating circle at a given point are called center of curvature and radius of curvature of the curve at that point. The osculating circle is also known as the Circle of Curvature. In other words the osculating circle of a curve at a given point is the circle that has the same tangent as that of the curve as well as the same curvature at that point.

4. A Mathematical analysis of locus of centre of osculating circles of a circular helix:

First we shall show that the tangent to the locus of centre of osculating circle makes a constant angle with the axis of circular helix. Let C₁ be the locus of centre of curvature of osculating circle on curve C. let us denote all the quantities associated with curve C₁by using suffix unity. So if r₁is position vector of any point on C₁ then

r_{1 =}r + ρ n (1)

Where r is current point’s position vector on curve C and ρ is radius of curvature of the curve at that point.

On differentiating (1) w.r.t. s₁

5. CONCLUSION:

From above analysis using differential geometry, it is clear that the tangent to the locus of centre of osculating circle makes a constant angle with the axis of circular helix. Further we have proved that the locus C₁ of centre of osculating circle has constant curvatureand its torsion varies inversely as that of C. Also the locus of centre of curvature of osculating circle is a helix as the necessary and sufficient condition for a curve to be helix is that its curvature and torsion are in constant ratio. Finally the pitch of new helix so formed is same as the former helix.

6. REFERENCES:

1. Thinking Geometrically, A Survey of Geometries, Thomas Q. Sibley 2015 The Mathematical Association of America pp 409-410

2. Companion Encyclopedia of History and Philosophy of Mathematical Sciences, I. Grattan- Guinness, The John Hopkins University Press, pp 331-333

3. A treatise on Analytic Geometry of Three Dimensions , George Salmon , second edition , Hodges Smith and co. Dublin 1865 pp 287-288

4. Differential geometry of three dimensions, C. E. Weather burn, Cambridge University Press 1964 pp16

5. Theory and problems of differential geometry, Martin M. Lipschutz Schaum’s outline series 1969 pp 46

6. An introduction to differential geometry, T. J. Will more , Oxford at the Clarendon press 1959 pp 26

7. Differential geometry of three dimensions, C. E. Weather burn, Cambridge University Press 1964 pp24

8. Three dimensions differential geometry Bansi Lal Atma Ram and sons, Delhi-6 , 1966 pp 29-30

9. Differential geometry of three dimensions, C. E. Weather burn, Cambridge University Press 1964 pp26-27

Received on 27.04.2017 Accepted on 22.05.2017

Research J. Engineering and Tech. 2017; 8(2): 122-124.

DOI: 10.5958/2321-581X.2017.00018.6