5.  Curves

 

In this section, we apply our software to the graphical representation of some results from the theory of curves.

5.1. Curves in two-dimensional space

Curves in two-dimensional space may be given by an equation 

f(x1,x2) = 0

(4)
or by a parametric representation 

x (t) = (x1(t),x2(t))      (t Î I Ì R).

(5)

 

Example 4.  Some algebraic curves

Figure 10.  The geometric definition of a double egg line

 

Figure 11. The geometric definition of a rosette

 

Now we study envelopes of families of curves. 

Let   I Ì R be an interval and G = {gc:c Î I} be a family of curves in a plane, given by the equations

f(x1,x2; c) = 0     for   c Î I.

(6)
A curve g* which has the property that it is tangent to a curve gc Î G  at each of its points is called an envelope of  G.  An equation of  g* is obtained by eliminating the parameter c from equation (6) and 

F/c (x1,x2; c)  =  0.

(7)

 

Example 5.   The envelope of a family of ellipses

Figure 12. A family of curves (p-norm) and their envelope; a = 2

 

Figure 13. A family of curves (p-norm) and their envelope; a = 4

 

Figure 14. A family of curves (p-norm) and their envelope; a = 3/4

 

Now we consider orthogonal trajectories. Let G = {gc: c Î I} be a family of curves that cover a domain D Ì R2 such that there is one and only one curve through every point of D. The curves g^ that intersect every curve gc Î G at a right angle are called orthogonal trajectories of G.  If the curves gc are given by the equation f(x1,x2; c) = 0, then the orthogonal trajectories are given by the solutions of the differential equations 

 

dx2/dx1  f /x1 (x1,x2; c)  =  f /x2(x1,x2; c).

(8)

 

Example 6.  Orthogonal trajectories of generalized circle lines

Figure 15. A family of curves (p-norm) and their orthogonal trajectories; a  = 1/2

 

Figure 16. A family of curves (p-norm) and their orthogonal trajectories; a  = 1

 

Figure 17. A family of curves (p-norm) and their orthogonal trajectories; a  = 4/3

 

Figure 18. A family of curves (metric) and their orthogonal trajectories

 

 

5.2.  Some results from the general theory of curves

In this part we deal with the graphical representation of some results from the general theory of curves.

Let g be a curve in three-dimensional Euclidean space  R3 with a parametric representation

x(s) = (x1(s),x2(x),x3(s))      with   s  the arc length along g.

 

Then the vectors

v1(s) =  x'(s)
v2(s) =  x''(s)  /  || x''(s) ||      and
v3(s) =  v1(s) × v2(s).

are called tangent, normal and binormal vectors of g at s; they are the vectors of the trihedra of g. The planes at every point of a curve that are  orthogonal to the vectors v3, v1  and  v2 are called the osculating, normal and rectifying planes, respectively.

Example 7.   The vectors of the trihedra of a helix with a parametric representation

x(s) = (r cos(w s), r sin(w s), h w s)      with   s Î R  where  r > 0,  h Î R  and  w = 1 / (r2+h2)1/2  arc constants.

Figure 19. The vectors of trihedra of a helix

The vector  x''(s)  and its length  k(s) = || x''(s) ||  are called the vector of curvature and the curvature of  g  at  s. The curvature is a measure of the deviation of a curve from a straight line. The uniquely defined circle in the osculating plane of a curve  g  at   s, which is a second order approximation of  g  at  s, is called the osculating circle of  g  at  s.

Figure 20. Vectors of curvature and osculating circles

The value  t(s)  =  v'2(s) ·  v3(s)  is called the torsion of  g  at  s. The torsion is a measure of the deviation of a curve from a plane.

Figure 21. The torsion along a non-planar curve on a cone

Example 8.  The curvature and torsion of the helix in Example 7 are given by  k(s) = rw2  and  t(s) = hw2. The third order approximations of the helix in the osculating, normal and rectifying planes are given by the equations

y=1/2  r w2x2,
z2 = 2/9 h2 / r · w 2y3      and
z = 1/6  r h w4x3,  

that is, they are a quadratic, a Neil's and a cubic parabola.

Figure 22. Third order approximations of a helix in the osculation, normal and rectifying planes

 

Let  g  be a curve with a parametric representation  x(s). Then the osculating sphere of g  at s is the sphere that is a third order approximation of g at s. The osculating sphere of a curve g at s is uniquely defined whenever k(s), t(s) ¹ 0;  its centre and radius are given by 

m(s)  =  x(s)  +  v2(s) / k(s)  - k'(s)v3(s) / t(s)

and

r(s) = ( 1/k2(s) + (k'(s))2/ (t2(s) k4(s)) )1/2 

 

Example 9.  The osculating sphere of the helix in Example 7 is given by

m(s) = ( -h2/r  cos(ws), -h2/r  sin(ws),  hws )
 and 
r(s) = (r2+h2)/r.

Thus the centres of the osculating spheres of a helix are on a helix.

Figure 23. Helix with osculating plane and osculating sphere

The fundamental theorem of curves states that the shape of a curve is uniquely defined by its curvature and torsion.

Example 10.  Planar curves with  k(s) = c·s  where c is a constant are klothoids. Planar curves with k(s) = c/Ös where c is a constant are the lines of intersections of a helix with a plane orthogonal to the axis of the helix.

Figure 24. A curve with k(s) = c·s

 

Figure 25. A curve with k(s) = c/Ös

Curves with the property that their curvature and torsion are proportional are called lines of constant slope; they have a constant angle with a given direction in space.

Example 11.  Lines of constant slope on surfaces of revolution.

Figure 26. Orthogonal projections of lines of constant slope on a sphere and a paraboloid of rotation on to planes

 

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