# 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; = 1/2 Figure 16. A family of curves (p-norm) and their orthogonal trajectories; = 1 Figure 17. A family of curves (p-norm) and their orthogonal trajectories; = 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, vand  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 > 0h 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