In this section, we apply our software to the graphical representation of some results from the theory of curves.
Curves in two-dimensional space may be given by an equation
f(x^{1},x^{2}) = 0 |
x (t) = (x^{1}(t),x^{2}(t)) (t Î I Ì R). |
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 = {g_{c}:c Î I} be a family of curves in a plane, given by the equations
f(x^{1},x^{2}; c) = 0 for c Î I. |
¶F/¶c (x^{1},x^{2}; c) = 0. |
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 = {g_{c}: c Î I} be a family of curves that cover a domain D Ì R^{2} such that there is one and only one curve through every point of D. The curves g^{^} that intersect every curve g_{c} Î G at a right angle are called orthogonal trajectories of G. If the curves g_{c} are given by the equation f(x^{1},x^{2}; c) = 0, then the orthogonal trajectories are given by the solutions of the differential equations
dx^{2}/dx^{1 }¶f /¶x^{1} (x^{1},x^{2}; c) = ¶f /¶x^{2}(x^{1},x^{2}; c). |
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
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 R^{3} with a parametric representation
x(s) = (x^{1}(s),x^{2}(x),x^{3}(s)) with s the arc length along g.
Then the vectors
v_{1}(s) = x'(s)
v_{2}(s) = x''(s)
/ || x''(s) || and
v_{3}(s) = v_{1}(s)
× v_{2}(s).
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 / (r^{2}+h^{2})^{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) · v_{3}(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) = rw^{2} and t(s) = hw^{2}. The third order approximations of the helix in the osculating, normal and rectifying planes are given by the equations
y=1/2 r w^{2}x^{2},
z^{2} = 2/9 h^{2 }/ r · w
^{2}y^{3} and
z = 1/6 r h w^{4}x^{3},
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) + v_{2}(s) / k(s) - k'(s)v_{3}(s) / t(s)
and
r(s) = ( 1/k^{2}(s) + (k'(s))^{2}/ (t^{2}(s) k^{4}(s)) )^{1/2}
Example 9. The osculating sphere of the helix in Example 7 is given by
m(s) =
( -h^{2}/r cos(ws),
-h^{2}/r sin(ws),
hws )
and
r(s) = (r^{2}+h^{2})/r.
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