Geometry of Curves
We will study some constructions and visualizations of the differential geometry of curves: the tangent line and Frenet frame (Section 6.1), the singular points (Section 6.2), the length and center of mass (Section 6.3), and the curvature and torsion (Sec
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Geometry of Curves
We will study some constructions and visualizations of the differential geometry of curves: the tangent line and Frenet frame (Section 6.1), the singular points (Section 6.2), the length and center of mass (Section 6.3), and the curvature and torsion (Section 6.4).
6.1 Tangent Lines We will plot the (moving) tangent line and Frenet frame for various types of presentations of a curve. Then we will study some applications of tangent and normal lines to a curve. Curves can appear as envelopes of families of lines, circles, etc. Using R MATLAB , we will realize some methods of “mathematical embroidery” from the literature, and study evolutes, caustics, and parallel curves.
6.1.1 Tangent lines, normals, and the Frenet frame The tangent line to the curve γ of class C1 at the point P ∈ γ is the limit position of the secant PQ as the second point of intersection Q approaches P along the curve. Semi-tangent lines from the right and from the left are defined analogously. Since the secant through the points at t and t + Δ t is parallel to the vector Δ r = r(t + Δ t) − r(t), and by the definition of the derivative r (t) = limΔ t→0 ΔΔ rt , the line through the point of the regular curve r(t) ∈ γ in the direction of the velocity vector r (t) = [x (t), y (t), z (t)] is the tangent line (Figure 6.1). The unit tangent and normal vectors of the plane curve r(t) are derived as follows:
R V. Rovenski, Modeling of Curves and Surfaces with MATLAB , Springer Undergraduate Texts in Mathematics and Technology 7, DOI 10.1007/978-0-387-71278-9 6, c Springer Science+Business Media, LLC 2010
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6 Geometry of Curves
τ=
x
y
, x2 + y2 x2 + y2
,
ν=
−y
x
, . x2 + y2 x2 + y2
The tangent line at the point t of the parametric curve r(t) ⊂ R3 has the following equation: r˜ (s) = r(t) + sr (t) (s ∈ R) ⎧ ⎪ ⎨ x = x(t) + sx (t), y = y(t) + sy(t), ⇐⇒ ⎪ ⎩ z = z(t) + sz(t).
Fig. 6.1 Derivative of a vector-valued function and a tangent line. Equations of the tangent line to a plane curve of class C1 (for various types of equations of the curve) are the following (here X , Y are coordinates of points on the tangent line, and s ∈ R): graph
Y = y + f (x)(X − x),
parametric
X = x(t) + sx(t), Y = y(t) + sy(t),
implicit
∂F ∂F ∂ x (x, y)(X − x) + ∂ y (x, y)(Y
− y) = 0.
Equations of the normal vector to the plane curve are the following: graph
X = x − s f (x), Y = y + s,
parametric
X = x(t) − sy(t), Y = y(t) + sx(t),
implicit
X = x + s ∂∂Fx (x, y), Y = y + s ∂∂Fy (x, y).
For an implicitly defined plane curve, F(x, y) = 0, the vector gradient ∇F = is parallel to its normal vector.
"
∂F ∂F ∂x , ∂y
#
Examples. 1. A dog runs along the axis OY starting from O = (0, 0), and its owner (initially staying on the axis OX ) follows the dog, pulling on the leash of a length a. What is the trajectory of the owner? The curve is called the tractrix (Figure 6.3(c)). We deduce its equation using the tangent line (see also Exercise 1, p. 279, with MATLAB). Solution. From a
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