2. PRELIMINARIES

In this section, we recall some important notions and also give some properties of curves and surfaces in Euclidean 3-space ${\bf {\opf E}}^3 $. For more details, we refer to Babaarslan and Munteanu (Reference Munteanu2013), Babaarslan and Yayli (Reference Babaarslan and Yayli2014), Guler et al. (Reference Guler, Yayli and Hacisalihoglu2010), Ikawa (Reference Ikawa2000), O'Neill (Reference O'Neill1966) and Do Carmo (Reference Do Carmo1976).

The inner product of two vectors x = (x ₁, x ₂, x ₃) and y = (y ₁, y ₂, y ₃) in ${\bf {\opf E}}^3 $ is given by:

(1)$$\left\langle {x,y} \right\rangle = x_1 y_1 + x_2 y_2 + x_3 y_3 $$

The norm (length) of a vector $x \in {\bf {\opf E}}^3 $ is $\vert\vert x \vert\vert = \sqrt {\left\langle {x,x} \right\rangle} $ and it is called a unit vector if ||x||=1. The angle θ between x and y is given by

(2)$$\cos \theta = \displaystyle{{\left\langle {x,y} \right\rangle} \over {\vert\vert x \vert\vert \vert\vert y \vert\vert}}$$

where 0 < θ < π. In particular, if θ = π/2, then 〈x, y〉 = 0. Thus we can define vectors to be orthogonal provided their inner product is zero.

Let $\beta :I \subset {\bf {\opf R}} \to {\bf {\opf E}}^3 $ be a regular curve in ${\bf {\opf E}}^3 $ (i.e., $\dot \beta \ne 0$ for all t ∈ I). The arc-length of β between t ₀ and t is given by

(3)$$s(t) = \int_{t_0} ^t {\vert\vert {\dot \beta (t)} \vert\vert} dt$$

Then the parameter $s \in J \subset {\bf {\opf R}}$ is determined such as ||β′(s)|| = 1. Thus β is called a unit speed curve if ||β′(s)|| = 1.

Now we give the definitions of rotational surfaces and helicoidal surfaces and also some important formulas in ${\bf {\opf E}}^3 $.

Let $\beta :I \subset {\bf {\opf R}} \to P$ be a regular curve in a plane $P \subset {\bf {\opf E}}^3 $ and l be a straight line in P. If this profile curve β is rotated about the axis l, then it sweeps out a rotational surface in ${\bf {\opf E}}^3 $. Similarly, we assume that when the profile β rotates about the axis l, it simultaneously displaces parallel to l so that the speed of displacement is proportional to the speed of rotation. As a result it sweeps out a helicoidal surface in ${\bf {\opf E}}^3 $.

Let us assume that l = span{(0, 0, 1)}. Then the rotation which leaves the axis l invariant is given by the following rotational matrix

(4)$$\left[ {\matrix{ {\cos v} & { - \sin v} & 0 \cr {\sin v} & {\cos v} & 0 \cr 0 & 0 & 1 \cr}} \right],v \in I \subset {\bf {\opf R}}$$

If the rotation axis is l, then there is a Euclidean transformation where the axis l is transformed to the x ₃-axis of ${\bf {\opf E}}^3 $. Rotation leaves the planes that are orthogonal to the rotation axis l invariant. Thus we can choose the profile curve in a plane containing the rotation axis l. As a result we can take the profile curve β in the (x ₁x ₃)-plane that can be parameterised by β(u) = (f(u), 0, g(u)), $u \in I \subset {\bf {\opf R}},$ where f(u) ≠ 0.

A helicoidal surface with the rotation axis l and the pitch $\lambda \in {\rm {\opf R}}\backslash \{ 0\} $ can be parameterised as follows

$$H(u,v) = \left[ {\matrix{ {\cos v} & { - \sin v} & 0 \cr {\sin v} & {\cos v} & 0 \cr 0 & 0 & 1 \cr}} \right]\left[ {\matrix{ {\,f\,(u)} \cr 0 \cr {g(u)} \cr}} \right] + \lambda v\left[ {\matrix{ 0 \cr 0 \cr 1 \cr}} \right],$$

or

(5)$$H(u,v) = \left( {\,f\,(u)\cos v,f\,(u)\sin v,g(u) + \lambda v} \right)$$

When g is a constant function, the helicoidal surface is called the right helicoidal surface. Also, when λ = 0, the helicoidal surfaces reduce to rotational surfaces.

For a surface X(u, v), the first fundamental form or line-element in the base {X _u, X_v} is given by

(6)$$ds^2 = Edu^2 + 2Fdudv + Gdv^2 $$

where E = 〈X _u, X_u〉, F = 〈X _u, X_v〉 and G = 〈X _v, X_v〉 are the coefficients of the first fundamental form.

The angle θ between two regular curves $\beta :I \subset {\bf {\opf R}} \to X$ and $\gamma :I \subset {\bf {\opf R}} \to X$ which intersect at t = t ₀ is given by

(7)$$\cos \theta = \displaystyle{{\left\langle {\dot \beta (t_0 ),\dot \gamma (t_0 )} \right\rangle} \over {\vert\vert {\dot \beta (t_0 )} \vert\vert \vert\vert {\dot \gamma (t_0 )} \vert\vert}}$$

In particular, the angle ϕ between the constant parameter curves of X(u, v) is given by

(8)$$\cos \varphi = \displaystyle{{\left\langle {X_u, X_v} \right\rangle} \over {\vert\vert {X_u} \vert\vert \vert\vert {X_v} \vert\vert}} = \displaystyle{F \over {\sqrt {EG}}} $$

Thus the constant parameter curves are orthogonal if and only if F(u, v) = 0 for all (u, v).

3. DIFFERENTIAL EQUATION OF THE LOXODROME

Let us consider the helicoidal surface H that is given by Equation (5). To simplify the calculations, we assume that f′²(u) + g′²(u) = 1 for all $u \in J \subset {\bf {\opf R}},$ that is, the profile curve β is parameterised by arc-length parameter. The coefficients of first fundamental form of the helicoidal surface are

(9)$$E = \left\langle {H_u, H_u} \right\rangle = 1,F = \left\langle {H_u, H_v} \right\rangle = {\lambda} g'(u)\; and\,G = \left\langle {H_v, H_v} \right\rangle = f^2 (u) + \lambda ^2 $$

Obviously, the constant parameter curves of H(u, v) are orthogonal if and only if H(u, v) is either a right helicoidal surface or a rotational surface. Substituting Equation (9) into (6), the first fundamental form of the helicoidal surface is determined by the following equation

(10)$$ds^2 = du^2 + 2\lambda g'(u)dudv + (\,f^2 (u) + \lambda ^2 )dv^2 $$

Also the arc-length of any curve on the helicoidal surface between u ₁ and u ₂ is given by

(11)$$s = \int_{u_1} ^{u_2} {\sqrt {1 + 2\lambda g'(u)\displaystyle{{dv} \over {du}} + (\,f^2 (u) + \lambda ^2 )(\displaystyle{{dv} \over {du}})^2} du} $$

We may assume that the loxodrome α(t) is the image of a curve (u(t), v(t)) in the (uv)-plane by H. Since in the basis {H _u, H_v}, the vector α′(t) has the coordinates (u′, v′) and the vector H _u has the coordinates (1, 0). At the point H(u, v) where the loxodrome intersects the meridian at a constant azimuth θ, we have

(12)$$\matrix{ {\cos \theta} \hfill & = \hfill & {\displaystyle{{\left\langle {\alpha^{\prime}(t),H_u} \right\rangle} \over {\vert\vert {\alpha^{\prime}(t)} \vert\vert \vert\vert {H_u} \vert\vert}}} \hfill & = \hfill & {\displaystyle{{Edu + Fdv} \over {\sqrt {E^2 du^2 + 2EFdudv + EGdv^2}}}} \hfill \cr {} \hfill & {} \hfill & {} \hfill & = \hfill & {\displaystyle{{du + \lambda g^{\prime}(u)dv} \over {\sqrt {du^2 + 2\lambda g^{\prime}(u)dudv + (\,f^2 (u) + \lambda^2 )dv^2}}}} \hfill \cr} $$

From Equation (12), we obtain the following differential equation of the loxodrome on the helicoidal surface:

(13)$$\left( {\cos ^2 \theta (\,f^2 (u) + \lambda^2 ) - \lambda^2 g^{\prime 2} (u)} \right)(\displaystyle{{dv} \over {du}})^2 - 2\lambda \sin ^2 \theta g^{\prime}(u)\displaystyle{{dv} \over {du}} = \sin ^2 \theta $$

Thus the general solution of the differential equation of the loxodrome on the helicoidal surface is

(14)$$v = \int_{u_0} ^u {\displaystyle{{2\lambda \sin ^2 \theta g^{\prime}(u) + \varepsilon \sqrt {\sin ^2 2\theta \left( {\,f^2 (u) - \lambda ^2 (g^{\prime 2} (u) - 1)} \right)}} \over {2\cos ^2 \theta (\,f^2 (u) + \lambda ^2 ) - 2\lambda ^2 g^{\prime 2} (u)}}} du$$

where ε = ±1.

For the loxodrome on the right helicoidal surface, Equation (13) reduces to Equation (15):

(15)$$\cos ^2 \theta (\,f^2 (u) + \lambda ^2 )(\displaystyle{{dv} \over {du}})^2 = \sin ^2 \theta $$

With the solution of this differential equation, we find the general solution of the differential equation of the loxodrome on the right helicoidal surface as

(16)$$v = \varepsilon \tan \theta \int_{u_0} ^u {\displaystyle{{du} \over {\sqrt {\,f^2 (u) + \lambda ^2}}}} $$

where ε = ± 1.

Noble (Reference Noble1905) and Kos et al. (Reference Kos, Filjar and Hess2009) found the differential equation of the loxodrome on a rotational surface. If we take λ = 0 in Equation (16), we find the general solution of the differential equation of the loxodrome on rotational surface as follows

(17)$$v = \varepsilon \tan \theta \int_{u_0} ^u {\displaystyle{{du} \over {\,f\,(u)}}} $$

where ε = ± 1. This equation coincides with the equation in (Noble, Reference Noble1905; Kos et al., Reference Kos, Filjar and Hess2009).

Next, we give the arc-length of the loxodrome on the right helicoidal surface. The arc-length of any curve on the right helicoidal surface between u ₁ and u ₂ is given by

(18)$$s = {\int}_{u_1}^{u_2} {\sqrt {1 + (\,f^2 (u) + \lambda ^2 )(\displaystyle{{dv} \over {du}})^2} du} $$

By using Equation (15), the arc-length of the loxodrome is defined as follows

(19)$$s = \displaystyle{{u_2 - u_1} \over {\cos \theta}} $$

Similarly, the arc-length of the loxodrome on the rotational surface coincides with Equation (19).

Now we give three examples of loxodromes that lie on the helicoidal surfaces to strengthen our main results.

Figure 2. The helicoidal surface; loxodrome (blue), meridian (green).

Figure 3. The helicoidal surface; loxodrome (blue), meridian (green).

Due to the energy minimisation principle, minimal surfaces are omnipresent in nature and science. Thus their study has been a fascinating notion for centuries. One of the non-trivial examples of minimal surfaces is the helicoid found by Meusnier (Bates et al., Reference Bates, Wei and Zhao2008). We can construct it by using Equation (5). The parallel (u = constant) on the helicoid is a helix and it is perpendicular to the meridians (v = constant). Thus a helix on the helicoid is also a loxodrome. This is an important property since helices are strongly related to navigation. Thus we can give the following example:

Figure 4. The right helicoidal surface (helicoid); loxodrome (blue), meridian (green), parallel (red).