论文标题
在欧几里得3-空间中旋转表面上的Loxodromes的表征
Characterizations of Loxodromes on Rotational Surfaces in Euclidean 3--Space
论文作者
论文摘要
在本文中，我们研究了旋转表面上的Loxodromes的特征，这些旋转表面满足了一些特殊的几何特性，例如在欧几里得3空间中具有恒定的高斯曲率，平坦和最小的曲线。首先，我们给出$ \ Mathbb {e}^{3} $的任何旋转表面上通过弧形长度参数参数参数的loxodromes的参数化，然后，我们计算了此类Loxodromes的曲率和扭曲。然后，我们给出具有恒定高斯曲率的旋转表面上的Loxodromes的参数化。特别是，我们证明平坦旋转表面上的Loxodrome是一般螺旋。此外，我们以恒定的主曲线比（CRPC旋转表面）的比例研究了旋转表面上的叶状细胞。此外，我们在最小旋转表面上给出了Loxodromes的参数化，这是CRPC旋转表面的特殊情况。然后，我们表明，Loxodrome通过角度$π/{4} $与最小旋转表面的子午线相交，成为渐近曲线。最后，我们提供了一些视觉示例，以通过Wolfram Mathematica来增强我们的主要结果。
In this paper, we study on the characterizations of loxodromes on the rotational surfaces satisfying some special geometric properties such as having constant Gaussian curvature, flat and minimality in Euclidean 3-space. First, we give the parametrizations of loxodromes parametrized by arc-length parameter on any rotational surfaces in $\mathbb{E}^{3}$ and then, we calculate the curvature and the torsion of such loxodromes. Then, we give the parametrizations of loxodromes on rotational surfaces with constant Gaussian curvature. In particular, we prove that the loxodrome on the flat rotational surface is a general helix. Also, we investigate the loxodromes on the rotational surfaces with a constant ratio of principal curvatures (CRPC rotational surfaces). Moreover, we give the parametrizations of loxodromes on the minimal rotational surface which is a special case of CRPC rotational surfaces. Then, we show that the loxodrome intersects the meridians of minimal rotational surface by the angle $π/{4}$ becomes an asymptotic curve. Finally, we give some visual examples to strengthen our main results via Wolfram Mathematica.
