Sweep is a natural, intuitive, and convenient 3D modeling method in computer-aided design. Sweep surface can be obtained by extruding a 2D cross-sectional profile along a guide curve $x=x\left(t\right),t\in [a,b]$. A small segment of the sweep volume can also be understood by rotating a 2D sectorial generatrix curve around the guide curve. We assume that sweep surfaces have a parametric form $\Phi =\Phi (t,\theta ),$ where $\Phi ([t,t+\text{d}t],\theta )$ defines the sectorial generatrix curve segment at the angle of $\theta$ while $r_t\left(\theta \right)=\Phi (t,\theta ),\theta \in [0,2\pi ],$ defines the circumferential closed curve. Geodesic computation on sweep surfaces is a fundamental geometric operation in many scenarios like the manufacturing process of filament winding. In order to compute a geodesic path between two points on sweep surfaces, we propose a variational framework that works on the 2D parametric domain, without the step of discretizing the surface into a polygonal mesh. The solution to the objective function is a polyline curve of $n$ equally spaced vertices that approximates the real geodesic path, where $n$ is a user-specified parameter for accuracy control. We prove that the polyline approaches the real geodesic in quadratic order. Furthermore, it can be easily extended to compute $N$-round geodesic helix curves. We also discuss various configurations of $r_t\left(\theta \right)$: (1) $r_t\left(\theta \right)$ is a constant, indepedent of $t$ and $\theta$, (2) $r_t\left(\theta \right)$ depends on only $t$, indepedent of $\theta$, and (3) $r_t\left(\theta \right)$ depends on both $t$ and $\theta$. We validate the effectiveness and high performance of our method through extensive experimental results.

Sweep 是计算机辅助设计中一种自然、直观且方便的 3D 建模方法。可以通过沿引导曲线挤出 2D 横截面轮廓来获得扫掠面$X=X\left(吨\right),吨\in [\mathrm{一种},乙]$. 通过围绕引导曲线旋转二维扇形母线曲线，也可以理解扫描体积的一小部分。我们假设扫描曲面具有参数化形式$\Phi =\Phi (吨,\theta ),$ 在哪里 $\Phi ([吨,吨+\text{d}吨],\theta )$ 定义扇形母线曲线段的角度为 $\theta$ 尽管 $r_{吨}\left(\theta \right)=\Phi (吨,\theta ),\theta \in [0,2\pi ],$定义圆周闭合曲线。扫描表面上的测地线计算是许多场景中的基本几何运算，例如纤维缠绕的制造过程。为了计算扫描表面上两点之间的测地线路径，我们提出了一种适用于 2D 参数域的变分框架，无需将表面离散为多边形网格的步骤。目标函数的解是一条折线曲线$n$ 近似于真实测地线路径的等距顶点，其中 $n$是用户指定的精度控制参数。我们证明了折线以二次阶逼近真实测地线。此外，它可以很容易地扩展到计算$N$-圆形测地线螺旋曲线。我们还讨论了各种配置$r_{吨}\left(\theta \right)$: (1) $r_{吨}\left(\theta \right)$ 是一个常数，独立于 $吨$ 和 $\theta$, (2) $r_{吨}\left(\theta \right)$ 只取决于 $吨$，独立于 $\theta$, 和 (3) $r_{吨}\left(\theta \right)$ 取决于两者 $吨$ 和 $\theta$. 我们通过广泛的实验结果验证了我们方法的有效性和高性能。