由空间中两点之间的弧长参数化的螺旋方程 [英] Equation of a helix parametrized by arc length between two points in space

问题描述

空间中任意两个点之间由弧长参数化的螺旋(即弧长的函数)的方程是什么?有什么功能吗?如何使用Matlab或Mathematica实现相同的功能?

What is the equation of a helix parametrized by arc length (i.e. a function of arc length) between any two points in space? Is there any function for this ? How do i implement the same using matlab or mathematica ?

推荐答案

只是为了添加到米奇·惠特(Mitch Wheat)的答案中，螺旋并不是唯一的.对于给定的轴，自由度是转弯，半径和相位(下面的P，A和phi)之间的距离

just to add to Mitch Wheat's answer, helices are not unique; for a given axis, the degrees of freedom are distance between turns, radius, and phase (P, A, and phi below)

如果概括为

w = 2*pi/P
r(t) = (A cos (wt-phi)) i + (A sin (wt-phi)) j + (t) k

分析弧长与t的函数的一种方法(不必显式计算弧长积分)是要认识到速度的大小是恒定的.平行于半径的速度分量为0，平行于轴的速度分量为1，垂直于半径和轴的速度分量为Aw，因此速度的大小为速度= sqrt( 1 + A ² w ² )，=>弧长s = sqrt(1 + A ² w ² ) t

then one way to analyze the arclength as a function of t (without having to compute the arclength integral explicitly) is to realize that the magnitude of velocity is constant; the component of velocity parallel to the radius is 0, the component of velocity parallel to the axis is 1, the component of velocity perpendicular to both radius and axis is Aw, so therefore the magnitude of velocity is speed = sqrt(1 + A²w²), => arclength s = sqrt(1 + A²w²)t

您需要某种方式来定义轴P，A和phi作为输入的基础.仅端点和弧长是不够的.

You'd need some way of defining the axis, P, A and phi as a function of whatever inputs you are given. Just the endpoints and arclength wouldn't be enough.

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