1

Given is a point $(x, y)$ in Cartesian 2D space and a parametric curve of which we know the following:

  • The curve starts a $(0, 0)$ and extends in positive $x$ direction.
  • The curve has an angular velocity that starts at 0 and increases linearly, i.e. $a(t) = a_c t$ where $a_c$ is the constant angular acceleration.

I would like to find the constant acceleration factor $a_c$ that is required for the curve to go through $(x, y)$. Mutiple solutions might exist – I am only interest in the first one with minimal $a_c$. Once $a_c$ is found, I am also interested in the parametric/arc length $t$ of the curve.

I have tried constructing formulae for $x$ and $y$ by integrating $sin(a(t))$ and $cos(a(t))$, which yields solutions including Fresnel integrals like $S(\sqrt{a}t)$ and $C(\sqrt{a}t)$. I am unsure whether these equations could be solved for $a$ analytically.

Am I maybe overthinking this problem? If not, does an analytical solution exist, or should I rely on numerical methods?

Daerst
  • 143
  • 1
  • 6
  • The solution is, in fact, rather simple: Construct a circle that lies on the x=0 line and goes through both (0,0) and (x,y). The curvature is 1/r of this circle. – Daerst Nov 21 '24 at 07:02

0 Answers0