The parametrization w.r.t. arc length will tell the shape of the curve. In two dimensions ## \kappa=d \phi / ds=1/r ## where ## r ## is the instantaneous radius of curvature. ## ds/dt ## gives the speed that the path is being traversed. Even in two dimensions the TNB formulation is quite useful where it just uses T and N. I first saw the TNB formalism in 2-D in a calculus book by Purcell. It always helps when a useful application is found that uses the formalism. I can give you a problem you might find of interest in two dimensions that I came up with that I solved using the TNB. Start with object mass m that has velocity ## v_0 ## (in the x-direction) and experiences a force perpendicular to its path that increases linearly with time, so that acceleration ## \vec{a}=(bt) \, \hat{N} ## for some constant b. ## \ ## ## \ ## ## \ ## 1) Does the speed of the object change? and ## \ ## 2) Determine its path. Starting with ## \vec{v}=(ds/dt) \, \hat{T} ##, ## \ ## where ## \hat{T}=\cos(\phi) \hat{i}+\sin(\phi) \hat{j} ##, and writing out the expression for ## \vec{a}=d \vec{v} /dt ##, the TNB formalism allowed for a simple solution. In particular, ## d \hat{T}/dt=(d \hat{T}/ds)(ds/dt)=(d \hat{T}/d \phi)( d \phi /ds)(ds/dt) =(\hat{N}) \kappa (ds/dt) ## is a very useful result that I first saw in the Purcell calculus book. Thereby ## \vec{a}=(d^2 s/dt^2) \hat{T}+(ds/dt)^2 \kappa \hat{N} ##. A couple months ago, someone posted about the Frenet equations: https://www.physicsforums.com/threa...t-equations-using-the-vector-gradient.876724/Sho Kano said:Is it convenient mathematically speaking? Why is TNB given with respect to arc length and not time?
To date, I have found the 2-D formalism of more use than the complete 3-D formalism. In any case, I think you would find the calculation I presented in post #4 of interest if you tried working it. If you are doing airplane or rocket trajectories the 3-D formalism is likely to be useful, but the 2-D is easier to solve, and a very practical problem is the one that has a centripetal force that increases linearly with time (presented in post #4). The solution is a spiral. Perhaps you might try solving it.Sho Kano said:OK, so calculating TNB gives us a coordinate system wrt to the cur, and we can get at the TNB using only the equation of the curve. Basically, the curve at any instant in time is resolved into 3 components (TNB), and the derivatives of these components gives us the information about how the curve is changing with arc length or time? Wouldn't defining the Frenet apparatus with respect to time be easier because some arc length integrals are hard to solve?
Sho Kano said:What is the purpose of arc length parameterization?
Arc-Length Parameterization is a mathematical technique used to describe the position and orientation of points along a curve or surface. It is often used in computer graphics and animation to create smooth and realistic motion.
The purpose of Arc-Length Parameterization is to evenly distribute points along a curve or surface based on their arc length. This results in smoother and more accurate representations of the curve or surface, making it ideal for animation and other visual applications.
Arc-Length Parameterization differs from other parameterization methods in that it takes into account the actual length of the curve or surface when determining the position of points. This allows for a more accurate representation of the curve or surface, especially when it is highly curved or irregular.
The main benefit of using Arc-Length Parameterization is that it creates a more visually appealing and natural-looking representation of a curve or surface. It also allows for smoother and more accurate animations, as points are evenly distributed along the curve or surface.
While Arc-Length Parameterization is a useful technique, it does have some limitations. It may not work well for highly complex curves or surfaces, and it can be computationally expensive to calculate. Additionally, it may not be suitable for all applications and other parameterization methods may be more appropriate.