---
title: "Curvature & Frenet Track Designer Interactive Lab | Sandbox Physics"
description: "How do the bends in a track and the speed along it set the normal acceleration? Normal acceleration is curvature times speed squared."
canonical_url: "https://sandboxphysics.com/labs/frenet-track-designer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/frenet-track-designer/"
last_modified: "2026-10-27"
---

M008 · Moving bases

# Curvature & Frenet Track Designer

**How do the bends in a track and the speed along it set the normal acceleration?** Normal acceleration is curvature times speed squared. The shape of the path fixes the curvature at each point; the timing only sets how fast you pass through. Going twice as fast through the same bend gives four times the normal acceleration.

Shape a cubic curve in space and look at its tangent, its normal and its osculating circle. Then retime the motion until the sampled normal acceleration meets a target, and recover the motion from raw position samples.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The motion is prescribed and the position observations are simulated. The Lab does not certify forces or safety.

Physics tutorial

## Geometry, timing and acquired evidence

**Background** Each motion here has three parts: the shape of the path, the timing along it and a record of observed positions.

**Why it matters** Compare the vectors drawn on the track with the derivatives you recover from measured positions.

Start with the essentials

- **Focus question**: Which part of the acceleration bends the path?
- **One-sentence intuition**: Changing the timing does not change the curvature of the path.

### Core mathematical model

#### The cubic path you edit

$$
\mathbf c(q)=\sum_{i=0}^3\binom3i(1-q)^{3-i}q^i\mathbf P_i
$$

The end points lie on the curve, and the inner points steer its tangent. The lines joining the control points are not part of the track.

#### Timing along the path

$$
\begin{aligned}u&=t/T\\q&=u+\frac{\epsilon_t}{2\pi}\sin(2\pi u)\\\mathbf v&=\mathbf c_q\dot q\\\mathbf a&=\mathbf c_{qq}\dot q^2+\mathbf c_q\ddot q\end{aligned}
$$

Progress along the path never goes backward. When the modulation reaches one, the motion pauses for a single instant.

#### Curvature depends only on the path

$$
\kappa=\frac{\lVert\mathbf c_q\times\mathbf c_{qq}\rVert}{\lVert\mathbf c_q\rVert^3},\qquad R=\kappa^{-1}
$$

The formula needs a nonzero derivative of the curve. Where the curvature is zero, there is no finite osculating circle.

#### Acceleration in the Frenet frame

$$
\begin{aligned}\mathbf a&=a_t\mathbf T+a_n\mathbf N\\a_t&=\dot{\lVert\mathbf v\rVert}\\a_n&=\kappa\lVert\mathbf v\rVert^2\end{aligned}
$$

The normal component is what bends the path. This is the net acceleration, not the load on a seat or the support force.

#### Retiming without changing the path

$$
T_{\mathrm{new}}=T\sqrt{\frac{a_{n,\mathrm{peak}}}{a_{n,\mathrm{target}}}}
$$

The peak comes from sampling. Checking on a grid twice as fine helps, but it cannot rule out every narrow maximum.

### Common difficulties

#### A smooth curve is not the true acceleration

**Typical misconception** A smooth reconstruction or a dense plot proves you have the true acceleration.

**Better mental model** Sampling can miss turns, local fits add bias, and the largest sampled value is not a guaranteed bound on the continuous curve.

### Run the experiment

1. 01

   #### Design a path in space

   Choose a control point; drag it in each coordinate view and adjust its height.

   **What to observe:** One path sets the 3D track, the frame, the curvature and the acceleration.

2. 02

   #### Change the timing, keep the shape

   Double the duration, compare the peak, then use the retiming action.

   **What to observe:** Speed and acceleration scale differently with duration. The sampled peak is compared directly with the target.

3. 03

   #### Look at the special cases

   Compare Straight path, Inflection, Cusp and Midpoint pause.

   **What to observe:** Where a direction is undefined, it is left out instead of being filled in with an arbitrary frame.

4. 04

   #### Measure the motion from samples

   Acquire, add noise, change the fit window and export raw positions.

   **What to observe:** The window size trades noise against bias, and near the endpoints the one-sided fits add their own error.

Physical conclusion

Different coordinate components describe the same motion in an inertial frame. Singular frames and sparse observations each have clear limits.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)