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.

Sampled peak normal acceleration $\text{Ready}$

Peak change at double density $\text{Ready}$

Sampled target check $\text{Ready}$

Frame at selected time $\text{Ready}$

Selected time $\text{Ready}$

Selected speed $\text{Ready}$

Acceleration magnitude $\text{Ready}$

Signed tangential acceleration $\text{Ready}$

Normal acceleration $\text{Ready}$

Geometric curvature $\text{Ready}$

Osculating radius $\text{Ready}$

Sampled path length $\text{Ready}$

Acquired positions $\text{Ready}$

Cartesian fit speed-vector RMS error $\text{Ready}$

Cartesian fit acceleration RMS error $\text{Ready}$

Set the path, its timing and the sampled positions

Design the curve; inspect the frame

Clock modulation

\epsilon_t

0.00\,

Sampled normal acceleration target

a_{n,\max}

2.00\,\frac{\mathrm m}{\mathrm{s^2}}

Control point to edit; zero to three

i

1.00\,

Point 0 · First horizontal coordinate

x_{0}

-4.00\,m

Point 0 · Second horizontal coordinate

y_{0}

-2.00\,m

Point 0 · Height

z_{0}

0.00\,m

Point 1 · First horizontal coordinate

x_{1}

-3.00\,m

Point 1 · Second horizontal coordinate

y_{1}

4.00\,m

Point 1 · Height

z_{1}

3.00\,m

Point 2 · First horizontal coordinate

x_{2}

3.00\,m

Point 2 · Second horizontal coordinate

y_{2}

-3.00\,m

Point 2 · Height

z_{2}

4.00\,m

Point 3 · First horizontal coordinate

x_{3}

4.00\,m

Point 3 · Second horizontal coordinate

y_{3}

2.00\,m

Point 3 · Height

z_{3}

0.00\,m

Traversal duration

T

6.00\,s

Selected time fraction

t/T

0.40\,

Position sampling rate

f_s

20.00\,Hz

Position noise bound

\eta

0.00\,mm

Quadratic fit window; odd stations

n_w

5.00\,

The curve is a cubic Bézier with four control points, and each point can move in all three coordinates. Pick a point with the slider, or drag it in either coordinate view. The track is a prescribed path; gravity does not drive the motion along it.

The curve parameter is a modulated function of time; it is not arc length. Doubling the duration halves the speed and cuts the acceleration to a quarter, and the curvature stays the same. With the clock modulation at one, the motion pauses at the midpoint.

The moving Frenet frame is undefined when the speed is zero. On a straight segment or at an inflection, the normal and binormal are undefined. A cusp has no regular tangent. In these places no substitute frame and no huge stand-in radius are drawn.

The gold osculating arc lies in the local osculating plane and has the true radius of curvature. Only a short piece of the arc is shown. A radius larger than four times the scene size is not drawn, but it still appears in the readout.

Retiming finds the peak from 801 sample points and reports how much it changes with 1601. The result is held within the duration slider's range, and the status says whether the target is met at the sample points. This is a design aid; it does not bound the motion between samples or certify passenger loads.

An acquisition records only Cartesian position. Times are exact, and each coordinate carries a bounded, repeatable noise measured in millimetres. Derivatives come from a local least-squares quadratic fit, with one-sided windows at the ends. The estimator never sees the formula of the trajectory.

The fit window has an odd number of points and shrinks when the record is short. Acceleration needs at least three positions. A larger window suppresses noise but biases the estimate where the motion curves or changes quickly. The RMS errors use the analytic trajectory only as a separate reference.

Lengths are in metres. The velocity and acceleration arrows are raised on a connector so they are easy to see; their directions and display scales are unchanged. Markers and supports are schematic. Nothing plays automatically; drag the gold clock to move through time. Changing the fit window keeps the acquired positions, but changing the trajectory or sensor requires a new acquisition.

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

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 misconceptionA smooth reconstruction or a dense plot proves you have the true acceleration.

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

Run the experiment

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.
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.
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.
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.