M008 · Moving bases

Curvature & Frenet Track Designer

Edit a spatial cubic path, inspect its tangent, normal and osculating circle, then retime motion against a sampled normal-acceleration target. Recover 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 $\text{Ready}$

PATH / CLOCK / RAW 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\,

Four cubic Bézier control points are editable in three coordinates. Choose a point with the slider, or drag a point in the two coordinate views. The track is a prescribed path, not a gravity-driven rail.

Time is a modulated curve parameter, not arc length. Doubling duration halves speed and quarters acceleration without changing geometric curvature. Unit clock modulation pauses at the midpoint.

At zero speed the moving Frenet frame is undefined. At a straight segment or inflection, its normal and binormal are undefined. A cusp has no regular geometric tangent; no replacement frame or enormous fake radius is drawn.

The gold osculating arc has the actual radius in the local osculating plane. Only a local arc is shown; radii above four scene extents are omitted and retained in the readout.

Retiming uses the peak on 801 stations and reports its change at 1601. The duration slider clamps the result to its range; the status reports whether the sampled target is met. This is a design aid, not a continuous bound or passenger-load certification.

Acquisition records Cartesian position only, with exact time and bounded deterministic noise per coordinate in millimetres. A local least-squares quadratic estimates derivatives; endpoint windows are one-sided. There is no access to trajectory laws inside the estimator.

The selected odd window uses fewer stations if the record is short. Fewer than three positions cannot estimate acceleration. Larger windows suppress noise but bias curved or rapidly changing motion. RMS errors use the analytic trajectory only as a separate reference.

Geometry is in metres. Acceleration and velocity glyphs are lifted on a connector for visibility, preserving their directions and display gains. Markers and supports are schematic. There is no autoplay; scrub the gold clock. Editing the fit window preserves the acquired positions, while trajectory or sensor changes require a new acquisition.

Prescribed kinematics and simulated position observations; no force or safety certification.

Start with the essentials

Focus question: Which acceleration bends the path?
One-sentence intuition: Changing time does not change geometric curvature.

Core mathematical model

Editable cubic path

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

Endpoints lie on the curve; inner points steer its tangent. Control polygon edges are not track segments.

A separate traversal clock

\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}

The clock is monotone, with an isolated pause when modulation reaches one.

Curvature is geometric

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

A nonzero curve derivative is required. Zero curvature has 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 bends the trajectory. This is net acceleration, not seat load or support force.

Retiming without changing the path

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

The peak is sampled; a doubled grid is a diagnostic and cannot exclude every narrow maximum.

Common difficulties

Model versus observation

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

Better mental modelSampling can miss turns, local fits introduce bias, and sampled maxima are not certified continuous bounds.

Run the experiment

01
Design a spatial path
Choose a control point; drag it in each coordinate view and adjust its height.
What to observe: The same path drives the 3D track, frame, curvature and acceleration.
02
Change time, keep geometry
Double duration, then compare the peak and use the retiming action.
What to observe: Speed and acceleration scale differently; the sampled target is checked explicitly.
03
Visit the exceptions
Compare Straight path, Inflection, Cusp and Midpoint pause.
What to observe: Undefined directions are omitted, rather than filled in with an arbitrary frame.
04
Measure instead of differentiating truth
Acquire, add noise, change the fit window and export raw positions.
What to observe: Window size trades noise against bias, including one-sided endpoint errors.