| vx | vy | vz | ax | ay | az | |v| | κ | R | k (x,y,z) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 0 | -1 | 1 | 0 | 1.414214 | 0.707107 | 1.414214 | (-0.5, 0.5, 0) |
- v = r′(t), a = r″(t), speed |v|.
- Curvature magnitude: κ = |v × a| / |v|³.
- Unit tangent: T = v / |v|.
- Curvature vector: k = dT/ds = ((v × a) × v) / |v|⁴.
- Radius of curvature: R = 1/κ (infinite when κ = 0).
- Pick a parameter value t on your curve r(t).
- Compute r′(t) and r″(t) using your method.
- Enter velocity (vx, vy, vz) and acceleration (ax, ay, az).
- Press Compute Curvature Vector to display results above.
- Export using CSV or PDF for documentation.
This calculator works from the first two derivatives of a parametric curve r(t). Enter the velocity vector v=r′(t) and acceleration vector a=r″(t) at the same parameter value t. Units may be meters and seconds, but any consistent system is valid. The speed |v| is computed first because curvature depends strongly on it.
Curvature magnitude κ as a turning rate
Curvature κ measures how rapidly the tangent direction changes per unit arc length. Numerically, κ increases when the curve bends sharply and decreases when the motion is nearly straight. Because κ=|v×a|/|v|³, doubling speed (with the same v×a) reduces κ by a factor of eight, which is important in dynamics and path planning.
Curvature vector k and principal normal
The curvature vector k=dT/ds points toward the instantaneous center of curvature. Its magnitude equals κ, and its direction aligns with the principal normal N when κ>0. The calculator uses k=((v×a)×v)/|v|⁴, which is stable for both planar and spatial curves.
Radius of curvature R for design thresholds
Radius R=1/κ summarizes bend tightness in the same units as position. For roads, tracks, robot paths, or spline segments, a minimum allowable radius can be enforced by limiting κ. When κ approaches zero, R grows large and the curve behaves locally like a straight line.
Quality checks and edge conditions
If |v| is zero, the tangent is undefined and curvature cannot be computed. If v×a is nearly zero, κ is near zero and the unit normal may become numerically unstable; the tool reports N as undefined in that case. Always verify that v and a come from the same t and not from adjacent samples.
Example values and export workflow
For the example v=(1,1,0) and a=(-1,1,0), the tool returns |v|≈1.414214, κ≈0.707107, and R≈1.414214, with curvature vector k≈(-0.5,0.5,0). Exporting CSV supports quick comparison across multiple t values, while the PDF is convenient for lab notes and documentation of a single evaluation point.
1) What inputs do I need if my curve is 2D?
Set vz=0 and az=0, then enter the x and y components of r′(t) and r″(t). The formulas remain valid and produce a planar curvature vector.
2) Why does curvature change when I re-parameterize the curve?
Curvature is geometric, but the raw derivatives depend on the parameter. The formula κ=|v×a|/|v|³ removes parameter speed, producing the same κ for any smooth re-parameterization.
3) When is the unit normal N undefined here?
If κ is zero (or extremely close), there is no unique turning direction at that point, so N is not reliable. The calculator marks N undefined to prevent misleading output.
4) What does the curvature vector direction mean physically?
k points toward the instantaneous center of curvature. In motion, it indicates the direction of normal acceleration required to follow the curve at unit speed along arc length.
5) Can I use this for space curves?
Yes. Provide all three components of v and a. The cross products capture 3D bending, and the Plotly chart helps you inspect the relative directions interactively.
6) How should I compare multiple points on the same curve?
Evaluate r′(t) and r″(t) at several t values, compute κ and k for each, then export CSV for a table. Trends in κ highlight where the curve tightens or relaxes.