Calculator Inputs
Enter velocity and acceleration components at a chosen parameter value. Point coordinates are optional graph anchors for the plotted vectors.
Example Data Table
| Curve Example | Point P(t) | Velocity v(t) | Acceleration a(t) | Principal Unit Normal N(t) | Curvature |
|---|---|---|---|---|---|
| Helix at t = 0 | (1, 0, 0) | (0, 1, 1) | (-1, 0, 0) | (-1, 0, 0) | 0.500000 |
| Unit circle at t = 0 | (1, 0, 0) | (0, 1, 0) | (-1, 0, 0) | (-1, 0, 0) | 1.000000 |
| Parabola r(t) = (t, t², 0), t = 1 | (1, 1, 0) | (1, 2, 0) | (0, 2, 0) | (-0.894427, 0.447214, 0) | 0.178885 |
Formula Used
T(t) = v(t) / |v(t)|
aT = a(t) · T(t)
aparallel = (a(t) · T(t)) T(t)
anormal = a(t) - aparallel
N(t) = anormal / |anormal|
κ = |v(t) × a(t)| / |v(t)|3
R = 1 / κ if κ = 0, the radius is infinite.
This method is ideal when you already know the first and second derivative vectors of a parametric curve at a selected parameter value.
How to Use This Calculator
- Enter a curve label and the parameter value if you want a named result.
- Provide the point coordinates to anchor the graph at the evaluated location.
- Enter velocity components from the first derivative vector r′(t).
- Enter acceleration components from the second derivative vector r″(t).
- Select the number of decimal places for reporting.
- Click the calculate button to show the result above the form.
- Review the tangent, normal, curvature, and radius values.
- Use the CSV or PDF buttons to export your result set.
FAQs
1) What is the principal unit normal vector?
It is the unit vector that points toward the instantaneous turning direction of a curve. It is perpendicular to the unit tangent and indicates where the path bends.
2) When is the principal unit normal undefined?
It becomes undefined when the normal component of acceleration is zero. That usually means the path is locally straight or acceleration is purely tangential at that point.
3) What is the difference between tangent and normal vectors?
The tangent vector gives the current direction of motion along the curve. The principal normal gives the direction in which the curve is turning at that same point.
4) Why are velocity and acceleration used here?
Velocity determines the tangent direction. Acceleration can be split into tangential and normal parts, and the normal part directly reveals the principal normal direction and curvature behavior.
5) Can I use this calculator for 2D curves?
Yes. Just set the z-components to zero. The calculator still works, and the resulting vectors will remain in the xy-plane.
6) What does curvature tell me?
Curvature measures how sharply the curve bends. Larger curvature means a tighter turn. Smaller curvature means the path is flatter and changes direction more gently.
7) Why can the radius of curvature be infinite?
If curvature becomes zero, the local path behaves like a straight line. A straight line has no finite turning radius, so the radius of curvature is infinite.
8) Do all input components need matching units?
Yes. Keep coordinate, velocity, and acceleration components internally consistent. Mixed units can distort curvature, radius, and plotted direction results.