Calculated Results
Results appear above the form after submission.
Position r(t)
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Velocity r′(t)
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Acceleration r″(t)
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Speed |r′(t)|
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Unit Tangent T
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Unit Normal N
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Unit Binormal B
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Curvature κ
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T · N
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B = T × N
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Calculator Inputs
Enter a space curve using t as the variable. Examples: cos(t), sin(t), t, t^2.
Example Data Table
This sample uses the helix r(t) = (cos t, sin t, t).
| Example Curve | Point t | r(t) | Expected T Trend | Expected B Trend |
|---|---|---|---|---|
| (cos t, sin t, t) | 1.00 | Rotating helix point | Tangent follows spiral direction | Binormal stays orthogonal to frame |
| (t, t^2, t^3) | 0.50 | Polynomial space curve | Tangent changes with growth rate | Binormal reflects twisting behavior |
| (3*cos(t), 2*sin(t), t/2) | 2.00 | Scaled elliptical helix | Tangent depends on unequal axes | Binormal varies with asymmetry |
Formula Used
r(t) = <x(t), y(t), z(t)>
r′(t) = <x′(t), y′(t), z′(t)>
T(t) = r′(t) / |r′(t)|
N(t) = T′(t) / |T′(t)|
B(t) = T(t) × N(t)
κ(t) = |r′(t) × r″(t)| / |r′(t)|³
This calculator evaluates derivatives numerically using a centered difference method. That approach works well for many smooth curves entered as standard expressions.
How to Use This Calculator
- Enter expressions for x(t), y(t), and z(t).
- Choose the point t where vectors should be evaluated.
- Set the plot interval and sample count.
- Adjust the derivative step if a curve is sensitive.
- Click Calculate Vectors to generate results.
- Review position, velocity, acceleration, T, N, B, and curvature.
- Inspect the 3D graph to understand local geometry.
- Export the output as CSV or PDF when needed.
Frequently Asked Questions
1) What does the unit tangent vector represent?
The unit tangent vector points in the instantaneous direction of motion along the curve. It is found by normalizing the velocity vector, so its magnitude becomes one.
2) Why can the normal vector fail to appear?
The unit normal vector depends on the derivative of the tangent vector. If the curve is locally straight or numerical change is too small, the normal direction can become undefined.
3) What is the binormal vector used for?
The binormal vector completes the orthonormal Frenet frame. It is useful in curve geometry, motion analysis, 3D path design, and studying twisting behavior in space.
4) Why does the calculator ask for a derivative step?
This tool uses numerical differentiation. The step size controls accuracy and stability. Smaller values may improve precision, but values that are too small can amplify rounding noise.
5) What kinds of expressions can I enter?
You can enter standard mathematical expressions using the variable t. Common functions include sin, cos, tan, exp, sqrt, and powers such as t^2.
6) What does curvature measure?
Curvature measures how quickly the direction of the curve changes. Larger curvature means stronger bending, while smaller curvature suggests a flatter local path.
7) Can this calculator handle helices and polynomial curves?
Yes. It works well for many smooth parametric curves, including helices, polynomial curves, scaled spirals, and other common space-curve examples used in mathematics courses.
8) Why are T, N, and B expected to be perpendicular?
For a regular smooth curve, the Frenet frame forms an orthonormal basis. That means the unit tangent, unit normal, and unit binormal are mutually perpendicular and each has unit length.