Enter Parametric Vector Function
Use t as the variable. Supported functions include
sin, cos, tan, sqrt,
abs, exp, ln, log10,
pow, min, and max.
Use * for multiplication.
Formula Used
For a parametric position vector
r(t) = <x(t), y(t), z(t)>,
curvature is calculated with this formula:
κ(t) = || r′(t) × r″(t) || / || r′(t) ||³
The radius of curvature is ρ(t) = 1 / κ(t). Speed is ||r′(t)||. Tangential acceleration is (r′ · r″) / ||r′||. Normal acceleration is κ||r′||².
This calculator uses central difference approximations. It estimates the first and second derivatives from nearby points. A smaller step may increase accuracy. Very small steps can increase round-off error.
How to Use This Calculator
- Enter the vector components x(t), y(t), and z(t).
- Use
tas the only variable. - Leave z(t) blank when the curve is in a plane.
- Enter the point where curvature should be evaluated.
- Set a derivative step. The default works for most smooth curves.
- Choose a graph range and sample count.
- Press calculate. Results appear above the form.
- Use the CSV or PDF button to save the output.
Example Data Table
| Curve | x(t) | y(t) | z(t) | t | Expected behavior |
|---|---|---|---|---|---|
| Unit circle | cos(t) | sin(t) | 0 | 1 | Curvature stays near 1. |
| Helix | cos(t) | sin(t) | t/4 | 1 | Curvature is steady and below 1. |
| Parabola | t | t^2 | 0 | 0 | Curvature is highest near the vertex. |
| Space curve | t | t^2 | t^3 | 1 | Curvature changes with position. |
Understanding Curvature of r(t)
What Curvature Means
Curvature measures how fast a curve changes direction. A straight line has zero curvature. A tight bend has high curvature. In physics, this idea helps describe particle motion. The vector r(t) gives the position of a particle at time t. Its first derivative gives velocity. Its second derivative gives acceleration. Curvature combines both derivatives to measure turning.
Why It Matters in Physics
A moving object may travel quickly and still turn gently. Another object may move slowly and turn sharply. Curvature separates turning from speed. This makes it useful in mechanics, robotics, orbital paths, and track design. It also helps estimate normal acceleration. Normal acceleration points toward the bending side of the path.
How the Calculator Works
The calculator accepts two-dimensional and three-dimensional vector functions. It evaluates the position at your chosen t value. Then it estimates velocity and acceleration with nearby t values. After that, it computes the cross product of velocity and acceleration. The magnitude of that cross product is divided by the cube of speed. The final value is curvature.
Reading the Output
The curvature value shows bending strength. The radius of curvature is its reciprocal. A large radius means a broad turn. A small radius means a sharp turn. The speed value shows how fast the parameterized point moves along the curve. Tangential acceleration shows speed change. Normal acceleration shows direction change.
Practical Tips
Use clear multiplication signs, such as 3*t. Use radians for trigonometric functions. Choose a graph range that covers the important motion. If the curve has sharp changes, increase sample count. If results look unstable, adjust the derivative step. Avoid points where speed becomes zero. At those points, curvature may be undefined.
FAQs
1. What is curvature of r(t)?
It is a measure of how sharply a parametric curve bends at a given t value. Higher curvature means a tighter turn. Lower curvature means a flatter path.
2. Can I use this for 2D curves?
Yes. Enter x(t) and y(t), then leave z(t) blank or enter 0. The same cross product formula works by treating the curve as a space curve with z equal to zero.
3. Which angle unit is used?
Trigonometric functions use radians. For example, pi represents 180 degrees. You can type pi directly inside expressions, such as cos(pi*t).
4. Why is speed important?
Curvature uses the magnitude of velocity in the denominator. If speed is zero or extremely small, curvature can become undefined or numerically unstable.
5. What does radius of curvature mean?
It is the reciprocal of curvature. A small radius indicates a sharp bend. A large radius indicates a wide and gentle turn.
6. Why are my results unstable?
The derivative step may be too large or too small. Try a value between 0.001 and 0.00001. Also check whether your curve has a cusp or zero speed.
7. Can I download the results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a formatted report with key results and sample rows.
8. Which functions are supported?
You can use common functions like sin, cos, tan, sqrt, abs, exp, ln, log10, pow, min, and max. Always write multiplication with an asterisk.