Curvature Result
Step Details
Advanced Calculator
Use functions such as sin(t), cos(t), t^2, exp(t), and sqrt(t).
For a 2D curve, leave z(t) as zero.
Example Data Table
| Curve | r(t) | t | Expected idea |
|---|---|---|---|
| Unit circle | <cos(t), sin(t), 0> | 1 | Curvature is close to 1. |
| Helix | <cos(t), sin(t), t> | 1 | Curvature is close to 0.5. |
| Parabola | <t, t^2, 0> | 1 | Curvature changes with t. |
Generated Sample Values
| t | x(t) | y(t) | z(t) | Curvature | Radius |
|---|---|---|---|---|---|
| Submit the calculator to generate sample values. | |||||
Formula Used
For a space curve r(t) = <x(t), y(t), z(t)>, curvature is found with:
κ(t) = |r′(t) × r″(t)| / |r′(t)|³
Here, r′(t) is the velocity vector. The vector r″(t) is the acceleration vector.
The numerator measures how strongly the path bends away from the tangent direction.
The denominator scales the result by speed cubed.
For a plane curve r(t) = <x(t), y(t)>, the same idea becomes:
κ(t) = |x′(t)y″(t) − y′(t)x″(t)| / (x′(t)² + y′(t)²)3/2
The radius of curvature is R = 1 / κ. A larger curvature means a sharper turn.
A smaller curvature means a flatter path.
How to Use This Calculator
Enter the x(t), y(t), and z(t) components of your vector function. Select 2D when the motion is in a plane. Enter the value of t where you want curvature. Choose the number of decimal places. Press the calculate button. The result appears above the form and below the header.
Use the sample range fields to create a small table of curvature values. This helps compare bending across a path. Download the CSV for spreadsheets. Download the PDF for reports, assignments, or study notes.
Article: Curvature of a Vector Function
What Curvature Means
Curvature describes how quickly a path changes direction. It is not the same as slope. Slope tells how a graph rises or falls. Curvature tells how strongly a moving object turns. In physics, this matters for circular motion, particle paths, rails, roads, and field line studies.
Why r(t) Is Useful
A vector function r(t) gives position at each parameter value. The parameter is often time. The first derivative gives velocity. The second derivative gives acceleration. Curvature connects these two derivatives. It shows how much acceleration bends the path instead of only changing speed.
Reading the Result
A curvature of zero means the path is locally straight. A large value means the curve turns sharply near that point. The radius of curvature is the reciprocal of curvature. A small radius means a tight bend. A large radius means a gentle bend.
Practical Physics Use
Curvature helps study motion along tracks, projectile paths, orbit shapes, and guided motion. Engineers use it to check comfort and safety in paths. Students use it to connect calculus with real motion. This calculator also reports speed and acceleration so the result has physical context.
Symbolic and Numeric Work
Symbolic derivatives are useful when expressions are simple and exact. Numeric derivatives are helpful when expressions are complex. This tool offers both methods. Always check that speed is not zero. Curvature is undefined when the velocity vector has no magnitude.
Better Study Workflow
Try several t values and compare the sample table. Notice where curvature rises or falls. Export the results for later review. Use the formula section to verify each step. This makes the calculator useful for homework, lab notes, and concept practice.
FAQs
1. What is curvature of r(t)?
It measures how fast a vector path changes direction at a given parameter value. Higher curvature means a sharper local turn.
2. Can this calculator handle 3D curves?
Yes. Enter x(t), y(t), and z(t). The calculator uses the cross product formula for space curves.
3. What should I enter for a 2D curve?
Use x(t) and y(t), set z(t) to zero, and choose the 2D curve option. The plane curve formula is then used.
4. Why is speed shown?
Speed is the magnitude of r′(t). Curvature needs speed in the denominator, so zero speed makes curvature undefined.
5. What is radius of curvature?
Radius of curvature is the reciprocal of curvature. It estimates the radius of the circle that best matches the curve locally.
6. Which derivative method should I use?
Use symbolic derivatives for standard formulas. Use numerical derivatives when symbolic parsing fails or the expression is too complicated.
7. What functions can I type?
You can use common functions like sin(t), cos(t), tan(t), sqrt(t), log(t), exp(t), abs(t), and powers such as t^2.
8. Why did I get an undefined result?
The result is undefined when the velocity magnitude is zero or the expression cannot be evaluated at the selected t value.