Radius of Curvature Summary
This section appears above the form after calculation, as requested.
| Metric | Value |
|---|
Calculator Inputs
The page stays in a single stacked layout, while calculator inputs use 3 columns on large screens, 2 on smaller screens, and 1 on mobile.
Plotly Graph
The graph plots the local curve and, when available, its osculating circle.
Example Data Table
| Method | Inputs | Curvature | Radius of Curvature | Notes |
|---|---|---|---|---|
| Explicit Curve | x=1, y=2, dy/dx=1.5, d²y/dx²=0.8 | 0.1365 | 7.3260 | Useful for standard function analysis. |
| Parametric Curve | x=1, y=1, x'=3, y'=4, x''=-2, y''=1 | 0.0880 | 11.3636 | Works well for trajectories and paths. |
| Polar Curve | θ=45°, r=5, dr/dθ=1, d²r/dθ²=0.5 | 0.1848 | 5.4112 | Ideal for polar geometry problems. |
| Arc Geometry | Chord=8, Sagitta=2 | 0.2000 | 5.0000 | Fast solution for circular arc estimation. |
Formula Used
1) Explicit Curve y(x)
For a point on a curve written as y = f(x):
R = [(1 + (dy/dx)²)^(3/2)] / |d²y/dx²|Curvature is κ = 1 / R. If d²y/dx² = 0, the point is locally straight and R becomes infinite.
2) Parametric Curve x(t), y(t)
For a curve defined using a parameter t:
R = [(x'² + y'²)^(3/2)] / |x' y'' - y' x''|This form is common in mechanics, motion design, and vector geometry.
3) Polar Curve r(θ)
For a polar curve with derivatives taken with respect to θ:
R = [(r² + (dr/dθ)²)^(3/2)] / |r² + 2(dr/dθ)² - r(d²r/dθ²)|This mode helps with spiral, radial, and angle-based curve analysis.
4) Arc Geometry from Chord and Sagitta
For a circular arc with chord c and sagitta s:
R = c² / (8s) + s / 2Curvature is κ = 1 / R. This is practical when the arc comes from measured geometry.
How to Use This Calculator
- Choose the calculation method that matches your available data.
- Enter the required values for the selected method.
- Set decimal precision and an optional curve label for exported files.
- Press Calculate Radius of Curvature to show the result above the form.
- Review radius, curvature, diameter, center data, and interpretation.
- Use the Plotly graph to inspect local bending and the osculating circle.
- Download the current result as CSV or PDF when needed.
- Reset the form at any time to restore the default sample values.
Frequently Asked Questions
What is radius of curvature?
Radius of curvature is the radius of the osculating circle at a chosen point on a curve. A smaller value means the curve bends more sharply there.
What is the difference between curvature and radius of curvature?
Curvature measures bending directly, while radius of curvature is its reciprocal. If curvature increases, the radius of curvature decreases for the same point.
Which units does this calculator return?
The radius and diameter use the same length units as your coordinates, chord, or sagitta inputs. Curvature is returned in inverse length units.
When does the radius become infinite?
If curvature is zero at the selected point, the curve is locally straight there. In that case, the radius of curvature becomes infinite.
Can I use this for parametric motion paths?
Yes. The parametric mode uses first and second derivatives of x and y, making it useful for motion paths, trajectories, and vector-defined curves.
How does polar mode work?
Polar mode uses r, dr/dθ, and d²r/dθ² at a chosen angle. It converts local behavior into curvature and radius using the standard polar formula.
What do chord and sagitta represent?
The chord is the straight distance between two arc endpoints. The sagitta is the arc height measured from the chord midpoint to the arc.
Why does the graph show a circle?
The graph displays the local curve and its osculating circle when possible. That circle best matches the curve's bending at the chosen point.