Calculator Inputs
Choose a method, enter values, and compute the local radius of curvature.
Example Data Table
Sample inputs and outputs for the four supported methods.
| Method | Inputs | Radius | Curvature | Extra Output |
|---|---|---|---|---|
| Explicit | y′ = 0.75, y″ = 0.20 | 9.7656 units | 0.1024 1/units | Diameter = 19.5313 units |
| Parametric | x′ = 3, y′ = 4, x″ = -1, y″ = 2 | 12.5000 units | 0.0800 1/units | Positive turning orientation |
| Chord and Sagitta | c = 8, s = 1 | 8.5000 units | 0.1176 1/units | Central angle = 56.1450° |
| Curvature | κ = 0.08 | 12.5000 units | 0.0800 1/units | Diameter = 25.0000 units |
Formula Used
1) Explicit curve from derivatives
R = ((1 + (y′)^2)^(3/2)) / |y″|
Use this when the first and second derivatives of y = f(x) are known at a specific point. The sign of y″ indicates local convex or concave behavior.
2) Parametric curve from derivatives
R = ((x′² + y′²)^(3/2)) / |x′y″ − y′x″|
Use this for curves defined by a parameter, such as motion paths, splines, or geometric trajectories. It works even when the curve cannot be written neatly as a single explicit function.
3) Chord and sagitta for circular arcs
R = c²/(8s) + s/2
Here, c is the chord length and s is the sagitta. This method is practical when you measure a convex arc directly from a drawing, part, template, or image.
4) Directly from curvature
R = 1 / |κ|
If curvature is already available, the radius follows immediately. Larger curvature means tighter bending. Smaller curvature means a flatter shape and a larger radius.
How to Use This Calculator
FAQs
1) What is convexity radius?
It is the local radius of curvature of a curve. A smaller radius means a tighter bend, while a larger radius means a flatter shape.
2) Why does this calculator offer multiple methods?
Different problems provide different data. Some give derivatives, some give curvature directly, and some only provide geometric measurements like chord length and sagitta.
3) Can the radius become infinite?
Yes. If curvature is zero, the curve is locally flat, so the radius becomes infinite. That usually happens at a straight segment or flat tangent behavior.
4) What is the difference between curvature and radius?
Curvature measures how strongly a curve bends. Radius measures the size of the matching osculating circle. They are inverses in magnitude: R = 1 / |κ|.
5) Does the sign of curvature matter?
Yes. The sign shows turning direction or local convex versus concave behavior, depending on the method. The radius itself is reported as a positive magnitude.
6) When should I use the chord and sagitta method?
Use it when you can measure the arc directly but do not know derivatives. It is especially useful for drafting, inspection, templates, and circular segment work.
7) Is this only for circles?
No. The derivative and parametric methods apply to general smooth curves. Only the chord and sagitta method assumes a circular arc approximation or circular segment.
8) Which units should I use?
Use any consistent length unit. The calculator does not convert units automatically, so all entered geometric values should use the same scale.