Analyze meridian and circumferential curvature from revolving profiles. Input derivatives to compare both principal curvatures. Export results, review formulas, and learn every calculation step.
Use the profile curve form X(t, θ) = (r(t)cosθ, r(t)sinθ, z(t)). Enter local values at the selected point t0.
| Surface Example | r(t0) | r'(t0) | r''(t0) | z'(t0) | z''(t0) | k1 | k2 |
|---|---|---|---|---|---|---|---|
| Paraboloid at t0 = 1 with r=t and z=t² | 1 | 1 | 0 | 2 | 2 | 0.17888544 | 0.89442719 |
| Cylinder with r=3 and z=t | 3 | 0 | 0 | 1 | 0 | 0.00000000 | 0.33333333 |
| Cone with r=t and z=2t at t0 = 2 | 2 | 1 | 0 | 2 | 0 | 0.00000000 | 0.44721360 |
For a surface of revolution parameterized by X(t, θ) = (r(t)cosθ, r(t)sinθ, z(t)):
E = r'(t)2 + z'(t)2
k1 = (r'(t)z''(t) - z'(t)r''(t)) / (E)3/2
k2 = z'(t) / (r(t)√E)
H = (k1 + k2) / 2
K = k1 × k2
The sign depends on the chosen surface normal. The calculator follows one standard orientation. Magnitudes still show bending strength clearly.
This principal curvatures of a surface of revolution calculator helps students, engineers, and geometry learners evaluate local bending on rotational surfaces. A surface of revolution is created when a profile curve spins around an axis. At each point, the surface bends in two independent principal directions. One direction follows the meridian curve. The other follows the circular parallel. These bend measures are the principal curvatures. They describe how sharply the surface turns near a chosen point.
Understanding principal curvature supports differential geometry, CAD modeling, mechanical design, optics, and shape analysis. It helps compare convex, flat, and saddle-like regions. It also supports Gaussian curvature and mean curvature calculations. Those values are useful in surface classification and advanced mathematical modeling. When the profile derivatives are known, local curvature can be computed directly. This calculator reduces repeated hand work and lets you test many profiles quickly.
Enter the radius value r at the selected parameter, then add first and second derivatives for the profile radius and height. The tool uses these local quantities to compute meridian curvature and parallel curvature. It also returns mean curvature, Gaussian curvature, speed term, and curvature radii. Signs depend on the chosen normal direction. Magnitudes show bending strength. A zero curvature indicates a locally straight direction on the surface.
Use this calculator for paraboloids, cylinders, cones, spheres, and custom rotational shapes. It is helpful for coursework, exam preparation, numerical checks, and design review. You can export results for reports, save values as CSV, and create a PDF snapshot for documentation. The example table shows typical inputs and outputs. Review the formulas section to verify each step. Then use the guide section to enter reliable derivatives and interpret the final curvature values with confidence.
To get stable results, use derivatives from a consistent parameterization. Keep the same axis definition throughout the problem. The radius must stay positive away from the axis. If the speed term becomes very small, curvature values can change rapidly and numerical error may increase. For symbolic work, simplify the profile first. For numeric work, verify derivatives carefully. Good inputs produce trustworthy curvature estimates and clearer geometric insight. This saves time during assignments and design validation.
A surface of revolution is formed by rotating a planar profile curve around a fixed axis. Common examples include spheres, cones, cylinders, and paraboloids.
They measure local bending in two perpendicular surface directions. For rotational surfaces, one direction follows the meridian and the other follows the circular parallel.
The sign depends on the orientation of the surface normal. Reversing the normal flips curvature signs, while the absolute magnitudes remain the same.
Yes. Arc length often simplifies the formulas because the speed term becomes one. This calculator also works with any regular parameterization.
The parallel curvature formula divides by r(t0). At the axis, that local expression breaks down, so you should use a limiting argument or another formulation.
A zero principal curvature means the surface is locally straight in that principal direction. Cylinders are a classic example because one principal curvature is zero.
Yes. It works for many rotational surfaces if you provide the local radius and derivative values correctly. The example table includes cone and cylinder cases.
CSV files are useful for spreadsheets and repeated analysis. PDF files help with reports, homework submission, and quick documentation of computed curvature values.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.