Analyze torsion from first, second, and third derivative vectors. See determinants, norms, and curve behavior. Save outputs, sample tables, and graphs for later review.
Enter the first, second, and third derivative vectors at the same parameter value.
This sample uses a helix-like derivative set at one parameter value. It gives a clean nonzero torsion result.
| Case | x′ | y′ | z′ | x″ | y″ | z″ | x‴ | y‴ | z‴ | τ |
|---|---|---|---|---|---|---|---|---|---|---|
| Helix at t = 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | -1 | 0 | 0.5 |
Let r′(t), r″(t), and r‴(t) be the first three derivative vectors of a space curve.
Cross vector: r′ × r″
Scalar triple product: (r′ × r″) · r‴
Torsion: τ = ((r′ × r″) · r‴) / |r′ × r″|²
Curvature: κ = |r′ × r″| / |r′|³
Unit binormal: B = (r′ × r″) / |r′ × r″|
This calculator also reports the cross vector, determinant value, vector magnitudes, and the local binormal direction.
This method is useful for helixes, particle paths, parametric curves, and geometry problems involving local twisting in three dimensions.
Vector torsion measures how a space curve twists out of its osculating plane. Curvature shows bending. Torsion shows twisting. Both are needed for a full local description of three-dimensional motion.
In mechanics, a trajectory can bend and twist at the same time. The first derivative gives direction and speed scaling. The second derivative changes direction. The third derivative helps detect how the curve leaves its current bending plane.
The scalar triple product is central here. It compares the plane formed by the first two derivatives with the third derivative. A nonzero value indicates genuine spatial twisting. A zero value suggests no local twist at that point.
The denominator uses the squared magnitude of r′ × r″. This stabilizes the torsion formula around the local osculating plane. When that cross product becomes zero, torsion is not defined because the curve is locally degenerate or straight for that evaluation step.
This page is useful for advanced physics students, applied mathematicians, and anyone studying differential geometry, particle motion, or three-dimensional parametric curves.
Torsion measures how a curve twists in three dimensions. It tells you how rapidly the osculating plane rotates as you move along the curve.
The torsion formula depends on r′, r″, and r‴. The first two define the bending plane, and the third tests how the curve twists away from that plane.
Torsion becomes undefined. The curve is locally straight or degenerate there, so the osculating plane cannot be established for a valid torsion calculation.
No. Torsion here is a geometric property of a curve. Torque is a physical moment related to force and rotation.
Yes. The sign depends on orientation. Positive and negative values indicate opposite local twisting directions under the chosen parameterization and coordinate order.
Curvature and torsion work together. Curvature explains bending in the local plane, while torsion explains twisting out of that plane.
Yes. A helix is a classic torsion example. It has constant nonzero curvature and constant nonzero torsion under a standard parameterization.
Torsion typically has inverse length units when the parameter is arc length. Otherwise, unit interpretation depends on your curve parameterization.
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.