Analyze space curves with tangent-based vector geometry tools. Calculate unit normal and binormal components quickly. Verify orthogonality, curvature behavior, and directional change with confidence.
| Curve | t₀ | Point r(t₀) | r′(t₀) | r″(t₀) | T | N | B | κ |
|---|---|---|---|---|---|---|---|---|
| r(t) = (t, t², t³) | 1 | (1.000000, 1.000000, 1.000000) | (1.000000, 2.000000, 3.000000) | (0.000000, 2.000000, 6.000000) | (0.267261, 0.534522, 0.801784) | (-0.674453, -0.490511, 0.551825) | (0.688247, -0.688247, 0.229416) | 0.166424 |
Unit tangent: T = r′(t) / |r′(t)|
Unit binormal: B = (r′(t) × r″(t)) / |r′(t) × r″(t)|
Unit normal: N = B × T
Curvature: κ = |r′(t) × r″(t)| / |r′(t)|³
Normal plane: T · [(x, y, z) - r(t₀)] = 0
Osculating plane: B · [(x, y, z) - r(t₀)] = 0
Rectifying plane: N · [(x, y, z) - r(t₀)] = 0
Normal and binormal vectors describe how a space curve turns. They are core parts of the Frenet frame. This calculator helps students, teachers, and analysts study three-dimensional motion with less manual work. You enter a point, the first derivative, and the second derivative. The tool then computes the unit tangent vector, principal normal vector, unit binormal vector, and curvature. It also builds related plane equations. This makes the page useful for geometry lessons, multivariable calculus practice, and applied modelling.
The first derivative gives the direction of motion. Its magnitude gives speed along the curve parameter. The cross product of the first and second derivatives measures directional twisting away from a straight path. When that cross product is nonzero, the curve has a defined osculating plane. The unit binormal vector comes from normalizing that cross product. The principal normal vector is then found from the binormal and tangent vectors. Because these vectors are mutually perpendicular, the calculator also helps verify orthogonality.
The output is practical. The tangent vector points along the curve. The normal vector points toward the local turning direction. The binormal vector is perpendicular to the osculating plane. Curvature shows how sharply the curve bends at the chosen parameter value. Larger curvature means tighter turning. Smaller curvature means flatter motion. The normal plane, osculating plane, and rectifying plane give extra geometric context. These equations support sketching, proof work, and conceptual review. They also help compare theoretical results with software output and classroom notes.
This page is useful when checking homework, building examples, or reviewing curve behavior before an exam. It saves time and reduces algebra mistakes. It also shows whether the curve point is regular and whether the normal and binormal directions are defined. If the speed is zero, the tangent cannot be formed. If the cross product is zero, the curve is locally straight, so the normal and binormal are not uniquely determined. The result table and exports make revision easier, and the example dataset gives a fast starting point for practice. Students can also test many curve samples quickly. That strengthens intuition and improves geometric interpretation. This supports cleaner study sessions, tutoring examples, and more confident verification.
The principal normal vector points toward the local turning direction of the curve. It shows where the path bends at the selected point. It is always perpendicular to the tangent when the curve is regular and curved.
The binormal vector is perpendicular to both the tangent and the normal vectors. It defines the orientation of the osculating plane. In three-dimensional curve analysis, it helps describe spatial twisting and local frame direction.
The first derivative gives direction and speed information. The second derivative helps measure how that direction changes. Together, they allow the calculator to form the Frenet vectors and compute curvature at the chosen parameter value.
They are undefined when the first derivative is zero or when the cross product of the first and second derivatives is zero. In those cases, the curve point is not suitable for a unique Frenet frame.
Curvature measures how sharply the curve bends at the selected point. A larger curvature means stronger turning. A smaller curvature means the curve is flatter there. The tool uses the standard space-curve curvature formula.
Yes. Evaluate the curve, first derivative, and second derivative at your chosen parameter first. Then enter those numerical values here. This is useful when checking textbook examples, homework, and worked lecture problems.
The normal, osculating, and rectifying planes give extra geometric meaning. They show how the local curve frame sits in space. These equations are helpful for sketches, proofs, and understanding the relationship between T, N, and B.
Enter the parameter at which your derivatives were evaluated, such as t = 1 or u = 0.5. It appears in the result summary. The actual calculations use the coordinate and derivative values you provide.
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.