The formula this page uses
N(t) = T′(t) / |T′(t)| κ = |T′(t)| / |r′(t)| radius of curvature = 1/κ
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Unit Normal Vector Calculator
Solve unit normal vector problems with clear steps, notation, and a final check.
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Unit Normal Vector Calculator explained
The short version
- The principal unit normal N(t) points from the curve toward the inside of its bend.
- You get it by differentiating the unit tangent T(t) and then normalizing that derivative.
- The same calculation hands you the curvature, since κ = |T′(t)| / |r′(t)|.
What each part means
SymbolWhat it means
T(t) — Unit tangentDirection of travel, length 1, computed first.
T′(t) — Derivative of the tangentMeasures how fast the direction turns. Always perpendicular to T.
N(t) — Principal unit normalT′ divided by its own length. Points toward the centre of the bend.
κ — CurvatureHow sharply the curve bends, in units of 1 over length. A straight line has κ = 0.
Show your work: a full example
- The curve, a circle of radius 3r(t) = (3 cos t, 3 sin t)
- Velocity and speedr′(t) = (−3 sin t, 3 cos t), and |r′(t)| = 3
- Unit tangentT(t) = (−sin t, cos t)
- Differentiate TT′(t) = (−cos t, −sin t)
- Its length|T′(t)| = √(cos²t + sin²t) = 1
- Divide to get NN(t) = (−cos t, −sin t)
- Check at t = 0the point is (3, 0) and N(0) = (−1, 0), pointing straight back at the centre, and κ = 1/3 which is 1 over the radius
A second, different case
- A different case: a helix, where the curve also risesr(t) = (3 cos t, 3 sin t, 4t)
- Unit tangent, from |r′| = 5T(t) = (−0.6 sin t, 0.6 cos t, 0.8)
- Differentiate TT′(t) = (−0.6 cos t, −0.6 sin t, 0)
- Its length|T′(t)| = 0.6√(cos²t + sin²t) = 0.6
- DivideN(t) = (−cos t, −sin t, 0), which has no z part at all
- Curvatureκ = 0.6 ÷ 5 = 0.12
- Radius of curvature1 ÷ 0.12 ≈ 8.33, larger than the helix's own radius of 3 because the climb stretches the bend out
Copy-ready example
r(t) = (3 cos t, 3 sin t)
The curve, a circle of radius 3
Questions about the Unit Normal Vector Calculator
Why is N always perpendicular to T?
Because T has constant length 1, so T · T = 1 always. Differentiating both sides gives 2 T · T′ = 0, which forces T′ and therefore N to be perpendicular to T. It is a consequence of the length never changing.
What does the curvature number actually mean?
It is 1 over the radius of the circle that hugs the curve best at that point. A circle of radius 3 has κ = 1/3 everywhere, and the helix's κ = 0.12 means it bends like a circle of radius 8.33.
Why does a straight line have no unit normal?
Its tangent never changes, so T′ is the zero vector and the division has nothing to divide by. That matches κ = 0: with no bending there is no inside of a bend to point at.
What is B, the third frame vector?
The binormal, B = T × N. It is perpendicular to both and completes a right-handed set that travels with the point. For the helix, B tilts out of the horizontal plane, and its turning rate is called torsion.