Triedro de Frenet

Inspect a moving frame

Choose one of the six curves and calculate. Radius R is used by the helix and circle; rise a is used only by the helix. The exact parameter field and slider select a point on the prepared curve. Changing the view changes only its projection. Coordinates and model parameters are dimensionless; u is a curve parameter, not generally arc length.

Curves and domains

For 0 ≤ u ≤ 2π: helix r = (R cos u, R sin u, a u); circle r = (R cos u, R sin u, 0); trefoil r = (sin u + 2 sin 2u, cos u − 2 cos 2u, −sin 3u). For −1 ≤ u ≤ 1: line r = (u, 0, 0); cubic inflection r = (u, u³, 0); cusp r = (u², u³, 0). A helix rises by 2πa in one turn, including negative a. Circle and trefoil close exactly; the helix keeps its full height at 2π.

Curvature and torsion

All derivatives with respect to u are analytic. T = r′/|r′|, B = (r′ × r″)/|r′ × r″| and N = B × T. The frame is right-handed: B = T × N. Curvature is κ = |r′ × r″|/|r′|³ and is nonnegative. Torsion is τ = ((r′ × r″) · r‴)/|r′ × r″|² and retains its sign, with the convention dB/ds = −τN.

For a helix, κ = R/(R² + a²) and τ = a/(R² + a²). With R = 1 and a = 0.25, these are 16/17 and 4/17. Changing a to −0.25 preserves curvature and changes torsion to −4/17. The unit circle has κ = 1 and τ = 0.

Undefined frames

On the straight line and at u = 0 on the cubic inflection, κ = 0 and T exists, but N, B and τ are undefined. At the cusp with u = 0, r′ = 0, so T and κ are also undefined. A dash means undefined, never zero. Extremely ill-conditioned numerical cases are labelled separately; only values that remain computable and finite are retained. No substitute normal frame is invented.

Reading the projection

The drawing joins 401 curve points into 400 straight segments; the exact probe is evaluated independently. T, N and B each have model length 1, although projection can make a vector look shorter or reduce it to a point. All four views use the same spatial scale and include the complete curve and its unit arrows. The xyz key shows orientation. This is a transparent wire drawing; crossing lines do not indicate which is hidden behind another.

The oblique view uses azimuth 45° and elevation 25°. Other views look onto the xy, xz or yz plane. Invalid probe input hides the point and frame while retaining the curve. Changing curve parameters clears the result; clearing the result retains inputs. Navigation preserves the current result and view. Reloading or changing language restores valid settings and waits for Calculate.

Reference

Michael Taylor, Multivariable Calculus, §3.4: Frenet frame conventions, curvature, torsion and the helix.