Frenet–Serret frame
Explore curvature, signed torsion and the tangent, normal and binormal of six space curves.
About this tool
Choose one of six dimensionless curves. Radius applies to helix/circle, rise a only to the helix. Valid edits update directly. Play moves the selected point and frame; nominal traversal takes8 seconds and then restarts, including on open curves. Settings contains exact parameter u, its slider, projection, curvature/torsion, coordinates, vector components and legend.
The 3D view is orthographic. Drag with a mouse to rotate freely, use arrow keys with the view focused, or Home to reset its orientation/zoom. Ctrl/⌘+wheel zooms around the centre; normal wheel scrolls the page. On touch, horizontal movement rotates and vertical movement scrolls. xy/xz/yz presets give exact plane projections. A shortened projected arrow is still a unit vector in space. All arrows share the same world scale; rotations/projections do not change the model.
For a regular curve r(u), T=r′/|r′|, B=(r′×r″)/|r′×r″|, N=B×T. Curvature κ=|r′×r″|/|r′|³ and signed torsion τ=((r′×r″)·r‴)/|r′×r″|². These are analytic derivatives. The displayed blue curve is sampled at401 positions; the selected frame is evaluated separately at the exact u.
Helix r=(Rcosu,Rsinu,au), R0.1…10, a−2…2, u0…2π: κ=R/(R²+a²), τ=a/(R²+a²). The circle is a=0. Trefoil r=(sinu+2sin2u,cosu−2cos2u,−sin3u), u0…2π. The straight line r=(u,0,0), cubic inflection r=(u,u³,0) and cusp r=(u²,u³,0) use u−1…1.
At zero curvature, T exists but classical N, B and torsion do not. At the stationary cusp u=0 the complete frame is undefined. Ill-conditioned tiny values are marked; missing vectors are hidden instead of inventing a replacement frame. T×N=B where the regular frame exists. Changing helix rise changes the sign of torsion.
A fixed conservative bounding sphere includes the entire curve and all unit-arrow endpoints over its domain, leaving some margin; zooming can clip the view. The drawing is wireframe, without hidden-line removal. PNG exports the current view and numerical parameters. Raw source/parameter drafts persist, including unfinished or inactive inputs. Computed results, camera/zoom and animated positions are not stored. Navigation/reload recalculates at the last manual u in the default view. Reset restores the default helix. Hidden retained views freeze and resume only existing playback intent.