Parametric Curves in ℝ³

Plot γ(t), animate a moving point, and attach velocity, acceleration, and Frenet-frame vectors.

Curve standard math syntax

Example

γ₁(t) =

γ₂(t) =

γ₃(t) =

t min

t max

Samples

Supported: sin, cos, tan, exp, log, sqrt, abs, pi, e, powers with ^, and implicit products such as 2t.

Motionpaused

Position on parameter interval

Animation speed

1.0×

Moving vectors3 selected

Arrow display size

1.00×

Normalize each arrow Best for comparing directions across quantities with different units.

Current pointregular

t 0

speed |v| 0

γ(t) (0, 0, 0)

curvature κ —

torsion τ —

The Frenet normal/binormal is undefined or numerically unstable here because speed or curvature is near zero.

Definitions

v = γ′, 𝔱 = v/|v|, a = γ″.

a^⊥ = a − (a·𝔱)𝔱, 𝔫 = a^⊥/|a^⊥|, 𝔟 = 𝔱 × 𝔫.

Primes on 𝔱, 𝔫, 𝔟 mean derivatives with respect to the input parameter t, not arc length. Derivatives are computed numerically from the sampled curve.

Current curve

γ(t) = (cos(t), sin(t), 0.18t)

t = 0.000

Drag to rotate · wheel/pinch to zoom · double-click to reset view

Perspective view