∑ Mathematics LabParametric Curves · v4.0.4

MATHEMATICS LAB · GEOMETRY

Parametric Curves in 3D Space

Explore helices and spatial curves, observe the Frenet frame, and learn how to calculate arc length, curvature and torsion.

Coordinates use arbitrary units (u). The parameter t is dimensionless; for trigonometric curves it represents an angle in radians. Curvature and torsion have units u⁻¹. The Frenet frame is undefined where curvature vanishes.

2. Interactive 3D construction

Rotatable orthographic view
Circular helix and Frenet frame.
The graphic requires Canvas 2D. Results remain available below.

Drag the graph to rotate the view; use the mouse wheel or + and − to zoom. Select “Animate P” to move the point along the curve, or enter a value of t on the left.

Curve T tangent N principal normal B binormal Osculating plane

3. How to read the construction

4. Results

Live updates

5. Step-by-step calculations

Derivatives, curvature, torsion

What you can study

Curves in space

Parametrizations, regularity, moving points, circles, helices and closed curves.

Differential geometry

Tangent vectors, principal normals, binormals, osculating planes, curvature and torsion.

Arc length

Parametric speed, distance along a curve and osculating circles.

Curvature and torsion are geometric invariants of regular curves under arc-length reparametrization; the chosen orientation affects the sign of torsion. The curve is rendered using a finite sample of points.