Torsion

Differential Geometry · Axiom Academy

Understanding how curves twist out of the osculating plane in three-dimensional space The osculating plane at any point on a curve is the plane that "best fits" the curve at that point - it contains both the tangent vector T and the principal normal vector N . If a curve stays within this plane, it has zero torsion. When the curve leaves this plane, we have non-zero torsion. For a parametric curve r (t) in three-dimensional space, the torsion is given by: r' is the first derivative (velocity vector) r'' is the second derivative (acceleration vector) r''' is the third derivative (jerk vector) r' × r'' is the cross product (binormal direction) The dot product (r' × r'') · r''' gives the signed measure of twisting 3. Planar Curves Have Zero Torsion A fundamental property: a curve lies entirely in a plane if and only if its torsion is zero everywhere . Why? If a curve stays in a single plane, the osculating plane never rotates - it's always the same plane. Thus, there's no twisting out of the plane, and torsion is zero. Examples of planar curves with τ = 0: Any 2D curve (even if embedded in 3D space) 4. The Sign of Torsion: Right-Handed vs Left-Handed Twisting Unlike curvature (which is always non-negative), torsion can be positive or negative. The sign tells us the direction of twisting:

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