The curvature, tangent, and normal for a circle in two dimensions:

The curvature, torsion, and associated basis for a helix expressed in cylindrical coordinates:

A straight line is degenerate, and so the basis is padded with zero vectors:

Curvature, torsion, and associated basis of a spiral restricted to a plane:

Construct the unit normal and tangent to the parabola :

The Frenet–Serret system specifying metric, coordinate system, and parameters:

A curve with nonzero tangent, normal, binormal, and trinormal:

FrenetSerretSystem works in curved spaces:

Determine if two space curves intersect in a plane by checking the equality of their binormals:

Two curves in Euclidean space can be overlaid by rigid motion if and only if their curvatures are equal as functions of arc length:

As parameterized, the curves appear to have different curvatures:

But re-expressing the curvatures in terms of arc length shows that the curves are related by a rigid motion:

Construct an osculating circle, which is the circle that best approximates the curve at a point:

The radius of the osculating circle is inverse to the curvature:

The center of the circle lies along the normal to the curve at the contact point:

Plot the curve with two osculating circles and points of contact:

In dimension , the first curvatures are always non-negative, but the last can be negative:

In two dimensions, the curvature is signed:

In dimensions three and higher, the ArcCurvature is the first generalized curvature:

In dimension two, the ArcCurvature is the absolute value of the single generalized curvature:

In two dimensions, the normal is always rotated counterclockwise relative to the tangent:

When the curve is embedded in three-space, the normal can be rotated in either direction:

The change in direction happens when the curvature crosses zero:

The normal and binormal are undefined at the crossing, then reverse direction:

In Euclidean space, if only the last curvature is identically 0, the curve lies in a hyperplane:

That hyperplane is perpendicular to the last basis vector, in this case the binormal:

Plot the plane containing the curve, using a point on the curve and the tangent and normal at that point:

The curve, hyperplane, and binormal:

For a curve embedded in nonplanar surface, no basis vector other than the tangent needs to maintain a constant angle to the surface:

The plot of a function with the unit tangent and normal is shown. Click a point to see the unit and tangent at that point:

The trefoil knot with an animation of the tangent, normal, and binormal moves along the curve: