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Topology and differential geometry

Miklós Hoffmann

Kempelen Farkas Hallgatói Információs Központ

Evolvent, evolute

Evolvent, evolute

Given a curve, now we study an important family of curve generated by the given curve, and an envelope of great importance.

Definition 7.3. Given a planar curve consider its tangent lines. The curve, which intersects all the tangent lines orthogonally, is called the evolvent (or involute) of the given curve.

As it follows from the definition, there are infinitely many evolvents of a given curve. These evolvents form a one-parameter family of curves, where the family-parameter reflects the point of the original curve, from which the actual evolvent starts.

Those curves, which intersect orthogonally all the elements of a family of curves are called orthogonal trajectories. Thus each evolvent of a curve is an orthogonal trajectory of the tangent lines (as family of curves). Given a curve, the evolvent can be constructed by attaching an imaginary taut string to the given curve and tracing its free end as if it unwinds. This construction can be expressed by the following equation of the evolvent.

Theorem 7.4. Given a curve parameterized by arc-length, then the evolvent starting from the curve point can be expressed as

Proof. We have to prove, that the tangent of the curve is orthogonal to the corresponding tangent of the curve in each point.

But in arc-length parameterization , from which , that is . □

Figure 7.2. The evolvent as the envelope of the normals

For a given curve another important curve can be defined as the locus of all its centers of curvature.

Definition 7.5. Given a curve and its curvature function then the curve

is called evolute of the given curve (see Figure 7.3 and the next video).

Figure 7.3. Evolute of the ellipse. Cusps of the evolute can be constructed by the method we have seen at the osculating curves

  V I D E O  

Theorem 7.6. Given a curve with non-vanishing curvature and derivative of curvature, the evolute is the envelope of the normal lines (parallel lines to the vector ) of the curve.

Proof. Since all the centers of curvature are on the normals of the given curve, it is obvious, that each member of the family of lines has common point to the evolute. For being an envelope, the evolute must have parallel tangent lines to the normals. But the direction of the normals is the actual vector , while the direction of the tangent of the evolute is

which proves the statement. □

Furthermore it is true, the evolvent of the evolute of the given curve is nothing else than the originally given curve .