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DIFFERENTIAL GEOMETRY COURSE

7. *CURVES OF BERTRAND*

*Definition 67:* If a curve __x__(t) has the property that its principal normals are also principal normals of an other curve __y__(t), we call it a curve of Bertrand.
Then both curves together form a pair of Bertrand curves.

*Proposition 68:* The curve __x__(t) is a curve of Bertrand if and only if it satisfies a natural equation of the form aτ+bκ=c (a,b,c≠0) or τ=0
(planar curve with an infinite number of partners) or κ=c (then there is an infinite number of partners if τ is also constant; in the other case, __y__(t) exists of the curvature
centers of __x__(t)).

*Proff:* We must have __y__(s)=__x__(s)+λ(s)__n__(s) (s is arc length of __x__(s), but in general not arc length of __y__(s)).

Now __y__ '(s) must be perpendicular to __n__(s), and the three vectors __y__ '(s),__y__''(s) and __n__(s) must lie in one plane.

Now __y__ ' = __x__ ^{.} + λ ^{.} __n__ + λ(-κ__t__+τ__b__). If __y__ ' is perpendicular
to __n__, we find λ ^{.} =0, so λ is constant.

Then __y__'' = (-λκ ^{.})__t__ + (κ-λκ^{2}-λτ^{2})__n__ +
(λτ ^{.})__b__. We have

, so τ ^{.} (1-λκ) = -λκ ^{.} τ.

So τ=0 or κ is constant (and τ is constant or λ = 1/κ) or τ ^{.} /τ = (λκ ^{.} )/(λκ-1) (then τ =
d(λκ-1) with d≠0).

*Example 69:* The circular helix (see 16 and 46) has partners __y__(t) = ((a-λ)cos(t), (a-λ)sin(t), bt).

*Problem 70 :* Prove that if __x__(t) and __y__(t) are corresponding curves of Bertrand, then the angle between corresponding tangents is constant. For which curves is
this constant angle 90 degrees?

*Problem 71 :* Prove that, if a curve __x__(s) has constant curvature κ(s), then the curve __y__(s) of the curvature centers of __x__(s) has a
constant curvature, too. In this case, __x__(s) is the curve existing of the curvature centers of __y__(s). Also provre that the product of the torsions in corresponding points is
constant.

*Problem 72 :* In each point *P* of a curve *K* we choose on the principal normal a point *Q*. If *P* runs through the curve *K*, then *Q* describes the curve
*C*.
In each point *Q*, the osculating plane of *C* coincides with the normal plane of *K* in *P*. Prove that *K* has constant curvature.

*Problem 73 :* A necessary and sufficient condition that the binormals of a curve be principal normals of another curve, is that there exists a constant λ≠0 such that
λ^{2}κτ^{2} = -κ + λτ ^{.}. Prove this.

*Opgave 74 :* Suppose we have a curve whose binormals are also binormals of a second curve. Prove the curve is planar and both curves are congruent.

*Opgave 75 :* The rectifying planes of a curve k are parallel to a fixed straight line *l*. Give an example of such a curve.

Prove that for all n in N_{o} we have τ^{(n)}κ^{(n+1)}=κ^{(n)}τ^{(n+1)}. Here f^{(m)} denotes the m-th derivative of f as a function of s.

answers

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