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DIFFERENTIAL GEOMETRY COURSE

9. *HELICES*

*Definition 81:* A helix is a curve (but not a straight line) whose tangents make a fixed angle with a fixed straight line (the axis).

*Proposition 82:* The helices are the curves with a natural equation of the form τ/κ = c (c constant)
(and the circular helices are the curves whose τ and κ are both constant and not 0).

*Proof:* Let __v__ be a direction vector of the fixed straight line with lenth 1.

From __t__.__v__ = c we find by differentiation __n__.__v__ = 0, so __v__ = __t__ cos(α) + __b__ sin(α) for a fixed
α.

Differentiating again, we get __0__ = (κ cos(α) - τ sin(α)) __n__, so τ/κ = c met c = cotg(α).

On the other hand, let τ/κ = cotg(α) (α constant).

Then __b__ ^{.} = -τ __n__ = (-τ/κ) __t__ ^{.} =
-cotg(α) __t__ ^{.}, si the vector __t__ cos(α) + __b__ sin(α) with length 1 is constant.

If we call this vector __v__, then we have __v__.__t__ = cos(α) = constant.

*Proposition 83:* We can give any helix a parametrisation of the form (x_{1}(σ),x_{2}(σ),σ cotg(α)), where σ is arc length of
(x_{1}(σ),x_{2}(σ), 0), and σ = s sin(α).

Furthermore, κ^{2} = κ_{1}^{2} sin^{4}(α), where κ_{1} is the curvature of (x_{1}(σ),x_{2}(σ), 0).

*Proof:* Choose coordinates in such a way that x_{3}-as is the axis of the helix.

Then we have (with __e___{3} = (0,0,1)): __t__.__e___{3} = cos(α), α constant. Hence x_{3}^{.} = cos(α), so
x_{3} = s cos(α).

The relation between s and σ follows from:
1 = (dx_{1}/dσ)^{2} + (dx_{2}/dσ)^{2} = (x_{1}^{.} ^{2} + x_{2}^{.} ^{2})(ds/dσ)^{2} =
(1 - x_{3}^{.} ^{2})(ds/dσ)^{2} = (ds/dσ)^{2} sin^{2}(α), so σ = s sin(α). Then we find the alleged parametrisation.

Furthermore, κ_{1}^{2} = (d^{2}x_{1}/dσ^{2})^{2} + (d^{2}x_{2}/dσ^{2})^{2} =
(x_{1}^{..}^{2} + x_{2}^{..}^{2} + x_{3}^{..}^{2})(ds/dσ)^{4} =
κ^{2} sin^{-4}(α). (Here we use that dx_{3}/dσ and ds/dσ are constant, so d^{2}x_{3}/dσ^{2} and d^{2}s/dσ^{2}
are 0,and also that x_{3}^{..} = 0.)

*Example 84:* We determine a parameter trepresentation of the helix with natural equations κ = τ = (2+s^{2})^{-1}.

We find tg(α)=1, σ = (1/2)s√2 and κ_{1} = 2 κ = (1 + σ^{2})^{-1}.

As in example 27, we find (x_{1}(σ), x_{2}(σ)) = (∫_{0}^{σ} cos(arctan(σ ') dσ ',
∫_{0}^{σ} sin(arctan(σ ') dσ ') = (∫_{0}^{σ} (1 + σ ' ^{2})^{-1/2} dσ ',
∫_{0}^{σ} σ ' (1 + σ ' ^{2})^{-1/2} dσ ' ) = (ln(σ + √(1+σ^{2})), √(1+σ^{2})).

Hence we get the parameter representation (ln(σ + √(1+σ^{2})), √(1+σ^{2}), σ) = (ln((1/2)s√2 + √(1+(1/2)s^{2})), √(1+(1/2)s^{2}),
(1/2)s√2).

*Problem 85:* Prove that __x__(s) is a helix if and only if
det(__x__ ^{..}, __x__ ^{...}, __x__ ^{....}) = 0, and κ ≠ 0.

*Problem 86:* Determine natural equations κ = κ(s) and τ = τ(s) for a helix that lies on a sphere.

*Problem 87:* Prove that __z__(t) from 47 is a helix, and determine the direction of the axis.

answers

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