Differential Geometry of Curves and Surfaces Curvature of Curves, guaranteed and verified 100% Pass

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Curvature of Curves


Let’s start with curves in ℝ2 . The curvature of a curve should measure the extent to
which it is contained in a line (i.e. a line should have zero curvature).



Let 𝛾(𝑡) be a unit speed curve in ℝ2 . Notice that as 𝑡 goes to 𝑡 + Δ𝑡 the amount the
curve 𝛾 moves away from the tangent line at 𝑡 is given by:
|(𝛾(𝑡 + Δ𝑡) − 𝛾(𝑡 )) ∙ 𝑛⃗|
where 𝑛
⃗ is a unit vector perpendicular to the tangent vector, 𝛾′(𝑡), of 𝛾(𝑡) at 𝑡.



𝑛⃗




𝑅 𝛾(𝑡 + ∆𝑡)
⃗⃗⃗⃗⃗ | = |𝑛⃗ ∙ 𝑃𝑅
|𝑄𝑅 ⃗⃗⃗⃗⃗ |

𝑄
𝛾(𝑡)

𝑃

, 2


By Taylor’s Theorem:
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𝛾(𝑡 + Δ𝑡 ) = 𝛾(𝑡 ) + 𝛾 ′ (𝑡 )(Δ𝑡) + 2 (𝛾 ′′ (𝑡 ))(Δ𝑡)2 + remainder
(remainder)
where
(Δ𝑡)2
goes to zero as Δ𝑡 goes to zero.


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𝛾(𝑡 + Δ𝑡 ) − 𝛾 (𝑡) = 𝛾 ′ (𝑡)(Δ𝑡 ) + 2 (𝛾 ′′(𝑡 ))(Δ𝑡 )2 + remainder


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(𝛾(𝑡 + Δ𝑡) − 𝛾(𝑡)) ∙ 𝑛⃗ = (𝛾 ′ (𝑡) ∙ 𝑛⃗)(Δ𝑡) + (𝛾 ′′ (𝑡) ∙ 𝑛⃗)(Δ𝑡)2 +remainder ∙ 𝑛⃗
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𝛾 ′ (𝑡) is tangent to 𝛾(𝑡 ) at 𝑡 so 𝛾 ′ (𝑡) ∙ 𝑛⃗ = 0.


Since 𝛾(𝑡) is unit speed: 𝛾 ′ ∙ 𝛾 ′ = 1.
Differentiating this equation we get:

𝛾 ′ ∙ 𝛾 ′′ + 𝛾 ′′ ∙ 𝛾 ′ = 0
𝛾 ′ ∙ 𝛾 ′′ = 0.

So 𝛾 ′′ (𝑡) is also perpendicular to 𝛾 ′ (𝑡) so it’s parallel to 𝑛
⃗ . Thus:
𝛾 ′′ (𝑡 ) = ±‖𝛾 ′′ (𝑡)‖𝑛⃗.


Thus we have:
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(𝛾(𝑡 + Δ𝑡 ) − 𝛾(𝑡)) ∙ 𝑛⃗ = ± 2 ‖𝛾 ′′ (𝑡)‖(Δ𝑡)2 + remainder∙ 𝑛⃗.