Arc Length, Curvature and Torsion with 1 example in each section

1. SUBJECT :- V.C.L.A.- 2110015
TOPIC :- Arc Length, Curvature and Torsion
Prepared by: Vaani Pathak
Branch: CE
Division: C1
Enrollment No.: 170120107131
Guided by:

2. Topics to be Covered
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 Arc Length
 Curvature
 Torsion

3. Arc Length
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 For plane curves r(t) = f(t) i + g(t) j
 |r’(t) = |𝑓’(t)i + g’(t)j| = [𝑓’(t)]2+[g′(t)]2
 For space curves r(t) = f(t) i + g(t) j + h(t) k
 |r’(t) = |𝑓’(t)i + g’(t)j + h’(t)k|
= [𝑓’(t)]2+[g′(t)]2+[h′(t)]2

4. Gandhinagar Institute Of Technology4
Example 1: Determine the length of the curve
𝒓 (t)=(2t,3sin(2t),3cos(2t)) on the interval 0≤t≤2𝝅
Solution
We swill first need the tangent vector and its magnitude
𝑟′(t)=(2,6cos(2t),-6sin(2t))
|| 𝑟′ 𝑡 || = 4+36cos2 (2t)+36sin2(2t)
= 4+36 = 2 10
The length is then,
L = 𝑎
𝑏
|| 𝑟′ 𝑡 ||dt
= 0
2𝝅
2 10 dt
= 4 𝝅 10

5. Curvature
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 The curvature measures how fast a curve is changing
direction at a given point.
 It is natural to define the curvature of a straight line to be
identically zero.
 Given any curve C and a point P on it,
there is a unique circle or line which most
closely approximates the curve near P,
the osculating circle at P.
 The curvature of C at P is
then defined to be the
curvature of that circle or
line.
 The radius of curvature is
defined as the reciprocal of
the curvature.

6. Curvature
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 The curvature is easier to compute if
it is expressed in terms of the parameter t instead of
s.
 So, we use the Chain Rule to write:
 However, ds/dt = |r’(t)|. So,

𝑑𝑇
𝑑𝑡
=
𝑑𝑇
𝑑𝑠
𝑑𝑠
𝑑𝑡
and k =
𝑑𝑇
𝑑𝑠
=
𝑑𝑇/𝑑𝑡
𝑑𝑠/𝑑𝑡
 k(t)=
𝑇′(𝑡)
𝑟′(𝑡)

7. Exercise: 1
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Determine the curvature for tkittr  2'
)(
In this case the second form of the curvature would probably be easiest.
Here are the first couple of derivatives
𝑟′ 𝑡 = 2𝑡 𝑖 + 𝑘 𝑟′′ = 2 𝑖
Next, we need the cross product.
𝑟′ 𝑡 × 𝑟′′(𝑡) =
𝑖 𝑗 𝑘
2𝑡 0 1
2 0 0
= 2 𝑗′
|| 𝑟′ 𝑡 × 𝑟′′ 𝑡 || = 2 | 𝑟′ 𝑡 | = 4𝑡2 + 2
The magnitudes are,
The curvature at any value of t is then,
𝑘 =
2
4𝑡2 + 1
3
2

8. Torsion of the curve
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 In the elementary differential geometry of
curves in three dimensions, the torsion of
a curve measures how sharply it is twisting out of
the plane of curvature.

9. Formula of Torsion
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 Let r = r(t) be the parametric equation of a space curve.
Assume that this is a regular parameterization and that
the curvature of the curve does not vanish.
Analytically, r(t) is a three times
differentiable function of t with values in R3 and the
vectors are linearly independent.
 Then the torsion can be computed from the following
formula:
 𝜏 =
det(𝑟′,𝑟′′,𝑟′′′)
𝑟′×𝑟′′ 2 =
𝑟′×𝑟′′ .𝑟′′′
𝑟′×𝑟′′ 2

10. Example
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11. THANK YOU