This document provides an overview of arc length, curvature, and torsion for plane and space curves. It defines arc length as the line integral of the magnitude of the tangent vector over an interval. Curvature is defined as the rate of change of the tangent vector and formulas are provided to calculate it in terms of derivatives of the position vector components. Torsion measures how sharply a space curve is twisting out of its plane of curvature. A formula is given for torsion in terms of derivatives of the position vector and the cross product of the tangent and normal vectors. An example problem calculates the arc length of a space curve and determines the curvature of a plane curve.

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