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Prove Frenet Seret formula

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Answer:

To prove the Frenet-Serret formulas, we start with a curve in three-dimensional space parameterized by the arc length, denoted as r(s) = (x(s), y(s), z(s)), where s is the arc length parameter.

We define three vectors associated with the curve:

1. Tangent vector T(s) = r'(s) represents the instantaneous direction of the curve at any point.

2. Principal normal vector N(s) is defined as N(s) = T'(s) / ||T'(s)||, where || || denotes the magnitude or length of the vector. N(s) represents the direction of the instantaneous curvature of the curve.

3. Binormal vector B(s) = T(s) x N(s) represents a vector perpendicular to both T(s) and N(s), completing the orthogonal frame.

Now we can derive the Frenet-Serret formulas:

1. Derivative of T(s):

Taking the derivative of T(s) = r'(s) with respect to s, we have:

T'(s) = r''(s)

2. Derivative of N(s):

Taking the derivative of N(s) = T'(s) / ||T'(s)||, we have:

N'(s) = (T''(s)||T'(s)|| - T'(s)(T''(s) · T'(s))) / ||T'(s)||²

3. Orthogonality of T(s), N(s), and B(s):

We can show that T'(s) · N(s) = 0, which means that T'(s) is perpendicular to N(s).

Similarly, N'(s) · N(s) = 0, showing that N'(s) is also perpendicular to N(s).

Because T(s), N(s), and B(s) form an orthogonal frame, we have B'(s) · T(s) = 0 and B'(s) · N(s) = 0.

4. Derivative of B(s):

Using the orthogonality relationships, we can express B'(s) in terms of T(s) and N(s):

B'(s) = αT(s) + βN(s), where α and β are some scalar functions to be determined.

Taking the dot product of B'(s) with T(s), we get:

0 = B'(s) · T(s) = α(T(s) · T(s)) + β(N(s) · T(s)) = α

Similarly, taking the dot product of B'(s) with N(s), we have:

0 = B'(s) · N(s) = α(T(s) · N(s)) + β(N(s) · N(s)) = β

Therefore, α = 0 and β = 0, which implies B'(s) = 0.

In conclusion, the Frenet-Serret formulas are:

T'(s) = κ(s)N(s)

N'(s) = - κ(s)T(s) + τ(s)B(s)

B'(s) = - τ(s)N(s)

where κ(s) represents the curvature of the curve at the point parameterized by s, and τ(s) represents the torsion of the curve at that point.

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