Exploring the Frenet Frame Equations for Curves in R_3

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In summary, the Frenet frame structural equation for a curve in R_3 can be written as T'(s) = W(s) x T(s), N'(s) = W(s) x N(s), and B'(s) = W(s) x B(s). The problem arises when defining T(s) as the unit tangent and parametrizing the curve by arc length, as N(s) becomes parallel to T'(s). This leaves only +/- B(s) as a possible choice for W(s), resulting in B'(s) being identically 0 for a general curve in R_3. However, W does not have to be orthogonal to T, and it is only when the torsion is zero that
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I'm trying to show that the Frenet frame structural equation for a curve in R_3 can be written in the following form for a vector W(s):

T'(s) = W(s) x T(s)
N'(s) = W(s) x N(s)
B'(s) = W(s) x B(s)

The problem I'm having here is that I define first that T(s) should be the unit tangent at the point s. I assume that my curve is parametrized by arc length so T'(s) is certainly orthogonal to T(s). The problem I have now, is that N(s) is parallel to T'(s). So my only possible choice for W(s) at this point is +/- B(s), as it must be orthogonal to both T(s) and N(s). But this gives that B'(s) is identically 0 for all s, which is fine for a plane curve I suppose because then the osculating plane isn't rolling and B'(s) is supposed to vanish because the torsion is identically 0. But this is supposed to be for a general curve in R_3.

Can anyone perhaps just point out a serious flaw in my logic so that I might continue on with this, or am I not mistaken in what I'm saying here?
 
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I do not see any mistake in what you said. Do u have reasons to think such a W exists?
 
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factor said:
I'm trying to show that the Frenet frame structural equation for a curve in R_3 can be written in the following form for a vector W(s):

T'(s) = W(s) x T(s)
N'(s) = W(s) x N(s)
B'(s) = W(s) x B(s)

The problem I'm having here is that I define first that T(s) should be the unit tangent at the point s. I assume that my curve is parametrized by arc length so T'(s) is certainly orthogonal to T(s). The problem I have now, is that N(s) is parallel to T'(s). So my only possible choice for W(s) at this point is +/- B(s), as it must be orthogonal to both T(s) and N(s). But this gives that B'(s) is identically 0 for all s, which is fine for a plane curve I suppose because then the osculating plane isn't rolling and B'(s) is supposed to vanish because the torsion is identically 0. But this is supposed to be for a general curve in R_3.

Can anyone perhaps just point out a serious flaw in my logic so that I might continue on with this, or am I not mistaken in what I'm saying here?

W must be a linear combination of T and B. It does not have to be orthogonal to T. I think it will be orthogonal only when the torsion is zero.
 

FAQ: Exploring the Frenet Frame Equations for Curves in R_3

What are the Frenet frame equations?

The Frenet frame equations are a set of differential equations that describe the curvature and torsion of a curve in three-dimensional space. They are used to calculate the position, orientation, and rate of change of a moving object along a curve.

What is the purpose of the Frenet frame equations?

The Frenet frame equations are used in differential geometry and physics to study the motion of objects along curves. They provide a mathematical framework for understanding the shape and behavior of curves and can be applied to a wide range of fields, including robotics, computer graphics, and fluid dynamics.

What is the relationship between the Frenet frame equations and the curvature and torsion of a curve?

The Frenet frame equations describe the rate of change of the tangent, normal, and binormal vectors along a curve, which are used to calculate the curvature and torsion of the curve. The curvature measures how much a curve deviates from a straight line, while the torsion measures how much it twists in space.

What are some applications of the Frenet frame equations?

The Frenet frame equations have many practical applications in physics and engineering. They are used in computer-aided design (CAD) software to model and manipulate curves, in robotics to control the motion of robotic arms, and in fluid dynamics to analyze the flow of fluids along curved surfaces.

How do the Frenet frame equations relate to the Frenet-Serret formulas?

The Frenet-Serret formulas are a special case of the Frenet frame equations for curves in three-dimensional space. They describe the behavior of a curve in terms of its curvature and torsion and are used to calculate the position, velocity, and acceleration of a moving object along the curve.

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