Frenet-Serret formulas Guide, Meaning , Facts, Information and Description
In
vector calculus, the
Frenet-Serret formulas describe the dynamic properties of a particle which moves along a continuous, differentiable
curve in three-dimensional space . More specifically, the formulas describe the
derivatives of the tangent, normal, and binormal unit vectors in terms of each other. The formulas are named after their (independent) discoverers: the Frenchmen
Jean Frédéric Frenet and
Joseph Alfred Serret.
Let s(t) represent the distance which the particle has moved along the curve. Let r(t) represent the position vector of the particle. Then the tangent unit vector T is defined as
-
The normal unit vector
N is defined as
-
and the binormal unit vector
B is defined as
-
From equation (2) it follows, since
T always has unit
magnitude, that
N is always perpendicular to
T. From equation (3) it follows that
B is always perpendicular to both
T and
N. Thus, the three unit vectors
T,
N, and
B are all perpendicular to each other.
It can be proven, from the above, that
-
and
- .
Then the Frenet-Serret formulas are:
The three vectors:
tangent,
normal, and
binormal, are collectively called the
Frenet vectors. Together they form a
basis for 3-space, which in turn defines a reference frame with its own
coordinate system. The
reference frame is called a
Frenet frame, and it is neither static nor inertial, since the Frenet frame moves tangentially to the (non-straight) curve, so that it is constantly accelerating. (The curve can be assumed to be
parametrically dependent on time: i.e. the Frenet frame can be visualized
kinematically).
Proof
Part One
First prove equations (4) and (5). From equation (3) it follows that
- .
An identity from vector calculus states that
-
Applying identity (7) to equation (6) yields
-
but N and T are perpendicular so that their dot product is zero. Also, T is always a unit vector, so the dot product of T with itself is always one. Therefore equation (8) simplifies to equation (4).
Now prove equation (5). From equation (4) it follows that
-
Applying identity (7) to equation (9) produces
-
but
T and
B are perpendicular so that their dot product is zero, and
B is always a unit vector so that the dot product of
B with itself is always unity. Therefore equation (10) simplifies to equation (5).
Equations (4) and (5) should be intuitively rather obvious.
Part Two
Let k equal the magnitude of dT / ds. Then the first Frenet-Serret formula follows from equation (2), where k is the curvature of the particle's path.
Now prove the third Frenet-Serret formula. From equation (3) it follows that
-
Applying the
chain rule for the
cross product to this last equation yields
-
Applying the first Frenet-Serret formula to this last equation yields
-
but the cross product of any vector with itself must be zero (due to an identity of vector calculus), so that equation (11) becomes
-
The cross product of two vectors must necessarily be perpendicular to both vectors, so that
dB/
ds must be perpendicular to
T, but it must also be perpendicular to
B, since
B is always a unit vector, so that
dB/
ds must be parallel to
N. Let
-
where
τ is
torsion, so that the third Frenet-Serret formula has been proven.
Lastly, prove the second Frenet-Serret formula. Equation (4) implies that
-
Applying the chain rule for the cross product to this last equation yields
-
Next, substitute the third and first Frenet-Serret formulas into equation (12), producing
-
Then, applying equations (3) and (5) to the last equation yields the second Frenet-Serret formula.
Q.E.D
See also: Frenet frame.
Reference
- Salas and Hille's Calculus -- One and Several Variables. Seventh Edition. Revised by Garret J. Etgen. John Wiley & Sons, 1995. p. 896.
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