Let __x__ = __x__(s) be a space curve and p be any point on this curve. If __t__, __b__, __n__ are *unit tangent, unit principle normal and unit binormal vectors* respectively of the curve at the point P, then the Serret – Frenet formula are given by,

(a)__ t__′ = K__n__ (K= kappa)

(b) __n__′ = -k__t__ +?__b__

(c) __b__′ = -?__n__

**Proof:** We have __t__, __n__, __b__ are mutually orthogonal unit vectors at a point p on the space curve __x__ = __x__(s), so

__t__.__t__ = __n__.__n__ = __b__.__b__ =1

__t__.__n__ = __n__.__b__=__b__.__t__ =0

__t__˄__n__ = __b__, __n__˄__b__ = __t__, __b__˄__t__ = __n__

If __x__ = __x__(u), then we have

Therefore t′ lies on the osculating plane at the point P on the space curve __x__ = __x__(u)

Again, __t__ . __t__ = 1

⇒ __t__′ . __t__ + __t__ . __t__′ = 0

⇒ 2__t__ . __t__′ = 0

⇒ __t__ . __t__′ = 0

That is __t__′ is perpendicular to __t__. Also,__ t__′ is along __n__.

Hence, we can write, __t__′ = K __n__ —————(i)

## Where K is a scalar function and K is called the curvature of __x__ = __x__(s) at S.

Consider, __t__. __b__ = 0

⇒ __t__′ . __b__+__b__′ . __t__ =0

⇒K__n__.__b__+__t__ . __b__′=0 [by (i)]

⇒ 0 + __t__ . __b__′ =0

⇒__t__ . __b__′ =0

⇒ __b__′ is perpendicular to__ t__.

Again consider, __b__ .__ b __=1

⇒ __b__′ . __b__ + __b__ . __b__′ =0

⇒ 2__b__ . __b__′ = 0

⇒ __b__ . __b__′ =0

⇒__ b′__ is perpendicular to __b__.

Thus we see that __b__′ is perpendicular to __b__ at t, so __b__′ is along __n__. Hence along we can write,

__b__′ = λ__n__

⇒|__b__′| =| λ __n__ |

⇒|?| = |λ| [∵?=|__b__| and |__n__| =1]

⇒ λ = ± ?

⇒ λ = -?

⇒ __b__′ = -?__n__ ————————-(ii)

Where ? is a scalar and ? is called torsion of x = x(s) at S.

Now, __n__ = __b__ ˄ __t__

⇒ __n__′ = __b__′ ˄ __t__ + __b__ ˄ __t__′

= -? __n__ ˄ __t__ + __b__ ˄ K __n__

= -? (-__b__) + K (-__t__)

= -K__t__ + ?__b__

*Matrix form of Serret – Frenet formula:* Serret – Frenet Formula can be written as

__t__′ = 0 . __t__ + K__ n __+ 0 . __b__

__n__′ = -K__t__ + 0. __n__ + ? __b__

__b__′ = 0 . __ t __– ? __n__ + 0 . __b__

This can be written in the matrix form,