Begin2.DVI

Normal and Binormal to Space Curve

Recall that the unit tangent vector to a space curve
r =
r (t), for any value of the

parameter t, is given by the equation

ˆet=

∣^1

∣d
rdt∣∣

d
r

dt

=

d
r

ds

(7 .12)

and satisfies ˆet·ˆet= 1. Differentiating this relation with respect to the arc length

parameter sone finds

d

ds

[ˆet·ˆet] = ˆet·dˆet

ds

+dˆet

ds

·ˆet= 0 or 2 ˆet·dˆet

ds

= 0 (7 .13)

The zero dot product in equation (7.13) demonstrates that the vector dˆet

ds

is per-

pendicular to the unit tangent vector ˆet. Note that this vector can be calculated

using the chain rule for differentiation dˆet

ds

ds

dt

= dˆet

dt

where ds

dt

is calculated using

the equation (7.1). Observe that there are an infinite number of vectors which are

perpendicular to the unit tangent vector eˆt. The unit vector with the same direction

as the vector dˆet
ds

is called the principal unit normal vector to the curve
r (t)for each

value of the parameter t. The principal unit normal vector in the direction of the

derivative vector

dˆet

ds is given the label

ˆen. The vector dˆet

ds has the same direction

as ˆen and so one can write

dˆet

ds =κˆen (7 .14)

where κis a scaling constant called the curvature of the curve
r (t). The curvature

κwill vary as the parameter tchanges. The quantity ρ=^1 κ is called the radius of

curvature at the point associated with the parameter value of t. The unit vector ˆeb

calculated from the cross product of ˆetand ˆen,ˆeb=eˆt׈en, is perpendicular to both

the unit tangent ˆetand unit normal ˆenand is called the unit binormal vector to the

curve as the parameter tchanges.

The vectors ˆet, eˆn, eˆb are called a moving

triad along the curve
r (t) because the unit vec-

tors eˆt, ˆen, eˆbgenerated a localized right-handed

coordinate system which changes as the parame-

ter tchanges. The plane which contains the unit

tangent ˆet and principal normal ˆen is called the

osculating plane. The plane containing the unit