Find the equation of osculating plane at the point u= π/4 of the curve x=(a cos u, a sin u, b u).
Solution: We know the equation of osculating plane is[(y-x) ẋ ẍ)=0
⟹[(y-x) (ẋ ˄ ẍ)]=0
Given that, x=(a cosu, a sin u, bu)
⟹ẋ = (-a sin u, a cos u, b)
⟹ẍ = (-a cos u, -a sin u)
Now, at the point u = π/4, we get
x= (a /√2, a /√2, b π/4)
ẋ = (-a /√2, a /√2, b)
ẍ = (-a /√2, -a /√2, 0)
= (a b /√2, -a b /√2, a2/2+a2/2)
= (a b /√2, -a b /√2, a2)
Again, y-x = (x, y, z) – (a /√2, a /√2, b π/4)
=(x- a /√2, y- a /√2, z- b π/4)
Hence the equation of osculating plane at u = π/4 is
(y-x) (ẋ ˄ ẍ)]=0
⟹(x- a /√2, y- a /√2, z- b π/4). (a b /√2, -a b /√2, a2)=0
⟹(x- a /√2) (a b /√2) + (y- a /√2) (-a b /√2)+ (z- b π/4)a2 =0.
Normal line: Any line perpendicular to the tangent at any point on a space curve is called a normal line at the point.
Normal plane: The plane through a point u on a space curve perpendicular to to the tangent at u is called the normal plane at u.
The principle normal: The normal lying in the osculating plane at any point on a space curve is called the principle normal at the point.
That is the principle normal in the intersection of the normal plane and the osculating plane. We shall denote the unit vector along the principle normal by n.
Binormal: The normal perpendicular to the principle normal is called the binormal.
That is the binormal is the line perpendicular to the osculating plane. we shall denote the unite vector along the binormal by b.
Curvature of the curve: The rate of change of the direction of tangent with respect to arc length S at the point P moves along the curve is called the curvature vector of the curve whose magnitude is denoted by K Is called the curvature at P. Thus, we have
Torsion of the curve: The rate of change of the direction of the binormal with respect to the arc length S at the point P moves along the curve Is called the torsion vector of the curve whose magnitude is denoted by ? is called the torsion at P.
Thus, we have,
Radius of curvature: The reciprocal of the curvature is called the radius of curvature and it is denoted by R or ?. Thus we have
R = ?= 1/K.
Radius of torsion: The reciprocal of the torsion is called the radius torsion and it is denoted by T or ?. Thus we have
T = ? = 1/ ?