Tangent Vector And Normal Vector


             Normal vector has perpendicular vector on surface.It is simply called normal.It has another name that is called as unit normal and it is use the unit vector.If it has without redundancy, it indicate normalized normal vector.It differentiate the inward pointing and outward pointing, when closed surface are used as normal.N or n is indicate the normal vector.Another vector is tangent vector.It is a curve at particular point or surface at any particular point. We can specify the that points.


Normal vector


Explanation about normal vector:

            Curvature vector is another name of normal vector.Straight line indicate curve deviance.It is denoted as,                 `bare_(2)`(t) = γ''(t)- (γ''(t), e1(t))e1(t).The second frenet vector e2(t) is unit normal vector.

                            That is e2(t)= `bare_(2)`(t) /||`bare_(2)`(t)||.    

Osculating plane is defined both tangent vector and normal vector.Euclidian is used to analyse the curve and its vectors.In optical medium, normal is perpendicular to the surface.Unique direction is not possible in normal vector.It has opposite direction.Right hand rule is determine the direction of vector.Abberation of surface determined by functions of vector.


Tangent vector


Explanation about tangent vector:

                     Curves geometry is defined as differential.Rn is context of curve.Manifolds theory is defined the tangent space.The vector is defined as,

                               γ'(t0) = `d/dt` γ(t) at t=t0.




The above  tangent vector express the velocity of particle at P when  γ represent particle path.Curve γ= γ(t) for t= t0.

The magnitude of tangent vector = ||γ'(t0)||.It is denote the speed at time t0. Unit tangent vector is defined as first frenet vector e1(t).It is in same direction.

     e1(t) = γ'(t) / ||γ'(t)||.It is in Gama regular point.For simplification, use the natural parameter t=s and it has unit length.Curve orientation determines by unit tangent vector.Length of surface divide the derivatives of vector.Frenet frame is most widely used technique to analyze the curve and both normal and tangent vectors.properties of vector is analyzed by differential geometry.