Recently, when I was doing my graduation thesis, I involved the normal vector of the space curve. I was embarrassed to find that everything I had learned before was returned to my teacher, so I consulted Differential Geometry edited by Mei Xiangming and Huang Jingzhi and made the following notes.

Here we don't give a specific mathematical description of the concept of curve. In this description, we mainly consider the following parametric curves:

It's all about functions of parameters.

Tangent: Intuitively, the tangent is the closest straight line to the curve among all the straight lines passing through the tangent point.

Tangent vector: If it is differentiable, there are the following restrictions:

Then this vector is called the tangent vector of this point on the curve.

Intuitively, the closed plane of a curve is the tangent plane closest to the curve.

At a certain point on the curve, let its corresponding parameter be, if it is a vector, then determine a plane, which is the closed plane of the curve at that point, and its equation is

Let the parameter corresponding to this point on the curve be, then the unit tangent vector of this point is defined as

The second normal vector is defined as

The principal normal vector (perpendicular to the compact plane) is defined as