In Cartesian coordinate system, if a point on curve C (regarded as a point set or a locus of points suitable for some conditions) has the following relationship with the real solution of a binary equation f(x, y)=0:

The coordinates of points on the (1) curve are all solutions of this equation.

(2) The points whose coordinates are the solution of the equation are all points on the curve.

Then, this equation is called the equation of curve, and this curve is called the curve of equation.

Related extension of curve knowledge;

According to the classical definition, the continuous mapping from (a, b) to R3 is a curve, which is equivalent to saying:

The curves in 1 and R3 are continuous images of one-dimensional space, so they are one-dimensional.

2. The curve of 2.R3 can be obtained by various deformations of a straight line.

3. A certain value of the parameter refers to a certain point on the curve, but the reverse is not necessarily true, because we can consider the self-intersecting curve.

Differential geometry is a subject that uses calculus to study geometry. In order to apply the knowledge of calculus, we can't consider all curves, even continuous curves, because continuity is not necessarily differentiable.

This requires us to consider differentiable curves. But differentiable curves are not very good, because there may be some curves whose tangent direction is uncertain at a certain point, which makes it impossible for us to start from the tangent. This requires us to study this kind of curve whose derivative is not zero everywhere, which we call regular curve.