Plane curve

Mathematical noun

This entry is made by Compilation and application of scientific encyclopedia entries of "Science Popularization China" to examine.

Plane curve is a mathematical term, which means Euclid Plane, affine plane, or Projection Curves in the plane. The most frequently studied cases are smooth planar curves (including subsection Smooth curves) and Algebra Plane curve.

Chinese name: Plane curve
Foreign name: Plane curve
Discipline: mathematics

Classification: Smooth plane curve and algebraic plane curve
Example: Line, circle, etc
Related nouns: Space curve

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Basic Introduction

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In mathematics, a plane curve can be Euclid Plane, affine plane, or Projection Curves in the plane. The most frequently studied cases are smooth planar curves (including subsection Smooth curves) and Algebra Plane curve.

Smooth Curve

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Smooth plane curves are actually Euclidean plane R two The curve in is a one-dimensional smooth streamline curve. This means that the smooth curve is a plane curve that "looks like a line locally". Near each point, it can be mapped to a line through the smoothing function. Similarly, the smooth plane curve can be given by the equation f (x, y)=0, where f: R two → R is a smooth function, and the partial derivatives partial f/partial x and partial f/partial y will not be 0 at the same point of the curve.^[1]

Algebraic curve

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An algebraic plane curve is a curve in an affine or projection plane given by a polynomial equation f (x, y)=0 (or F (x, y, z)=0), where F is a polynomial.)

Algebraic curves have been extensively studied since the 18th century.

Each algebraic plane curve has a certain dimension, defining the dimension of the equation, which is equal to the number of intersections of the curve and the line at the general position in the case of algebraic closed field. For example, by the formula x two + y two =1 The circle given is two-dimensional.

2-D nonsingular planar algebraic curve is called cone section, and its projection is the same as that of circle x two + y two =Projection of 1 (i.e. equation x two + y two - z two =0) are isomorphic. 3D planar curves are called cubic planar curves if they are nonsingular elliptic curves. Those four-dimensional plane curves are called quartic plane curves.^[2-3]