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Classifications of Curves

We intuivelly think of a curve as a one-dimensional configuration which has its origin in relatively simple examples, like the path of a moving point of particle. However, the general concepts of curves are very inclusive and require a precise and careful mathematical treatment. To avoid some of the complexities when handling the curves in general, some restrictions are necessary. For instance, in 1890 G.Peano [1] and one year later in 1891 D.Hilbert [2] described 'plane--filling' curves which meet every point in some given patch of the plane which demonstrate in a dramatic way that our naive ideas about representation of curves is very limited.

Affine plane curves

Fix a field , ordered couples whose coordinates belong to form an affine plane . A curve is an ordered configuration of points given by two continuous functions of a parameter :  

(1)

where is an interval (possibly infinite). The function , is often referred to as a point function.

A curve in this sense is still very general, and in many situations requires restrictions. For instance, an arc is a curve given by  (1) with a finite range of the parameter, say , , and  

• distinct ends, i.e. , and
• with no self-intersections, i.e. for .

Continuous deformations of an arc, such as bending, twisting, stretching or shrinking, of an arc result again in an arc, provided no originally distinct points are brought together.

If the points corresponding to and coincide, we say that the curve is closed. Simple closed curve or a closed Jordan curve is a closed curve without self-intersection. For instance, a circle is a closed Jordan curve, but a parabola is neither an arc nor a closed curve. Boundary of a square is a closed Jordan curve.

A curve given in terms of equation given in equation (1) is often referred to as a curved given parametrically. Other form to define a curve is to define it implicitly.  If is a function of two variables with coefficients in and is an overfield of then the locus of points  

(2)

is called curve defined by and is the equation of the curve over . Elements of are called points of the curve, and the points are called -rational points.  

Let be a subfield of the field of complex numbers. We say that a curve given by a point-valued function defines a smooth plane curve if

• both coordinate functions and have continuous derivatives on ,
• for all , or equivalently for all .

This means that the direction of a tangent line at each point of the curve varies continuously as the point moves along the curve.

Algebraic curves

The curves can be classified with respect to a variety of criteria.  If the curve is the locus  of some polynomial equation then it is called algebraic curves. Non-algebraic curves are called transcendental curves.

If , the  field of real numbers, is referred to as a real algebraic curve. If , the set is called a complex algebraic curve. If denotes a finite field, then defines an algebraic curve over a finite field.

Algebraic curve can be classified by their order. The order of an algebraic curve is the degree of the defining polynomial. Here the degree of is defined as the maximal integer in the set .

A point on an algebraic curve defined by the equation   is called singular if  partial derivatives ¹  of at   are both vanishing, otherwise it is called non-singular here. If is a singular point, then we say that the curve is singular at . The curve is non-singular it if contains no singular point.

Other classifications

The curves can be classified using various other aspects, as person’s names who invented or studied the curve, by construction methods, by shape, by a sailent property, by historical reasons, etc.   

Notes

1	If the field is not a subfield of  complex numbers, then the partial derivatives are computed formally using the usual rules of calculus.

References

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