What is a curve?

Geometry is an exact and complex science. To understand some of its concepts, it is necessary to penetrate into many axioms, theorems, concepts. We will try to make your life a little easier. In our article, the question of what a curve is, what they are, and how they are used in graphic editors is considered in detail.

Curve line

Imagine that we have a segment AB, which is located in a three-dimensional space, which is characterized by its height, width, and length. Here is a set of points in space, the coordinates of which are functions of one variable, and is called a curve. If one of the coordinates (for example, length) equals zero, then we get two-dimensional space.

Types of curves

Curves can be divided on several grounds. According to one of the classifications are distinguished:

  • closed;
  • flat;
  • simple curves;
  • way.

Closed are those lines where the source and end point are the same. A curve is called flat if all its points lie in two-dimensional space. A simple curve, or the Jordan curve, is a concept that is rarely used in the course of geometry.So called the image of a particular segment or circle in space. By path means continuous mapping of the segment AB (where A = 0, B = 1) to the topographic space.

In algebra, curves of the first, second, third, and subsequent orders are distinguished. The equations of curves of the first order coincide with straight lines, so they are not considered separately.

The simplest curves of the second order are parabola and hyperbole.

Bezier curves

In the 60s, independently from each other, representatives of two major automobile concerns, Renault and Citroen, developed a remarkable tool for designing bodies. These are the curves that are specified by the control points. This principle is used to work in graphic editors and programs for automatic design and creation of animation.

There are two, three, four and even more support points. If we define two control points, then the curve will look like a segment drawn through them. With three points - it will be an arc. The first point - its beginning, the third - the end, and the second - indicates the place of the fracture. If there are four points, then there will be two fracture lines: two points will be the beginning and the end, and anothertwo denote bends. It is clear that by increasing the number of points we can make our curve more and more tortuous.

The curves will always be in the middle of the convex hulls formed by the support points. This property is used to determine the possibility of intersection of different curves. If their shells do not intersect, then the curves will never intersect.

Bezier curves get their name from the name of Pierre Beziers, who developed them.

Using Curves in Graphic Editors

Also in graphic editors use curves. It is worth noting that all the curves in them are Bezier curves. Each letter represents a few curves. These editors include Coral Draw and Inkspace.

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