Curves and Surfaces > Curves > Types of Curves
Maps from R1 into R2 or R3.
 Curves in R2 vs. R3 Space curves and plane curves. Parametric vs. Implicit Curves can be defined by a parameter or implicitly. Rectangular vs. Polar Some curves are easier to represent in polar coordinates. Smooth vs. Non-smooth Smooth curves can be differentiated many times. Regular vs. Non-regular Regular curves have nonzero acceleration everywhere. Locally vs. Globally Defined A curve can be defined by local or global properties

### Curves in R2 vs. R3

There are two common types of curves: plane curves, which are maps from R1 to R2, and space curves, which are maps from R1 to R3. In a somewhat different sense, curves can also be viewed as 1-dimensional subsets of R2 or R3. For an example of a curve in R2, see the semicubical parabola; for an example of a curve in R3, see the twisted cubic.

### Parametric vs. Implicit

As was mentioned above, curves can be defined as maps or as subsets. Curves which are defined as maps that take a single variable t (called the parameter) and send it to a vector in R2 or R3 are called parameterized curves. On the other hand, a curve which is described as the set of solutions to an equation (usually in two variables) is said to be defined implicitly. The circle is an excellent example of each.

### Rectangular vs. Polar

Frequently, plane curves are specified in terms of the x and y coordinates of successive points on the curve, as functions of a parameter t. Occasionally, however, it is convenient to specify the distance of a point from the origin (denoted by r) and the angle a line from the point to the origin makes with the positive x axis (denoted by the Greek letter theta). This coordinate system is called polar, while the usual xy system is called rectangular. For an example of a curve in both rectangular and polar coordinates, see the cardioid.

### Smooth vs. Non-smooth

Curves which have derivatives of all orders at a given point are said to be smooth at that point. If all points on a given curve are smooth, then the curve itself is said to be smooth. An example of a smooth curve is the parabola. The absolute value curve, however, is not smooth because its derivative does not exist at the origin.

### Regular vs. Non-regular

Regular curves have nonzero acceleration at all points (i.e., the second derivative of the curve never vanishes.) An example of a regular curve is the circle. The cycloid, however, is non-regular at its cusps.

### Locally vs. Globally Defined

Some curves are interesting only for their local properties. The clothoid, for example, has a curvature function that varies linearly with the parameter t. Other local properties include torsion, tangent lines, and so forth. Many curves also have global properties, however. For example, a closed curve is either convex or non-convex based on its shape as a whole, completely independent of its local behavior. (A closed curve is said to be convex if, for any two points in the region bounded by the curve, a line between the points remains entirely within the region.) For example, the sine oval is convex, while the nephroid is not.