Closed Curves

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Closed Curves

A closed curve is a curve within a plane that has no endpoints and, thus, completely encloses an area within it. It can be drawn without lifting the pencil from the paper and that ends at the point where it began. In Figure 1, A, B, and C are closed curves; D, E, and F are not.

Curve A is a circle. Although the starting point is not indicated, any of its points can be chosen to serve that purpose. Curve B crosses itself and is therefore not a simple closed curve, but it is a closed curve. Curve C has a straight portion, but curve, as used in mathematics, includes straight lines as well as those that bend. Curve D, in which the tiny circle indicates a single missing point, fails for either of two reasons. If the starting point is chosen somewhere along the curve, then the one-point gap is a discontinuity. One cannot draw it without lifting the pencil. If one tries instead to start the curve at the point next to the gap, there is no point next to the gap. Between any two

distinct points on a continuous curve, there is always another point. Whatever point one chooses, there will always be an undrawn point between it and the gap— in fact an infinitude (endless number) of such points. Curve E has a closed portion, but the tails keep the curve as a whole from being closed. Curve F simply fails to end where it began.

Curves can be described statically as sets of points. For example, the sets {P: PC = r, where C is a fixed point and r is a positive constant} and {(x, y): x² + y² = r²} describe circles. The descriptions are static in the way a pile of bricks is static. Points either belong to the sets or they do not; except for set membership, there is no obvious connection between the points.

Although such descriptions are very useful, they have their limitations. For one thing, it is not obvious that they describe curves as opposed, say, to surfaces (in fact, the first describes a sphere or a circle, depending on the space in which one is working). For another, they omit any sense of continuity or movement. A planet’s path around the Sun is an ellipse, but it does not occupy every point on the ellipse simultaneously, and it does not hop from point to point. It moves continuously along the curve as a function of time. A dynamic description of its path is more useful than a static one. Curves are therefore often described as paths, as the position of a point, P(t), that varies continuously as a function of time (or some analogous variable), which increases from some value a to value b. If the variable t does not represent time, then it represents a variable which increases continuously as time does, such as the angle formed by the Earth, the Sun, and a reference star.

However it is described, a curve is one-dimensional. It may be drawn in three-dimensional space, or on a two-dimensional surface such as a plane, but the curve itself is one-dimensional. It has one degree of freedom (i.e., length).

A person on a roller coaster experiences this phenomenon. The roller coaster car loops, dives, and twists, but it does not leave the track. As time passes, it makes its way from the starting point, along the track, and finally back to the starting point. Its position is a function of a single variable, called a parameter, time.

A good way to describe a circle as a path is to use trigonometric functions: P(t) = (r cos t, r sin t), where P(t) is a point on the coordinate plane. This describes a circle of radius r with its center at the origin. If t is measured in degrees, varying over the closed interval [90, 450], the curve will start at (0, 1) and, since the sine and cosine of 90° are equal to the sine and cosine of 450°, will end there. If the interval over which t varies

KEY TERMS

Closed curve— A curve that begins and ends at the same point.

Curve— A one-dimensional continuous set of points in the plane or in space.

is the open interval (90, 450), it will be the circle with one point missing, as shown in Figure 1, and will not be closed.

Not all curves can be represented as neatly as circles, but from a topological point of view, that does not matter. There are properties that certain closed curves share regardless of the way in which P(t) is represented.

One of the properties is that of being a simple closed curve. In Figure 1, curve B was not a simple curve; it crossed itself. There were two values of t, call them t₁ and t₂, for which P(t₁ ) = P(t₂ ). If there are no such values—if different values of t always give different points, then the curve is simple.

Another property is a sense of direction along the curve. On the roller coaster, as time passes, the car gets closer and closer to the end of the ride (to the relief of some of its passengers). On the circle described above, P(t) moves counterclockwise around it. If, in Figure 2, a toy train were started so that it traversed the left loop in a counterclockwise direction, it would traverse the other loop in a clockwise direction. This can be indicated with arrowheads.

Associated with each closed part of a closed curve is a winding number. In the case of the circle above, the winding number would be +1. A person standing inside the circle watching P(t) move around the circle would have to make one complete revolution to keep the point in view. Since such a person would have to rotate counterclockwise, the winding number is arbitrarily considered positive.

In the same way, a person standing inside the left loop of Figure 2 would rotate once counterclockwise. Watching the point traverse the right loop would necessitate some turning first to the left and then to the right, but the partial revolutions would cancel out. The winding number for the left loop is therefore +1. The winding number for the right loop, based on the number of revolutions someone standing inside that loop would have to make, would be -1.

In Figure 3 the winding numbers are as shown.

The reader can check this. In Figure 4, although the curve is quite contorted, the winding number is +1.

This illustrates a fundamental fact about simple closed curves: their winding numbers are always +1 or -1.

Being or not being closed is a property of a curve that can survive a variety of geometrical transformations. One can rotate a figure, stretch it in one direction or two, shrink it, shear it, or reflect it without breaking it open or closing it. One transformation that is a notable exception to this is a projection. When one projects a circle, as with a slide projector, one can, by adjusting the angle of the screen, turn that circle into an open parabola or hyperbola.