What is the curve produced when a plane intersects one Nappe of a cone to form a bounded curve?

Introduction According to Kepler's First Law of Planetary Motion, the orbit of each planet is an ellipse, with one focus of that ellipse at the center of the Sun. Newton's reformulation of this Law states that the orbit of each planet is a conic section, with one focus of that conic section at the center of the Sun. To properly understand planetary orbits, we therefore need some understanding of ellipses in particular, and conic sections in general.

Conic Sections

A section is the surface or outline of that surface formed by cutting a solid figure with a plane. If the solid figure is a right circular cone, the resulting curve is called a conic section. The diagram below shows such a cone, formed by rotating a diagonal line around a vertical axis so that the axis, the diagonal and a horizontal line connecting the two form a right triangle. Four planes are shown, cutting through the cone at various angles, producing the curves shown in the following diagram. The intersection of each plane with the cone forms a conic section. The kind and shape of the conic section is determined by the angle of intersection of the plane with the axis and surface of the cone.

Angled view of a cone, with conic sections produced by cutting the cone at different angles. Cutting at right angles to the axis produces a circle. Cutting at less than a right angle to the axis but more than the angle made by the side of the cone produces an ellipse. Cutting parallel to a side of the cone produces a parabola. Cutting more nearly parallel to the axis than to the side produces a hyperbola (the hyperbola in the diagram represents a cut parallel to the axis of the cone).

View from above of, from left to right, a circle, an ellipse, a parabola and a hyperbola. A circle is a smooth, uniform curve, while an ellipse is "stretched" out along one axis, and "compressed" along the perpendicular axis. Circles and ellipses are closed curves, while parabolas and hyperbolas are open curves. For parabolas the two arms are parallel to each other at infinity, but for hyperbolas the two arms make an angle with each other even at infinity.

The Four Kinds of Conic Sections
When a plane cuts a cone at right angles to its axis a circle is formed. The axis passes through the center of the circle, and the focus (and the Sun, if the curve represents the orbit of a planet) is at the center of the circle.
When the plane cuts the cone parallel to the side of the cone a parabola is formed. Since the plane of the parabola is parallel to one side of the cone, the curve never cuts the side it is parallel to and as the arms of the parabola extend into infinity they become parallel to each other and the axis of the parabola. Since circles and parabolas are formed by cutting the cone at specific angles they have unique shapes. All circles are identical in shape, and all parabolas are identical in shape; only their size and orientation differ (there are bigger and smaller circles, and broader and narrower parabolas). For ellipses and hyperbolas however, there is a wide range of angles between the plane and the axis of the cone, so they have a wide range of shapes.

When the plane cuts the cone at an angle between a perpendicular to the axis (which would produce a circle) and an angle parallel to the side of the cone (which would produce a parabola), the curve formed is an ellipse. Since circles and parabolas are formed by angles just beyond the range of angles which produce ellipses, ellipses can vary in shape from very nearly circular to very nearly parabolic. The closer the plane is to a perpendicular to the axis the more nearly circular the ellipse is, and the closer its focus (which is the location of the Sun for objects moving around the Sun) lies to its center. The closer the plane is to being parallel to the side of the cone the more elongated the ellipse is, the closer its focus is to one end of the ellipse, and the more either end of the ellipse looks similar to the "near" end of a parabola (for very small segments of very elongated ellipses, such as the paths followed by objects falling to the surface of the Earth, the portion of the ellipse that is observed is essentially identical to a parabola, and in basic physics classes falling objects are said to follow parabolic paths, even though they are actually following very elongated elliptical paths).

When the plane cuts the cone at an angle closer to the axis than the side of the cone a hyperbola is formed. As in the case of a parabola the curve extends into infinity, as the plane can never reach the far side of the cone and in fact, gets further and further from it the further along the arms you go. Hyperbolas that are formed by angles close to the side of the cone look very nearly parabolic, while hyperbolas that are formed at steeper angles look less parabolic; but in every case there is a fundamental difference between a hyperbola and a parabola: the arms of a parabola eventually become parallel to each other, while the arms of a hyperbola always make an angle relative to each other. For a hyperbola which is very nearly parabolic this angle may be close to zero; but for some hyperbolas the angle may be close to 180 degrees, and the hyperbola can be almost a straight line.

Application to The Solar System

Kepler's First Law of Planetary Motion says that the orbits of the planets are ellipses with the Sun at one focus of the ellipse. As reformulated by Newton, the First Law says that the orbits may be any kind of conic section, with the Sun at one focus of the section. In practice, however, all planetary orbits must be ellipses, because objects in parabolic or hyperbolic orbits would go around the Sun once, go out into interstellar space, and never return. For an object to have been orbiting the Sun for 4.5 billion years as the planets have been, the orbit must be closed and repeating, so it must be a circle or an ellipse; and since a circle can be viewed as a special kind of ellipse (as explained below), all the more or less stable orbits in our Solar System are elliptical.
This does not mean that open orbits are forbidden. There is a possibility that something might approach our Solar System from interstellar space. As it does so, the Sun's gravity would bend its path, causing it to follow a hyperbolic orbit through the Solar System, curving around the Sun then returning to the interstellar space from which it came. We have never seen anything do this, but it would be a very exciting thing if we did (and we would study the object as much as we could while we had the chance).
It is also possible for orbits to change from one conic section to another through perturbations -- that is, gravitational interactions with objects other than the Sun. In recent centuries several comets have passed close enough to Jupiter to allow Jupiter's gravity to change their orbits from very long ellipses to hyperbolas, "flinging" them out of the Solar System and into interstellar space. We've always noticed this after the fact, and in each case for a short time there was excitement about the possibility that we were seeing an interstellar visitor; but so far, every time we've traced the orbit backwards we've found that Jupiter was the cause of the change in the orbit.

Generating Ellipses

Since ellipses can have various shapes, it is important to understand how the various shapes are related to each other, and the terms that are used to descirbe those relationships.

There are a large number of ways in which the generation or creation of ellipses can be accomplished. If we want to calculate the exact place where each part of the curve is located, we would use some kind of algebraic curve, such as

As useful as these formulae might be for calculations, unless you are familiar with the mathematics involved it is difficult to see what the resulting curves look like, or how changing the parameters (the constants) in the equations affects the appearance of the curve. It is easier to understand the nature of ellipses of various types by using graphical methods of creating them, as in the example of cutting a cone with a plane used above. One simple way to generate ellipses is to take a circle and rotate it about a diameter. As shown below the diameter which is used as an axis of rotation is unchanged by the rotation, but all the diameters that make an angle to that one are reduced in size (foreshortened) by the rotation. If the rotation is small the resulting ellipse is very nearly round, but if the rotation is large the ellipse becomes very flattened (or very elongated, depending upon how you look at the effect), and if the circle is rotated until it is edge-on to our line of sight the "ellipse" becomes just a straight line segment. (The same thing can be done with open curves such as parabolas and hyperbolas, but the results are not as useful for orbital mechanics.)

Above: Creating ellipses by rotating a circle about a diameter. As the circle rotates it becomes foreshortened. The axis of rotation stays the same as the original diameter, but the other axes become shorter and shorter. In the process the ellipse becomes flattened or stretched out, depending upon your viewpoint.

All of the curves produced by rotating a circle are ellipses. Even the extremes, a circle (on the left) and a straight line segment (on the right) are technically considered to be ellipses -- just ellipses of special type. As you will see, the shape (or more accurately, the position of the focus) of an ellipse can be expressed by a number called the eccentricity, which is somewhere between zero and one. If the eccentricity is zero the ellipse is a circle. If the eccentricity is one the ellipse is a straight line segment. "Normal" ellipses have values between these extremes. Later we will discuss how we measure the size of an ellipse. As you will see at that time, all the ellipses generated by rotating a circle are the same "size" (namely, they have a semi-major axis equal to the radius of the original circle).

Degenerate Ellipses

Sometimes we encounter situations where things become, in a very real sense, not what we think they should be. The straight line segment that is considered to be an ellipse of eccentricity one is an example of such a situation. Ellipses are curves, and a straight line segment is not a curve, so it is hard to imagine calling it an ellipse of any sort. However, since we can create such an "ellipse" in the same way as any other ellipse, it seems fair to consider it a special case, in the same way that a circle is a special case; and to indicate this we give it a special name -- a degenerate ellipse. The word degenerate does not, in physical science or mathematics, refer to some kind of moral failing; it means that by pushing something to an extreme limit it "degenerates" from something complex to something much simpler, and at first glance not at all the same.

An ellipse can also be created by attaching a string to two thumbtacks and stretching the string as taut as possible. The resulting curve is the set of all points for which the sum of the distances to the two foci (the plural of focus) is a constant. The length of the string is equal to the sum of the distances involved, and is also equal to the major axis of the resulting ellipse (proof to follow). If the two foci are in the same place the curve that results is a circle centered on the foci (drawing to follow); if the two foci are so far apart that the string just barely stretches between them the curve is a straight line (a degenerate ellipse) with the foci at the two ends of the line. The closer the foci are to each other, the more nearly circular the ellipse appears; and the further apart they are, the more elongated it appears. (more diagrams to follow)