Introduction to Conic Sections

What Are Conic Sections?

Conic sections are obtained by the intersection of the surface of a cone with a plane, and have certain features.

Learning Objectives

Describe the parts of a conic section and how conic sections can be thought of as cross-sections of a double-cone

Key Takeaways

Key Points

  • A conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane; the three types are parabolas, ellipses, and hyperbolas.
  • A conic section can be graphed on a coordinate plane.
  • Every conic section has certain features, including at least one focus and directrix. Parabolas have one focus and directrix, while ellipses and hyperbolas have two of each.
  • A conic section is the set of points [latex]P[/latex] whose
    distance to the focus is a constant multiple of the distance from [latex]P[/latex] to the directrix of the conic.

Key Terms

  • vertex: An extreme point on a conic section.
  • asymptote: A straight line which a curve approaches arbitrarily closely as it goes to infinity.
  • locus: The set of all points whose coordinates satisfy a given equation or condition.
  • focus: A point used to construct and define a conic section, at which rays reflected from the curve converge (plural: foci).
  • nappe: One half of a double cone.
  • conic section: Any curve formed by the intersection of a plane with a cone of two nappes.
  • directrix: A line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two (plural: directrices).

Defining Conic Sections

A conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic sections are the hyperbola, the parabola, and the ellipse. The circle is type of ellipse, and is sometimes considered to be a fourth type of conic section.

Conic sections can be generated by intersecting a plane with a cone. A cone has two identically shaped parts called nappes. One nappe is what most people mean by “cone,” and has the shape of a party hat.

Conic sections are generated by the intersection of a plane with a cone. If the plane is parallel to the axis of revolution (the [latex]y[/latex]-axis), then the conic section is a hyperbola. If the plane is parallel to the generating line, the conic section is a parabola. If the plane is perpendicular to the axis of revolution, the conic section is a circle. If the plane intersects one nappe at an angle to the axis (other than [latex]90^{\circ}[/latex]), then the conic section is an ellipse.

An hourglass-shaped double cone (two nappes). A horizontal plane through the cone makes a circle. A plane at an angle less than the edge of the cone makes an ellipse. A plane at the angle of the edge of the cone makes a parabola. A plane at an angle greater than the edge of the cone makes a hyperbola.

A cone and conic sections: The nappes and the four conic sections. Each conic is determined by the angle the plane makes with the axis of the cone.

Common Parts of Conic Sections

While each type of conic section looks very different, they have some features in common. For example, each type has at least one focus and directrix.

A focus is a point about which the conic section is constructed. In other words, it is a point about which rays reflected from the curve converge. A parabola has one focus about which the shape is constructed; an ellipse and hyperbola have two.

A directrix is a line used to construct and define a conic section. The distance of a directrix from a point on the conic section has a constant ratio to the distance from that point to the focus. As with the focus, a parabola has one directrix, while ellipses and hyperbolas have two.

These properties that the conic sections share are often presented as the following definition, which will be developed further in the following section. A conic section is the locus of points [latex]P[/latex] whose distance to the focus is a constant multiple of the distance from [latex]P[/latex] to the directrix of the conic. These distances are displayed as orange lines for each conic section in the following diagram.

An ellipse is an oblong closed shape and has one focus inside the ellipse to one side along its long axis and its directrix outside it perpendicular to the long axis of the ellipse. A parabola has a focus inside the parabola along its axis of symmetry and a directrix outside it perpendicular to the axis of symmetry. A hyperbola is comprised of two symmetrical open curves with similar shapes to parabolas, but whose branches go towards lines instead of increasing in slope forever. It has a focus inside each curve along the axis of symmetry and a directrix outside each curve perpendicular to the axis of symmetry.

Parts of conic sections: The three conic sections with foci and directrices labeled.

Each type of conic section is described in greater detail below.

Parabola

A parabola is the set of all points whose distance from a fixed point, called the focus, is equal to the distance from a fixed line, called the directrix. The point halfway between the focus and the directrix is called the vertex of the parabola.

In the next figure, four parabolas are graphed as they appear on the coordinate plane. They may open up, down, to the left, or to the right.

Parabola opening up with focus inside and directrix below. Parabola opening down with focus inside and directrix above. Parabola opening to the left with focus inside and directrix to the right. Parabola opening to the right with focus inside and directrix to the left.

Four parabolas, opening in various directions: The vertex lies at the midpoint between the directrix and the focus.

Ellipses

An ellipse is the set of all points for which the sum of the distances from two fixed points (the foci) is constant. In the case of an ellipse, there are two foci, and two directrices.

In the next figure, a typical ellipse is graphed as it appears on the coordinate plane.

An ellipse is a closed oblong shape with a long axis and a short axis. The foci lie along the long axis inside the shape.

Ellipse: The sum of the distances from any point on the ellipse to the foci is constant.

Hyperbolas

A hyperbola is the set of all points where the difference between their distances from two fixed points (the foci) is constant. In the case of a hyperbola, there are two foci and two directrices. Hyperbolas also have two asymptotes.

A graph of a typical hyperbola appears in the next figure.

A hyperbola is comprised of two open curves shaped somewhat like parabolas. They are opposite each other and open away from each other. The shape has two axes of symmetry, one through the vertices of the curve, called the transverse axis, and one perpendicular to the transverse axis between the curves, called the conjugate axes. The curves open out with their branches tending towards oblique asymptotes, unlike a parabola. The foci are inside the curves along the transverse axis.

Hyperbola: The difference of the distances from any point on the ellipse to the foci is constant. The transverse axis is also called the major axis, and the conjugate axis is also called the minor axis.

Applications of Conic Sections

Conic sections are used in many fields of study, particularly to describe shapes. For example, they are used in astronomy to describe the shapes of the orbits of objects in space. Two massive objects in space that interact according to Newton’s law of universal gravitation can move in orbits that are in the shape of conic sections. They could follow ellipses, parabolas, or hyperbolas, depending on their properties.

Eccentricity

Every conic section has a constant eccentricity that provides information about its shape.

Learning Objectives

Discuss how the eccentricity of a conic section describes its behavior

Key Takeaways

Key Points

  • Eccentricity is a parameter associated with every conic section, and can be thought
    of as a measure of how much the conic section deviates from being circular.
  • The eccentricity of a conic section is defined to be the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest directrix.
  • The value of [latex]e[/latex] can be used to determine the type of conic section. If [latex]e= 1[/latex] it is a parabola, if [latex]e < 1[/latex] it is an ellipse, and if [latex]e > 1[/latex] it is a hyperbola.

Key Terms

  • eccentricity: A parameter of a conic section that describes how much the conic section deviates from being circular.

Defining Eccentricity

The eccentricity, denoted [latex]e[/latex], is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.

The eccentricity of a conic section is defined to be the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest directrix. The value of [latex]e[/latex] is constant for any conic section. This property can be used as a general definition for conic sections. The value of [latex]e[/latex] can be used to determine the type of conic section as well:

  • If [latex]e = 1[/latex], the conic is a parabola
  • If [latex]e < 1[/latex], it is an ellipse
  • If [latex]e > 1[/latex], it is a hyperbola

The eccentricity of a circle is zero. Note that two conic sections are similar (identically shaped) if and only if they have the same eccentricity.

Recall that hyperbolas and non-circular ellipses have two foci and two associated directrices, while parabolas have one focus and one directrix. In the next figure, each type of conic section is graphed with a focus and directrix. The orange lines denote the distance between the focus and points on the conic section, as well as the distance between the same points and the directrix. These are the distances used to find the eccentricity.

The ellipse, an oblong closed shape, has a focus inside it along its longest axis with the distances to the edges of the shape radiating out from it. Each of these points has a line going from it to the directrix, a vertical line perpendicular to the longest axis of the ellipse outside the ellipse. All the lines to the directrix are parallel to each other and perpendicular to the directrix. The parabola has a focus inside it along its axis of symmetry, and a directrix outside it perpendicular to the axis of symmetry. Lines radiate out to the parabola from it, and from those points to the directrix perpendicularly. The hyperbola is another open curve that goes out towards linear asymptotes. It has an axis of symmetry down the middle like a parabola, with a focus inside the hyperbola. The directrix is outside the hyperbola perpendicular to the axis of symmetry. Lines radiate from the focus to the hyperbola, and from those points to the directrix perpendicularly.

Conic sections and their parts: Eccentricity is the ratio between the distance from any point on the conic section to its focus, and the perpendicular distance from that point to the nearest directrix.

Conceptualizing Eccentricity

From the definition of a parabola, the distance from any point on the parabola to the focus is equal to the distance from that same point to the directrix. Therefore, by definition, the eccentricity of a parabola must be [latex]1[/latex].

For an ellipse, the eccentricity is less than [latex]1[/latex]. This means that, in the ratio that defines eccentricity, the numerator is less than the denominator. In other words, the distance between a point on a conic section and its focus is less than the distance between that point and the nearest directrix.

Conversely, the eccentricity of a hyperbola is greater than [latex]1[/latex]. This indicates that the distance between a point on a conic section the nearest directrix is less than the distance between that point and the focus.

Types of Conic Sections

Conic sections are formed by the intersection of a plane with a cone, and their properties depend on how this intersection occurs.

Learning Objectives

Discuss the properties of different types of conic sections

Key Takeaways

Key Points

  • Conic sections are a particular type of shape formed by the intersection of a plane and a right circular cone. Depending on the angle between the plane and the cone, four different intersection shapes can be formed.
  • The types of conic sections are circles, ellipses, hyperbolas, and parabolas.
  • Each conic section also has a degenerate form; these take the form of points and lines.

Key Terms

  • degenerate: A conic section which does not fit the standard form of equation.
  • asymptote: A line which a curved function or shape approaches but never touches.
  • hyperbola: The conic section formed by the plane being perpendicular to the base of the cone.
  • focus: A point away from a curved line, around which the curve bends.
  • circle: The conic section formed by the plane being parallel to the base of the cone.
  • ellipse: The conic section formed by the plane being at an angle to the base of the cone.
  • eccentricity: A dimensionless parameter characterizing the shape of a conic section.
  • Parabola: The conic section formed by the plane being parallel to the cone.
  • vertex: The turning point of a curved shape.

Conic sections are a particular type of shape formed by the intersection of a plane and a right circular cone. Depending on the angle between the plane and the cone, four different intersection shapes can be formed. Each shape also has a degenerate form. There is a property of all conic sections called eccentricity, which takes the form of a numerical parameter [latex]e[/latex]. The four conic section shapes each have different values of [latex]e[/latex].

Three double cones (two cones touching each other at their points, like an hourglass) with planes crossing through them. 1 has the plane perpendicuar to the edge of one cone, making a parabola. 2 has two planes crossing through it at angles less than the edge of the cone, making one closed ellipse and one circle when the plane is horizontal. 3 has a plane crossing through the cone vertically, making a hyperbola

Types of conic sections: This figure shows how the conic sections, in light blue, are the result of a plane intersecting a cone. Image 1 shows a parabola, image 2 shows a circle (bottom) and an ellipse (top), and image 3 shows a hyperbola.

Parabola

A parabola is formed when the plane is parallel to the surface of the cone, resulting in a U-shaped curve that lies on the plane. Every parabola has certain features:

  • A vertex, which is the point at which the curve turns around
  • A focus, which is a point not on the curve about which the curve bends
  • An axis of symmetry, which is a line connecting the vertex and the focus which divides the parabola into two equal halves

All parabolas possess an eccentricity value [latex]e=1[/latex]. As a direct result of having the same eccentricity, all parabolas are similar, meaning that any parabola can be transformed into any other with a change of position and scaling. The degenerate case of a parabola is when the plane just barely touches the outside surface of the cone, meaning that it is tangent to the cone. This creates a straight line intersection out of the cone’s diagonal.

Non-degenerate parabolas can be represented with quadratic functions such as

[latex]f(x) = x^2[/latex]

Circle

A circle is formed when the plane is parallel to the base of the cone. Its intersection with the cone is therefore a set of points equidistant from a common point (the central axis of the cone), which meets the definition of a circle. All circles have certain features:

  • A center point
  • A radius, which the distance from any point on the circle to the center point

All circles have an eccentricity [latex]e=0[/latex]. Thus, like the parabola, all circles are similar and can be transformed into one another. On a coordinate plane, the general form of the equation of the circle is

[latex](x-h)^2 + (y-k)^2 = r^2[/latex]

where [latex](h,k)[/latex] are the coordinates of the center of the circle, and [latex]r[/latex] is the radius.

The degenerate form of the circle occurs when the plane only intersects the very tip of the cone. This is a single point intersection, or equivalently a circle of zero radius.

image

Conic sections graphed by eccentricity: This graph shows an ellipse in red, with an example eccentricity value of [latex]0.5[/latex], a parabola in green with the required eccentricity of [latex]1[/latex], and a hyperbola in blue with an example eccentricity of [latex]2[/latex]. It also shows one of the degenerate hyperbola cases, the straight black line, corresponding to infinite eccentricity. The circle is on the inside of the parabola, which is on the inside of one side of the hyperbola, which has the horizontal line below it. In this way, increasing eccentricity can be identified with a kind of unfolding or opening up of the conic section.

Ellipse

When the plane’s angle relative to the cone is between the outside surface of the cone and the base of the cone, the resulting intersection is an ellipse. The definition of an ellipse includes being parallel to the base of the cone as well, so all circles are a special case of the ellipse. Ellipses have these features:

  • A major axis, which is the longest width across the ellipse
  • A minor axis, which is the shortest width across the ellipse
  • A center, which is the intersection of the two axes
  • Two focal points —for any point on the ellipse, the sum of the distances to both focal points is a constant

Ellipses can have a range of eccentricity values: [latex]0 \leq e < 1[/latex]. Notice that the value [latex]0[/latex] is included (a circle), but the value [latex]1[/latex] is not included (that would be a parabola). Since there is a range of eccentricity values, not all ellipses are similar. The general form of the equation of an ellipse with major axis parallel to the x-axis is:

[latex]\displaystyle{ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 }[/latex]

where [latex](h,k)[/latex] are the coordinates of the center, [latex]2a[/latex] is the length of the major axis, and [latex]2b[/latex] is the length of the minor axis. If the ellipse has a vertical major axis, the [latex]a[/latex] and [latex]b[/latex] labels will switch places.

The degenerate form of an ellipse is a point, or circle of zero radius, just as it was for the circle.

Hyperbola

A hyperbola is formed when the plane is parallel to the cone’s central axis, meaning it intersects both parts of the double cone. Hyperbolas have two branches, as well as these features:

  • Asymptote lines—these are two linear graphs that the curve of the hyperbola approaches, but never touches
  • A center, which is the intersection of the asymptotes
  • Two focal points, around which each of the two branches bend
  • Two vertices, one for each branch

The general equation for a hyperbola with vertices on a horizontal line is:

[latex]\displaystyle{ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 }[/latex]

where [latex](h,k)[/latex] are the coordinates of the center. Unlike an ellipse, [latex]a[/latex] is not necessarily the larger axis number. It is the axis length connecting the two vertices.

The eccentricity of a hyperbola is restricted to [latex]e > 1[/latex], and has no upper bound. If the eccentricity is allowed to go to the limit of [latex]+\infty[/latex] (positive infinity), the hyperbola becomes one of its degenerate cases—a straight line. The other degenerate case for a hyperbola is to become its two straight-line asymptotes. This happens when the plane intersects the apex of the double cone.