Introduction to Conic Sections

Conic Sections: Definition and Formation from a Double Cone

The study of conic sections (or simply conics) originated with the ancient Greek mathematician Menaechmus, who used them to solve the problem of duplicating the cube. Later, Apollonius of Perga wrote a comprehensive treatise on the subject. Conic sections are geometric curves of remarkable properties, defined as the curves obtained by intersecting a plane with a specific three-dimensional geometric figure: the double-napped right circular cone.

Double-Napped Right Circular Cone

To understand how conic sections are formed, we first need to define the geometric surface used for intersection.

Imagine a fixed vertical line, let's call it $l$. This line serves as the axis of the cone. Now, consider another line, $m$, which intersects the axis $l$ at a fixed point $V$. This point $V$ is called the vertex of the cone. The line $m$ makes a constant acute angle, denoted by $\alpha$, with the axis $l$. This angle $\alpha$ ($0^\circ < \alpha < 90^\circ$) is called the semi-vertical angle of the cone.

If we rotate the line $m$ (the generator) around the fixed axis $l$, keeping the vertex $V$ and the semi-vertical angle $\alpha$ constant, the surface generated in three-dimensional space is a double-napped right circular cone.

• Nappes: The surface consists of two identical parts joined at the vertex $V$. The part extending infinitely upwards from $V$ is called the upper nappe, and the part extending infinitely downwards from $V$ is called the lower nappe.

• Right Circular Cone: The term "right" signifies that the axis $l$ is perpendicular to the base if the cone is cut by a plane perpendicular to the axis (resulting in a circle). The term "circular" indicates that the cross-sections perpendicular to the axis are circles.

Formation by Intersection with a Plane

A conic section is the curve formed by the intersection of this double-napped cone and a plane. The type of conic section that results depends entirely on the orientation of the intersecting plane relative to the cone's axis and its semi-vertical angle $\alpha$.

Let $\beta$ be the angle between the intersecting plane and the axis of the cone ($l$). We measure $\beta$ as the acute angle between the plane and the axis, $0^\circ \le \beta \le 90^\circ$.

Assuming the plane does not pass through the vertex $V$ (which results in degenerate conics, discussed in the next subheading), the different types of non-degenerate conic sections are formed based on the comparison between the angle of the plane with the axis ($\beta$) and the semi-vertical angle of the cone ($\alpha$):

1. Circle: When the intersecting plane is perpendicular to the axis of the cone ($\beta = 90^\circ$). The plane cuts through only one nappe.

2. Ellipse: When the intersecting plane is tilted slightly and cuts through only one nappe, such that the angle the plane makes with the axis ($\beta$) is greater than the semi-vertical angle of the cone ($\alpha$), but less than $90^\circ$ ($\alpha < \beta < 90^\circ$).

3. Parabola: When the intersecting plane is parallel to the generator line ($m$). This occurs when the angle the plane makes with the axis ($\beta$) is exactly equal to the semi-vertical angle of the cone ($\alpha$). The plane cuts through only one nappe and is not parallel to the base.

4. Hyperbola: When the intersecting plane is steeper than the generator line relative to the axis. This occurs when the angle the plane makes with the axis ($\beta$) is less than the semi-vertical angle of the cone ($\alpha$) ($0 \le \beta < \alpha$). In this case, the plane cuts through both nappes of the double cone, resulting in two separate, unbounded curves (two branches).

These four curves - circle, ellipse, parabola, and hyperbola - are the non-degenerate conic sections. Their properties and equations in the Cartesian coordinate system are key areas of study in coordinate geometry.

Types of Conic Sections: Circle, Ellipse, Parabola, Hyperbola, Degenerate Conics

Conic sections are classified into different types based on the angle at which the intersecting plane cuts the double-napped right circular cone, relative to the cone's axis and semi-vertical angle ($\alpha$). The point of intersection (or lack thereof) with the vertex ($V$) also distinguishes between non-degenerate and degenerate conics.

Let $\beta$ be the angle between the intersecting plane and the axis of the cone ($0^\circ \le \beta \le 90^\circ$).

Non-Degenerate Conic Sections

These are the curves obtained when the intersecting plane cuts the cone but does not pass through the vertex ($V$). There are four main types:

1. Circle:

   • Condition: The plane intersects only one nappe and is positioned such that it is perpendicular to the axis of the cone.
   • Angle: The angle between the plane and the axis is exactly $90^\circ$ ($\beta = 90^\circ$).
   • Description: The resulting intersection curve is a circle. All points on the curve are equidistant from the axis of the cone at that specific height.
   

2. Ellipse:

   • Condition: The plane intersects only one nappe of the cone, and it is tilted such that the angle it makes with the axis is greater than the semi-vertical angle but less than $90^\circ$.
   • Angle: $\alpha < \beta < 90^\circ$.
   • Description: The resulting intersection curve is an ellipse, which is an oval shape. A circle can be considered a special case of an ellipse where the two focal points coincide.
   

3. Parabola:

   • Condition: The plane intersects only one nappe and is oriented exactly parallel to one of the generator lines of the cone.
   • Angle: The angle the plane makes with the axis is equal to the semi-vertical angle of the cone ($\beta = \alpha$).
   • Description: The resulting intersection curve is a parabola, a U-shaped curve that extends infinitely in one direction.
   

4. Hyperbola:

   • Condition: The plane intersects both nappes of the double cone.
   • Angle: The angle the plane makes with the axis is less than the semi-vertical angle of the cone ($0 \le \beta < \alpha$).
   • Description: The resulting intersection consists of two separate, unbounded curves, one from each nappe. Together, these two branches form a hyperbola.
   

Degenerate Conic Sections

These are special cases of conic sections that occur when the intersecting plane passes through the vertex ($V$) of the cone. The intersection results in simpler geometric figures:

1. Point:

   • Condition: The plane passes through the vertex $V$ and intersects the cone only at the vertex. This happens when the plane is not steep enough to cut through the cone's surface away from the vertex.
   • Angle: $\alpha < \beta \le 90^\circ$.
   • Description: The intersection is a single point, which is the vertex $V$. This can be considered a degenerate ellipse or circle.

2. Straight Line:

   • Condition: The plane passes through the vertex $V$ and is tangent to the surface of the cone.
   • Angle: $\beta = \alpha$.
   • Description: The intersection is a single straight line. This line is one of the generator lines of the cone. This can be seen as a degenerate parabola.

3. Pair of Intersecting Straight Lines:

   • Condition: The plane passes through the vertex $V$ and cuts through both nappes of the cone. The plane contains two generator lines.
   • Angle: $0 \le \beta < \alpha$.
   • Description: The intersection consists of two distinct straight lines that pass through the vertex $V$. This can be seen as a degenerate hyperbola.

Definition of a Conic Section (Focus, Directrix, Eccentricity)

While the cone intersection method provides a visual and fundamental understanding of conic sections, these curves can also be defined elegantly as a locus of points in a plane using distances to a fixed point and a fixed line. This analytical definition is particularly useful for deriving the algebraic equations of conic sections.

Definition using Focus, Directrix, and Eccentricity

A conic section is defined as the locus of all points $P$ in a plane such that the ratio of the distance from $P$ to a fixed point $F$ to the perpendicular distance from $P$ to a fixed straight line $L$ is a constant positive value. This fixed point $F$ is called the focus, and the fixed line $L$ is called the directrix. The directrix does not contain the focus ($F$ is not on $L$). The constant positive ratio is called the eccentricity of the conic section, and it is usually denoted by $e$.

Let $P(x, y)$ be any point on the conic section.

Let $F$ be the fixed focus point.

Let $L$ be the fixed directrix line.

Let $M$ be the foot of the perpendicular from point $P$ to the line $L$. So, $PM$ is the perpendicular distance from $P$ to the directrix $L$.

According to the definition, the locus of $P$ is a conic section if:

This can be written concisely as:

$\mathbf{\frac{|PF|}{|PM|} = e}$

or, equivalently,

$\mathbf{|PF| = e |PM|}$

Eccentricity and the Type of Conic Section

The value of the eccentricity $e$ plays a crucial role in determining the shape and type of the conic section formed by this locus definition. The three main types of non-degenerate conics are distinguished by the value of $e$:

1. Parabola:

   • Eccentricity: The eccentricity is exactly equal to 1.
   • $e = 1$
   • Locus Condition: $|PF| = 1 \cdot |PM| \implies \mathbf{|PF| = |PM|}$.
   • Description: A parabola is the locus of all points that are equidistant from the focus and the directrix.

2. Ellipse:

   • Eccentricity: The eccentricity is a positive value less than 1.
   • $0 < e < 1$
   • Locus Condition: $|PF| = e \cdot |PM| \implies \mathbf{|PF| < |PM|}$.
   • Description: An ellipse is the locus of all points where the distance to the focus is a constant fraction (less than 1) of the distance to the directrix. Ellipses have two foci and two corresponding directrices. The definition given here refers to one such pair.

3. Hyperbola:

   • Eccentricity: The eccentricity is a value greater than 1.
   • $e > 1$
   • Locus Condition: $|PF| = e \cdot |PM| \implies \mathbf{|PF| > |PM|}$.
   • Description: A hyperbola is the locus of all points where the distance to the focus is a constant multiple (greater than 1) of the distance to the directrix. Hyperbolas also have two foci and two corresponding directrices. The definition given here refers to one such pair.

Special Case: Circle

A circle is a special type of conic section. While it is clearly obtained from the cone intersection, its definition as a locus using a single focus and directrix is slightly problematic in the elementary sense:

• In the limit as the eccentricity $e$ approaches 0 ($e \to 0$), an ellipse becomes more and more circular.
• For a perfect circle, the eccentricity is defined as $e = 0$.
• If we plug $e=0$ into the condition $|PF| = e |PM|$, we get $|PF| = 0 \cdot |PM|$, which means $|PF| = 0$. This implies $P$ must coincide with the focus $F$. The locus is just the single point $F$. This is considered a degenerate ellipse (a point).
• The standard definition of a circle is the locus of points equidistant from a fixed point (the center). This definition does not involve a directrix in the same way. However, a circle can be considered as an ellipse where both foci coincide at the center. In this view, the directrix for a circle is a line at infinity.
• While a circle is a conic section, the focus-directrix definition with $e=0$ usually refers to the degenerate case of a point. Non-degenerate circles are commonly defined separately as the locus of points equidistant from a center, or are viewed as ellipses with $e=0$.

Axis of the Conic

For any conic section (except a circle based on the single focus-directrix definition), there is a unique line called the axis of the conic. This is the straight line that passes through the focus $F$ and is perpendicular to the directrix $L$. The vertex or vertices of the conic lie on the axis.