Conics
Conic Sections
Conic Sections
A conic section is the plane curve formed by the intersection of a plane and a right-circular, twonapped cone. Such a cone is shown in Figure 1.
The cone is the surface formed by all the lines passing through a circle and a point. The point must lie on a line, called the axis, which is perpendicular to the plane of the circle at the circle’s center. This point is called the vertex, and each line on the cone is called a generatrix. The two parts of the cone lying on either side of the vertex are nappes. When the intersecting plane is perpendicular to the axis, the conic section is a circle (Figure 2).
When the intersecting plane is tilted and cuts completely across one of the nappes, the section is an oval called an ellipse (Figure 3).
When the intersecting plane is parallel to one of the generatrices, it cuts only one nappe. The section is an open curve called a parabola (Figure 4).
When the intersecting plane cuts both nappes, the section is a hyperbola, a curve with two parts, called branches (Figure 5).
All these sections are curved. If the intersecting plane passes through the vertex, however, the section will be a single point, a single line, of a pair of crossed lines. Such sections are of minor importance and are known as “degenerate” conic sections.
Since ancient times, mathematicians have known that conic sections can be defined in ways that have no obvious connection with conic sections. One set of ways is the following:
Ellipse: The set of points P such that PF_{1} + PF_{2} equals a constant and F_{1} and F_{2} are fixed points called the foci (Figure 6).
Parabola: The set of points P such that PD= PF, where F is a fixed point called the focus and D is the foot of the perpendicular from P to a fixed line called the directrix (Figure 7).
Hyperbola: The set of points P such that PF_{1} – PF_{2} equals a constant and F_{1} and F_{2} are fixed points called the foci (Figure 8).
If P, F, and D are shown as in Figure 7, then the set of points P satisfying the equation PF/PD= e where e is a constant, is a conic section. If 0 < e < 1, then the section is an ellipse. If e= 1, then the section is a parabola. If e > 1, then the section is a hyperbola. The constant e is called the eccentricity of the conic section.
Because the ratio PF/PD is not changed by a change in the scale used to measure PF and PD, all
conic sections having the same eccentricity are geometrically similar.
Conic sections can also be defined analytically, that is, as points (x, y) that satisfy a suitable equation. An interesting way to accomplish this is to start with a suitably placed cone in coordinate space. A cone with its vertex at the origin and with its axis coinciding with the z-axis has the equation x^{2}+ y^{2}– kz^{2}= 0. The equation of a plane in space is ax+ by+ cz+ d= 0. If
one uses substitution to eliminate z from these equations, and combines like terms, the result is an equation of the form Ax^{2}+ Bxy+ Cy^{2}+ Dx+ Ey+ F= 0 where at least one of the coefficients A, B, and C will be different from zero.
For example if the cone x^{2}+ y^{2}– z^{2}= 0 is cut by the plane y+ z– 2= 0, the points common to both must satisfy the equation x^{2}+ 4y– 4= 0, which can be simplified by a translation of axes to x^{2}+ 4y= 0. Because, in this example, the plane is parallel to one of the generatrices of the cone, the section is a parabola (Figure 9).
One can follow this procedure with other intersecting planes. The plane z– 5= 0 produces the circle x^{2}+ y^{2}– 25= 0. The planes y+ 2z– 2= 0and 2y+ z– 2= 0 produce the ellipse 12x^{2}+ 9y^{2}– 16= 0 and the hyperbola 3x^{2}– 9y^{2}+ 4= 0 respectively (after a simplifying translation of the axes). These planes, looking down the x-axis, are shown in Figure 10.
As these examples illustrate, suitably placed conic sections have equations which can be put into the following forms:
Circle: x^{2} + y^{2} = r^{2}
Ellipse: A^{2}x^{2} + B^{2}y^{2} = C^{2}
Parabola: y = Kx^{2}
Hyperbola: A^{2}x^{2} − B^{2}y^{2} = + C^{2}
The equations above are “suitably placed.” When the equation is not in one of the forms above, it can be hard to tell exactly what kind of conic section the
equation represents. There is a simple test, however, which can do this. With the equation written Ax^{2}+ Bxy+ Cy^{2}+ Dx+ Ey+ F= 0, the discriminant B^{2}– 4AC will identify which conic section it is. If the discriminant is positive, the section is a hyperbola; if it is negative, the section is an ellipse; if it is zero, the section is a parabola. The discriminant will not distinguish between a proper conic section and a degenerate one such as x^{2}– y^{2}= 0; it will not distinguish between an equation that has real roots and one, such as x^{2}+ y^{2}+ 1= 0, that does not.
Students who are familiar with the quadratic formula will recognize the discriminant, and with good reason. It has to do with finding the points where the conic section crosses the line at infinity. If the discriminant is negative, there will be no solution, which is consistent with the fact that both circles and ellipses lie entirely within the finite part of the plane. Parabolas lead to a single root and are tangent to the line at infinity. Hyperbolas lead to two roots and cross it in two places.
Conic sections can also be described with polar coordinates. To do this most easily, one uses the focus-directrix definitions, placing the focus at the origin and the directrix at x = −k (in rectangular coordinates). Then the polar equation is r = Ke/(1− ecos θ) where e is the eccentricity (Figure 11).
The eccentricity in this equation is numerically equal to the eccentricity given by another ratio: CF/CV, where CF represents the distance from the geometric center of the conic section to the focus, and CV the distance from the center to the vertex. In the case of a circle, the center and the foci are the same; so CF and the eccentricity are both zero. In the case of the ellipse, the vertices are end points of the major axis, hence farther from the center than the foci. CV is therefore bigger than CF, and the eccentricity is less than 1. In the case of the hyperbola, the vertices lie on the transverse axis, between the foci, hence the eccentricity is greater than 1. In the case of the parabola, the “center” is infinitely far from both the focus and the vertex; so (for those who have a good imagination) the ratio CF/CV is 1.
KEY TERMS
Conic section— A figure that results from the intersection of a right circular cone with a plane. Conic sections are the circle, ellipse, parabola, and hyperbola.
Directrix— A line that, together with a focus, determines the shape of a conic section.
Eccentricity— The center-to-focus/center-to-vertex ratio in a conic section; or, the ratio distance-tofocus/distance-to-directrix, which is the same for all points on a conic section. These two definitions are mathematically equivalent.
Focus— A point, or one of a pair of points, whose position determines the shape of a conic section.
Resources
BOOKS
Finney, Ross L., et al. Calculus: Graphical, Numerical, Algebraic of a Single Variable. Reading, MA: Addison Wesley Publishing Co., 1994.
Gullberg, Jan, and Peter Hilton. Mathematics: From the Birth of Numbers. W.W. Norton & Company, 1997.
OTHER
Sellers, James A. “An Introduction to Conic Sections” Krell Institute. <http://www.krellinst.org/UCES/archive/resources/conics> (accessed October 7, 2006.).
J. Paul Moulton
Conic Sections
Conic sections
A conic section is the plane curve formed by the intersection of a plane and a right-circular, two-napped cone. Such a cone is shown in Figure 1.
The cone is the surface formed by all the lines passing through a circle and a point . The point must lie on a line, called the "axis," which is perpendicular to the plane of the circle at the circle's center. The point is called the "vertex," and each line on the cone is called a "generatrix." The two parts of the cone lying on either side of the vertex are called "nappes." When the intersecting plane is perpendicular to the axis, the conic section is a circle (Figure 2).
When the intersecting plane is tilted and cuts completely across one of the nappes, the section is an oval called an ellipse (Figure 3).
When the intersecting plane is parallel to one of the generatrices, it cuts only one nappe. The section is an open curve called a parabola (Figure 4).
When the intersecting plane cuts both nappes, the section is a hyperbola , a curve with two parts, called "branches" (Figure 5).
All these sections are curved. If the intersecting plane passes through the vertex, however, the section will be a single point, a single line, of a pair of crossed lines. Such sections are of minor importance and are known as "degenerate" conic sections.
Since ancient times, mathematicians have known that conic sections can be defined in ways that have no obvious connection with conic sections. One set of ways is the following:
Ellipse: The set of points P such that PF1 + PF2 equals a constant and F1 and F2 are fixed points called the "foci" (Figure 6).
Parabola: The set of points P such that PD = PF, where F is a fixed point called the "focus" and D is the foot of the perpendicular from P to a fixed line called the "directrix" (Figure 7).
Hyperbola: The set of points P such that PF1 -PF2 equals a constant and F1 and F2 are fixed points called the "foci" (Figure 8).
If P, F, and D are shown as in Figure 7, then the set of points P satisfying the equation PF/PD = e where e is a constant, is a conic section. If 0 < e < 1, then the section is an ellipse. If e = 1, then the section is a parabola. If e > 1, then the section is a hyperbola. The constant e is called the "eccentricity" of the conic section.
Because the ratio PF/PD is not changed by a change in the scale used to measure PF and PD, all conic sections having the same eccentricity are geometrically similar.
Conic sections can also be defined analytically, that is, as points (x,y) which satisfy a suitable equation.
An interesting way to accomplish this is to start with a suitably placed cone in coordinate space. A cone with its vertex at the origin and with its axis coinciding with the z-axis has the equation x2 + y2 -kz2 = 0. The equation of a plane in space is ax + by + cz + d = 0. If one uses substitution to eliminate z from these equations, and combines like terms, the result is an equation of the form Ax2+ Bxy + Cy2 + Dx + Ey + F = 0 where at least one of the coefficients A, B, and C will be different from zero .
For example if the cone x2 + y2 -z2 = 0 is cut by the plane y + z - 2 = 0, the points common to both must satisfy the equation x2 + 4y - 4 = 0, which can be simplified by a translation of axes to x2 + 4y = 0. Because, in this example, the plane is parallel to one of the generatrices of the cone, the section is a parabola (Figure 9).
One can follow this procedure with other intersecting planes. The plane z - 5 = 0 produces the circle x2 + y2 - 25 = 0. The planes y + 2z - 2 = 0 and 2y + z - 2 = 0 produce the ellipse 12x2 + 9y2 - 16 = 0 and the hyperbola 3x2 - 9y2 + 4 = 0 respectively (after a simplifying translation of the axes). These planes, looking down the x-axis are shown in Figure 10.
As these examples illustrate, suitably placed conic sections have equations which can be put into the following forms:
The equations above are "suitably placed." When the equation is not in one of the forms above, it can be hard to tell exactly what kind of conic section the equation represents. There is a simple test, however, which can do this. With the equation written Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, the discriminant B2 - 4AC will identify which conic section it is. If the discriminant is positive, the section is a hyperbola; if it is negative , the section is an ellipse; if it is zero, the section is a parabola. The discriminant will not distinguish between a proper conic section and a degenerate one such as x2 - y2 = 0; it will not distinguish between an equation that has real roots and one, such as x2 + y2 + 1 = 0, that does not.
Students who are familiar with the quadratic formula
will recognize the discriminant, and with good reason. It has to do with finding the points where the conic section crosses the line at infinity . If the discriminant is negative, there will be no solution, which is consistent with the fact that both circles and ellipses lie entirely within the finite part of the plane. Parabolas lead to a single root and are tangent to the line at infinity. Hyperbolas lead to two roots and cross it in two places.
Conic sections can also be described with polar coordinates . To do this most easily, one uses the focus-directrix definitions, placing the focus at the origin and the directrix at x = -k (in rectangular coordinates). Then the polar equation is r = Ke/(1 - e cos θ) where e is the eccentricity (Figure 11).
The eccentricity in this equation is numerically equal to the eccentricity given by another ratio: the ratio CF/CV, where CF represents the distance from the geometric center of the conic section to the focus and CV the distance from the center to the vertex. In the case of a circle, the center and the foci are one and the same point; so CF and the eccentricity are both zero. In the case of the ellipse, the vertices are end points of the major axis, hence are farther from the center than the foci. CV is therefore bigger than CF, and the eccentricity is less than 1. In the case of the hyperbola, the vertices lie on the transverse axis, between the foci, hence the eccentricity is greater than 1. In the case of the parabola, the "center" is infinitely far from both the focus and the vertex; so (for those who have a good imagination) the ratio CF/CV is 1.
Resources
books
Finney, Ross L., et al. Calculus: Graphical, Numerical, Algebraic of a Single Variable. Reading, MA: Addison Wesley Publishing Co., 1994.
Gullberg, Jan, and Peter Hilton. Mathematics: From the Birth of Numbers. W.W. Norton & Company, 1997.
J. Paul Moulton
KEY TERMS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .- Conic section
—A conic section is a figure that results from the intersection of a right circular cone with a plane. The conic sections are the circle, ellipse, parabola, and hyperbola.
- Directrix
—A line which, together with a focus, determines the shape of a conic section.
- Eccentricity
—The ratio center-to-focus/center-to-vertex in a conic section; or the ratio distance-tofocus/distance-to-directrix, which is the same for all points on a conic section. These two definitions are mathematically equivalent.
- Focus
—A point, or one of a pair of points, whose position determines the shape of a conic section.
Conic Sections
Conic Sections
Imagine there are two cone-shaped paper drinking cups, each fastened to the other at its point, or vertex. The figure that would result is described mathematically as a right circular cone (sometimes called a double cone), which is formed by a straight line that moves around the circumference of a circle while passing through a fixed point (the vertex) that is not in the plane of the circle.
If a right circular cone is cut, or intersected, by a plane at different locations, the intersections form a family of plane curves called conic sections (see the figure). If the intersecting plane is parallel to the base of the cone, the intersection is a circle—which shrinks to a point when the plane has moved toward the cone's tip and finally passes through the vertex. If the intersecting plane is not parallel to the base, passes through only one half of the cone, and is not parallel to the side of the cone, then the intersection is an ellipse . If the plane intersects both halves of the cone, and is not parallel
EQUATIONS FOR CONIC SECTION CURVES | |||
Curve | General Equation | Notes | Example |
Circle | (x − h)^{2} + (y − k)^{2} = r^{2} | center of circle = (h, k ) radius = r | x^{2} + y^{2} = 49 |
Ellipse | length of major axis = 2a foci at c and −c b ^{2} = a ^{2} − c ^{2} center = (h, k ) |
x^{2} + 25y^{2} = 49 | |
Hyperbola | foci at c and −c a ^{2} + b ^{2} = c ^{2} center = (h, k ) equation of asymptotes: y = ±(b /a )x |
||
Parabola | a(x − h)^{2} + k = y | axis of symmetry: x = h vertex = (h, k ) |
(x − 7)^{2} + 1 = y |
to the side of the cone, then the intersection is a curve that has two branches, called a hyperbola . If the plane intersects the cone so that the plane is parallel to the side of the cone, then the intersection is a curve called a parabola . The equations for the conic section curves can have the general forms summarized in the table.
These conic section curves—the circle, ellipse, hyperbola, and parabola—have been known and named for more than 2,000 years, and they occur in many applications. The path of a thrown ball, the arc of cables that support a bridge, and the arc of a fountain are examples of parabolas. The shape of a sonic boom, the path of comets, and the LORAN navigation system for ships involve the hyperbola. The paths traveled by the planets, dome-shaped ceilings, and the location of the listening points in a whispering gallery involve the ellipse. Of course, examples of the most familiar conic section—the circle—can be found everywhere: the shape of flowers, ripples in a pool, and water-worn stones.
see also Locus.
Lucia McKay
Bibliography
Jacobs, Harold R. Mathematics, A Human Endeavor. San Francisco: W.H. Freeman and Company, 1970.
FOCI
For every point on an ellipse, the sum of the distances from that point to two points called the foci is the same. These two points, the foci, are on the major (longest) axis of the ellipse. The value of this constant sum is 2a.
For every point on a hyperbola, the absolute value of the difference of the distances from that point to two points called the foci is the same. The absolute value of this constant difference is 2a.