Conic sections

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Conic sections are curves created by the intersection of a plane and a cone. There are six types of conic section: the circle, ellipse, hyperbola, parabola, a pair of intersecting straight lines and a single point.

All conics (as they are known) have at least two foci, although the two may coincide or one may be at infinity. They may also be defined as the locus of a point moving between a point and a line, a directrix, such that the ratio between the distances is constant. This ratio is known as "e", or eccentricity.

Ellipses[edit | edit source]

Animation showing distance from the foci of an ellipse to several different points on the ellipse.

An ellipse is a locus where the sum of the distances to two foci is kept constant. This sum is also equivalent to the major axis of the ellipse - the major axis being longer of the two lines of symmetry of the ellipse, running through both foci. The eccentricity of an ellipse is less than one.

In Cartesian coordinates, if an ellipse is centered at (h,k), the equation for the ellipse is

   (equation 1)

The lengths of the major and minor axes (also known as the conjugate and transverse) are "a" and "b" respectively.

Exercise 1. Derive equation 1.    (hint)

A circle circumscribed about the ellipse, touching at the two end points of the major axis, is known as the auxiliary circle. The latus rectum of an ellipse passes through the foci and is perpendicular to the major axis.

From a point P(, ) tangents will have the equation:

And normals:

Likewise for the parametric coordinates of P, (a , b ),

Properties of Ellipses[edit | edit source]

S and S' are typically regarded as the two foci of the ellipse. Where , these become (ae, 0) and (-ae, 0) respectively. Where these become (0, be) and (0, -be) respectively.

A point P on the ellipse will move about these two foci ut

Where a > b, which is to say the Ellipse will have a major-axes parallel to the x-axis:

The directrix will be:

Circles[edit | edit source]

A circle is a special type of the ellipse where the foci are the same point.

Hence, the equation becomes:

Where 'r' represents the radius. And the circle is centered at the origin (0,0)

Hyperbolas[edit | edit source]

A special case where the eccentricity of the conic shape is greater than one.

Centered at the origin, Hyperbolas have the general equation:

A point P on will move about the two foci ut

The equations for the tangent and normal to the hyperbola closely resemble that of the ellipse.

From a point P(, ) tangents will have the equation:

And normals:

The directrixes (singular directrix) and foci of hyperbolas are the same as those of ellipses, namely directrixes of and foci of

The asymptotes of a hyperbola lie at

Rectangular Hyperbolas[edit | edit source]

Rectangular Hyperbolas are special cases of hyperbolas where the asymptotes are perpendicular. These have the general equation:

Conic sections generally[edit | edit source]

Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the conic section, and also how to calculate the foci and directrices given the equation.

Deriving the equation[edit | edit source]

The curve is defined as a point whose distance to the focus and distance to a line, the directrix, have a fixed ratio, eccentricity Distance from focus to directrix must be non-zero.

Let the point have coordinates

Let the focus have coordinates

Let the directrix have equation where

Then

Square both sides:

Rearrange:

Expand simplify, gather like terms and result is:

where:

Implementation[edit | edit source]

# python code
import decimal

dD = decimal.Decimal # Decimal object is like a float with (almost) unlimited precision.
dgt = decimal.getcontext()
Precision = dgt.prec = 40

def ABCDEF_from_abc_epq (abc,epq,flag = 0) :
    '''
A,B,C,D,E,F = ABCDEF = ABCDEF_from_abc_epq (abc,epq[,flag]) 
    '''
    thisName = 'ABCDEF_from_abc_epq (abc,epq,flag = 0) :'
    a,b,c = [ dD(str(v)) for v in abc ]
    e,p,q = [ dD(str(v)) for v in epq ]
    divider = (a**2 + b**2)
    if divider != 1 :
        divider = divider.sqrt()
        a,b,c = [ (v/divider) for v in (a,b,c) ]
    distance_from_focus_to_directrix = a*p + b*q + c
    if distance_from_focus_to_directrix == 0 :
        print (thisName, 'distance_from_focus_to_directrix must be non-zero.')
        return None
    X = e*e
    A = X*a**2 - 1
    B = X*b**2 - 1
    C = 2*X*a*b
    D = 2*p + 2*X*a*c
    E = 2*q + 2*X*b*c
    F = X*c**2 - p*p - q*q
    Precision = dgt.prec = dgt.prec - 3
    A,B,C,D,E,F = [ v.normalize() for v in (A,B,C,D,E,F) ]
    Precision = dgt.prec = dgt.prec + 3
    if flag :
        str1 = '({})xx + ({})yy + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F)
        print (str1)
    return (A,B,C,D,E,F)

Examples[edit | edit source]

Parabola[edit | edit source]

Every parabola has eccentricity

Quadratic function complies with definition of parabola.
Distance from point to focus = distance from point to directrix = 10.
Distance from point to focus = distance from point to directrix = 1.

Simple quadratic function:

Let focus be point

Let directrix have equation: or

# python code

p,q = 0,1
a,b,c = abc = 0,1,q
epq = 1,p,q

ABCDEF = ABCDEF_from_abc_epq (abc,epq,1)
print ('ABCDEF =', ABCDEF)
(-1)xx + (0)yy + (0)xy + (0)x + (4)y + (0) = 0
ABCDEF = (Decimal('-1'), Decimal('0'), Decimal('0'), Decimal('0'), Decimal('4'), Decimal('0'))

As conic section curve has equation:

Curve is quadratic function: or

For a quick check select some random points on the curve:

# python code

for x in (-2,4,6) :
    y = x**2/4
    print ('\nFrom point ({}, {}):'.format(x,y))
    distance_to_focus = ((x-p)**2 + (y-q)**2)**.5
    distance_to_directrix = a*x + b*y + c
    s1 = 'distance_to_focus' ; print (s1, eval(s1))
    s1 = 'distance_to_directrix' ; print (s1, eval(s1))
From point (-2, 1.0):
distance_to_focus 2.0
distance_to_directrix 2.0

From point (4, 4.0):
distance_to_focus 5.0
distance_to_directrix 5.0

From point (6, 9.0):
distance_to_focus 10.0
distance_to_directrix 10.0

Gallery[edit | edit source]

Curve in Figure 1 below has:

  • Directrix:
  • Focus:
  • Equation: or

Curve in Figure 2 below has:

  • Directrix:
  • Focus:
  • Equation: or

Curve in Figure 3 below has:

  • Directrix:
  • Focus:
  • Equation:

Ellipse[edit | edit source]

Every ellipse has eccentricity

Ellipse with ecccentricity of 0.25 and center at origin.
Point1
Eccentricity

A simple ellipse:

Let focus be point where

Let directrix have equation: or

Let eccentricity

# python code
p,q = -1,0
e = 0.25
abc = a,b,c = 1,0,16
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
(-0.9375)xx + (-1)yy + (0)xy + (0)x + (0)y + (15) = 0

Ellipse has center at origin and equation:

Ellipses with ecccentricities from 0.1 to 0.9.
As eccentricity approaches shape of ellipse approaches shape of circle.

The effect of eccentricity.


All ellipses in diagram have:

  • Focus at point
  • Directrix with equation


Five ellipses are shown with eccentricities varying from to

Gallery[edit | edit source]

Curve in Figure 1 below has:

  • Directrix:
  • Focus:
  • Eccentricity:
  • Equation:

Curve in Figure 2 below has:

  • Directrix:
  • Focus:
  • Eccentricity:
  • Equation:

Curve in Figure 3 below has:

  • Directrix:
  • Focus:
  • Eccentricity:
  • Equation:

Hyperbola[edit | edit source]

Every hyperbola has eccentricity

Hyperbola with eccentricity of 1.5 and center at origin.
Point1
Eccentricity
For every point on curve,

A simple hyperbola:

Let focus be point where

Let directrix have equation: or

Let eccentricity

# python code
p,q = 0,-9
e = 1.5
abc = a,b,c = 0,1,4
epq = e,p,q
ABCDEF_from_abc_epq  (abc,epq,1)
(-1)xx + (1.25)yy + (0)xy + (0)x + (0)y + (-45) = 0

Hyperbola has center at origin and equation:

Some basic checking:

# python code

four_points = pt1,pt2,pt3,pt4 = (-7.5,9),(-7.5,-9),(22.5,21),(22.5,-21)
for (x,y) in four_points :
    # Verify that point is on curve.
    sum = 1.25*y**2 - x**2 - 45
    sum and 1/0 # Create exception if sum != 0.
    distance_to_focus = ( (x-p)**2 + (y-q)**2 )**.5
    distance_to_directrix = a*x + b*y + c
    e = distance_to_focus / distance_to_directrix
    s1 = 'x,y' ; print (s1, eval(s1))
    s1 = '    distance_to_focus, distance_to_directrix, e' ; print (s1, eval(s1))
x,y (-7.5, 9)
    distance_to_focus, distance_to_directrix, e (19.5, 13.0, 1.5)
x,y (-7.5, -9)
    distance_to_focus, distance_to_directrix, e (7.5, -5.0, -1.5)
x,y (22.5, 21)
    distance_to_focus, distance_to_directrix, e (37.5, 25.0, 1.5)
x,y (22.5, -21)
    distance_to_focus, distance_to_directrix, e (25.5, -17.0, -1.5)

Hyperbolas with ecccentricities from 1.5 to 20.
As eccentricity increases, curve approaches directrix:

The effect of eccentricity.


All hyperbolas in diagram have:

  • Focus at point
  • Directrix with equation


Six hyperbolas are shown with eccentricities varying from to

Gallery[edit | edit source]

Curve in Figure 1 below has:

  • Directrix:
  • Focus:
  • Eccentricity:
  • Equation:

Curve in Figure 2 below has:

  • Directrix:
  • Focus:
  • Eccentricity:
  • Equation:

Curve in Figure 3 below has:

  • Directrix:
  • Focus:
  • Eccentricity:
  • Equation:

Other resources[edit | edit source]

  • Should the contents of this Wikiversity page be merged into the related Wikibooks modules such as b:Conic Sections/Ellipse?