Conics (Under Construction)

Conics (Conic Sections, in full) are defined as the intersection of a plane and a cone (and, at a simple level, we just consider the four shapes obtained when the plane does not pass thru the vertex of the cone).

Alternatively : - If S is a fixed point and l is a fixed line which does not pass thru S, a conic is the locus of points such that the ratio of the distance from S to the distance from l is a constant. The fixed point is the focus, and the fixed line is the directrix. The constant ratio is called the eccentricity.

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Summary of Basic Properties

	Circle	Ellipse	Parabola	Hyperbola
Standard Cartesian Equation :	x2 + y2 = r2		y2 = 4ax
Eccentricity (e):	0	0 < e <1	1	1 < e
Relation between a,b and e	b = a	b2 = a2(1-e2)		b2 = a2(e2-1)
Parametric Representation			x = at2 y = 2at	or
Definition : It is the locus of all points which meet the condition...	distance to the origin is constant	sum of distances to each focus is constant	distance to focus = distance to directrix	difference between distances to each foci is constant

It might tidy the logic up to consider a circle to be a special case of an ellipse. Then there are two 'main' classes

• an ellipse, with e < 1
• a hyperbola, with e > 1

and a 'critical' class - the parabola with e = 1.

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The General Equation of a Conic

The General Equation for a Conic is

Ax² + Bxy + Cy² + Dx + Ey + F = 0

The actual type of conic can be found from the sign of B² - 4AC

If B2 - 4AC is...	then the curve is a...
< 0	ellipse, circle, point or no curve.
= 0	parabola, 2 parallel lines, 1 line or no curve.
> 0	hyperbola or 2 intersecting lines.

note : the above notation brings a close analogy with the formulas of quadratic equations. Sometimes, however, the formula is stated slightly differently

Ax² + 2Bxy + Cy² + Dx + Ey + F = 0

Here the type of conic must be found from the sign of B² - AC

If B2 - AC is...	then the curve is a...
< 0	ellipse, circle, point or no curve.
= 0	parabola, 2 parallel lines, 1 line or no curve.
> 0	hyperbola or 2 intersecting lines.

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Polar Form

For an origin at a focus, the general polar form (apart from a circle) is

where L is the semi latus rectum.

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Ellipse

The cartesian equation of an ellipse is

where a and b would give the lengths of the semi-major and semi-minor axes.

In its general form, with the origin at the center of coordinates

• the foci are at
  

• the directrix are at
  

• the major axis of of length 2a
• the minor axis is of length 2b
• the semi latus rectum is of length
  

From the general polar form, the equation for an ellipse is

For any point P on the perimeter, the sum

PF₁ + PF₂

will be constant, no matter which point is chosen as P.

Hence, an ellipse can also be defined as the locus of a point which moves in a plane so that the sum of its distances from two fixed points is constant.

According to Kepler's First law, the orbit of a planet is an ellipse.

The Earth is shaped like an ellipsoid.

Any signal from one of the foci will pass thru the other focus.

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Hyperbola

The cartesian equation of an hyperbola is

In its general form, with the origin at the center of coordinates

• the foci are at (+/- ae, 0)
• the directrix are at x = +/- a/e
• the transverse axis of of length 2a
• the conjugate axis is of length 2b
• the semi latus rectum is of length 2b²/a

Note the similarity in notation with ellipses; although now the eccentricity is greater than one

Also by analogy with an ellipse

For any point P on a hyperbola, the sum

PF₁ - PF₂

will be constant, no matter which point is chosen as P.

Hence, a hyperbola can also be defined as the locus of a point which moves in a plane so that the difference of its distances from two fixed points is constant.

Asymptotes of Hyperbola

Rejigging the hyperbola formula to

As x becomes larger, y tends to

these are the equations of the asymptotes.

Rectangular Hyperbola

A hyperbola is rectangular if its asymptotes are perpendicular.

From

this requires

b = a

Substituting this into the cartesian formula for a hyperbola produces

x² - y² = 1

which has an eccentricity equal to the square root of 2

Rotating a rectangular so as to makes its asymptotes into the coordinates axes, changes the formula to

xy = c²

where c² = (a²/2)

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Moire Patterns

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