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Conic Sections
© Copyright 2001, Jim Loy
The simple curves of geometry are called conic sections (or
conics). That is because they can be produced by cutting flat slices out of a
right circular cone (one that has a circular base, and is not slanting). On the
left we see two cones. The first shows how we can get a circle or an
Ellipse from a cone. The second cone
shows a parabola, produced with a cut that is parallel to a slanting height of
the cone. On the right, we have a Hyperbola. The cone is defined to be in two
pieces (two nappes), two cones, nose to nose. And the hyperbola also comes in
two pieces. The straight line is also a conic section, the cut just touching
the two nose to nose cones, and going through the vertex.
The picture on the left is from my article on
Long Period Comets. Two body orbits (like a
comet around the Sun) are conic sections. The regular ones are ellipses. Add a
third body (Jupiter) and the orbit becomes much more complicated.
Conic sections have the following equation (See Analytic Geometry), where e is the eccentricity,
(0,0) is the focus, and x=-a is a line called the directrix (and here, x^2
means x squared):
x^2+y^2=e^2(x+a)^2
The eccentricity is the shape, measuring the amount that the curve
varies from a circle:
- e=0 then the curve is a circle.
- e<1, the curve is an ellipse.
- e=1, the curve is a parabola.
- e>1, the curve is a hyperbola
All circles have the same shape, just different sizes. All parabolas
have the same shape. Hyperbolas and ellipses have different shapes. Five points
on a plane uniquely determine a conic section.
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