The
ellipse is a closed curve with two foci located
inside the curve. It has major and minor axes equal to the
length and width of the ellipse.
The
parabola is an open curve. It has one focus and
one axis, located on the line drawn between the vertex and the
focus.
The
hyperbola has two open curves. They are mirror
images of each other. There are two asymptotes with slopes =
+-(b / a). The hyperbola has two foci, each located inside
its own curve. It has major and a minor axes located between
the two curves.
Rectangular Coordinates - ellipse and hyperbola
with center C(x
_{o}, y
_{o}) and major axis
parallel to the x-axis, parabola with axis parallel to the
x-axis.
- Ellipse: (x - x_{o})^{2} /
a^{2} + (y - y_{o})^{2} /
b^{2} = 1, 2a = major axis, 2b = minor axis
- Parabola: (y - yo )^{2} = 4a (x -
x_{o}), parabola opening to the right, a is a
constant > 0
(y - y_{o})^{2} = -4a (x -
x_{o}), parabola opening to the left, a is a
constant > 0
- Hyperbola: (x - x_{o})^{2} /
a^{2} - (y - y_{o})^{2} /
b^{2} = 1, 2a = major axis and 2b = minor
axis
Polar Coordinates - ellipse and hyperbola with
center C(x
_{o}, y
_{o}) and major axis parallel
to the x-axis, parabola with axis parallel to the x-axis.
- Ellipse: r^{2} = (a^{2} b^{2})
/ (a^{2} sin^{2} θ + b^{2}
cos^{2} θ), the center is at the origin, 2a
= major axis, 2b = minor axis
r = (a(1 - ε^{2})) / (1 - ε cos
θ), the center is on the x-axis and one focus is at
the origin
- Parabola: r = 2a / (1 - cos θ), the focus is at
the origin
- Hyperbola: r^{2} = (a^{2}
b^{2}) / (a^{2} sin^{2} θ -
b^{2} cos2 θ), the center is at the origin,
2a = major axis, 2b = minor axis
r = (a(ε^{2} - 1)) / (1 - ε cos
θ), the center is on the x-axis and one focus is at
the origin
From Schaum's Outline Series
THE MATHEMATICAL HANDBOOK of
FORMULAS and TABLES.