:* The following pictures show examples in which the circle x < sup > 2 </ sup >+ y < sup > 2 </ sup >- 1 = 0 meets another ellipse in fewer intersection points because at least one of them has multiplicity greater than 1:

This is a curve very similar to the spitzer ogive, except that the circular arc is replaced by an ellipse defined in such a way that it meets the axis at exactly 90 °.

Anthemius assumes a property of an ellipse not found in Apollonius's work, that the equality of the angles subtended at a focus by two tangents drawn from a point, and having given the focus and a double ordinate he goes on to use the focus and directrix to obtain any number of points on a parabola — the first instance on record of the practical use of the directrix.

The distance between antipodal points on the ellipse, or pairs of points whose midpoint is at the center of the ellipse, is maximum along the major axis or transverse diameter, and a minimum along the perpendicular minor axis or conjugate diameter.

If we draw an ellipse twice as long as it is wide, and draw the circle centered at the ellipse's center with diameter equal to the ellipse's longer axis, then on any line parallel to the shorter axis the length within the circle is twice the length within the ellipse.

A parabola is an ellipse that is tangent to the line at infinity Ω, and the hyperbola is an ellipse that crosses Ω.

Translation of an ellipse centered at is expressed as

For an ellipse in canonical position ( center at origin, major axis along the X-axis ), the equation simplifies to

Formulae connecting a tangential angle, the angle anchored at the ellipse's center ( called also the polar angle from the ellipse center ), and the parametric angle t < ref > If the ellipse is illustrated as a meridional one for the earth, the tangential angle is equal to geodetic latitude, the angle is the geocentric latitude, and parametric angle t is a parametric ( or reduced ) latitude of auxiliary circle </ ref > are:

In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate measured from the major axis, the ellipse's equation is

In the slightly more general case of an ellipse with one focus at the origin and the other focus at angular coordinate, the polar form is

The following equation on the polar coordinates ( r, θ ) describes a general ellipse with semidiameters a and b, centered at a point ( r < sub > 0 </ sub >, θ < sub > 0 </ sub >), with the a axis rotated by φ relative to the polar axis:

A conjugate axis of length 2b, corresponding to the minor axis of an ellipse, is sometimes drawn on the non-transverse principal axis ; its endpoints ± b lie on the minor axis at the height of the asymptotes over / under the hyperbola's vertices.

25 minutes — sub-orbital spaceflight in an elliptic flightpath ; the flightpath is part of an ellipse with a vertical major axis ; the apogee ( halfway through the midcourse phase ) is at an altitude of approximately 1, 200 km ; the semi-major axis is between 3, 186 km and 6, 372 km ; the projection of the flightpath on the Earth's surface is close to a great circle, slightly displaced due to earth rotation during the time of flight ; the missile may release several independent warheads, and penetration aids such as metallic-coated balloons, aluminum chaff, and full-scale warhead decoys.

After approximately 40 failed attempts, in early 1605 he at last hit upon the idea of an ellipse, which he had previously assumed to be too simple a solution for earlier astronomers to have overlooked.

# The orbit of every planet is an ellipse with the Sun at one of the two foci.

:" The orbit of every planet is an ellipse with the Sun at one of the two foci.

Note that the Sun is not at the center of the ellipse, but at one of its foci.

In this sense, a parabola may be considered an ellipse that has one focus at infinity.

) This is similar to saying that a parabola is an ellipse, but with one focal point at infinity.

After a sufficient number of observations are recorded over a period of time, they are plotted in polar coordinates with the primary star at the origin, and the most probable ellipse is drawn through these points such that the Keplerian law of areas is satisfied.

An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant.

The foci of the ellipse are two special points F < sub > 1 </ sub > and F < sub > 2 </ sub > on the ellipse's major axis and are equidistant from the center point.

The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis ( PF < sub > 1 </ sub > + PF < sub > 2 </ sub > = 2a ).

Each of these two points is called a focus of the ellipse.

The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the two foci, to the length of the major axis or e = 2f / 2a = f / a.

Drawing an ellipse with two pins, a loop, and a pen.

The characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to a method of drawing one using two drawing pins, a length of string, and a pencil.

Using two pegs and a rope, this procedure is traditionally used by gardeners to outline an elliptical flower bed ; thus it is called the gardener's ellipse.

: Draw two perpendicular lines M, N on the paper ; these will be the major and minor axes of the ellipse.

In the parallelogram method, an ellipse is constructed point by point using equally spaced points on two horizontal lines and equally spaced points on two vertical lines.

In Euclidean geometry, the ellipse is usually defined as the bounded case of a conic section, or as the set of points such that the sum of the distances to two fixed points ( the foci ) is constant.

The ratio of these two distances is the eccentricity of the ellipse.

In projective geometry, an ellipse can be defined as the set of all points of intersection between corresponding lines of two pencils of lines which are related by a projective map.

By projective duality, an ellipse can be defined also as the envelope of all lines that connect corresponding points of two lines which are related by a projective map.

For comparison, the other two general conic sections, the ellipse and the parabola, derive from the corresponding Greek words for " deficient " and " comparable "; these terms may refer to the eccentricity of these curves, which is greater than one ( hyperbola ), less than one ( ellipse ) and exactly one ( parabola ), respectively.

In this way, an ellipse becomes a parabola when a focus moves toward infinity, and when two foci of an ellipse merge into one another, a circle is formed.

* In conic sections, it is said of two ellipses, two hyperbolas, or an ellipse and a hyperbola which share both foci with each other.

Such calculations generally result in an elliptical path on a plane defined by some point on the orbit, and the two foci of the ellipse.

Note that over the complex numbers factors as and is degenerate because reducible, while defines a non-degenerate curve, an ellipse / hyperbola ( these are not distinct over the complex numbers, because there is no sense of positive or negative ).

Mz 3 is a complex system composed of three nested pairs of bipolar lobes and an equatorial ellipse.