Ellipse

An ellipse can be defined geometrically as a set or locus of points in the Euclidean plane:

Using Dandelin spheres, one can prove that any section of a cone with a plane is an ellipse, assuming the plane does not contain the apex and has slope less than that of the lines on the cone.

The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of the ellipse, the x-axis is the major axis, and:

It follows from the equation that the ellipse is symmetric with respect to the coordinate axes and hence with respect to the origin.

Conversely, the canonical form parameters can be obtained from the general form coefficients by the equations:

This description of the tangents of an ellipse is an essential tool for the determination of the orthoptic of an ellipse. The orthoptic article contains another proof, without differential calculus and trigonometric formulae.

point to two conjugate points and the tools developed above are applicable.

The converse is also true and can be used to define an ellipse (in a manner similar to the definition of a parabola):

Ellipse: the tangent bisects the supplementary angle of the angle between the lines to the foci.
Rays from one focus reflect off the ellipse to pass through the other focus.

Because the tangent is perpendicular to the normal, the statement is true for the tangent and the supplementary angle of the angle between the lines to the foci (see diagram), too.

The rays from one focus are reflected by the ellipse to the second focus. This property has optical and acoustic applications similar to the reflective property of a parabola (see whispering gallery).

Orthogonal diameters of a circle with a square of tangents, midpoints of parallel chords and an affine image, which is an ellipse with conjugate diameters, a parallelogram of tangents and midpoints of chords.

An affine transformation preserves parallelism and midpoints of line segments, so this property is true for any ellipse. (Note that the parallel chords and the diameter are no longer orthogonal.)

Conjugate diameters in an ellipse generalize orthogonal diameters in a circle.

If there is no ellipsograph available, one can draw an ellipse using an .

For any method described below, knowledge of the axes and the semi-axes is necessary (or equivalently: the foci and the semi-major axis). If this presumption is not fulfilled one has to know at least two conjugate diameters. With help of Rytz's construction the axes and semi-axes can be retrieved.

This method is the base for several ellipsographs (see section below).

Similar to the variation of the paper strip method 1 a variation of the paper strip method 2 can be established (see diagram) by cutting the part between the axes into halves.

Most ellipsograph drafting instruments are based on the second paperstrip method.

The following method to construct single points of an ellipse relies on the Steiner generation of a conic section:

Steiner generation can also be defined for hyperbolas and parabolas. It is sometimes called a parallelogram method because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.

Usually one measures inscribed angles by a degree or radian θ, but here the following measurement is more convenient:

At first the measure is available only for chords not parallel to the y-axis, but the final formula works for any chord.

The radius is the distance between any of the three points and the center.

Like a circle, such an ellipse is determined by three points not on a line.

At first the measure is available only for chords which are not parallel to the y-axis. But the final formula works for any chord. The proof follows from a straightforward calculation. For the direction of proof given that the points are on an ellipse, one can assume that the center of the ellipse is the origin.

Analogously to the circle case, the equation can be written more clearly using vectors:

By calculation one can confirm the following properties of the pole-polar relation of the ellipse:

Wave pattern of a little droplet dropped into mercury in one focus of the ellipse

It is beneficial to use a parametric formulation in computer graphics because the density of points is greatest where there is the most curvature. Thus, the change in slope between each successive point is small, reducing the apparent "jaggedness" of the approximation.

It is sometimes useful to find the minimum bounding ellipse on a set of points. The ellipsoid method is quite useful for solving this problem.