Ellipse

An ellipse is a special type of curved shape in mathematics. It is like a circle that has been stretched out in one direction. Imagine taking a circle and gently pulling it to make it longer on one side and shorter on the other. The two points inside the ellipse, called focal points, are important because the total distance from any point on the ellipse to both focal points always stays the same.

Ellipses are very useful in science. For example, the path that planets take around the Sun is almost like an ellipse, with the Sun at one of the focal points. This idea helps scientists understand how objects move in space. Ellipses can also be found in engineering and physics, showing up in many different areas of study.

The shape of an ellipse can be described using numbers. The longest width of the ellipse is called the major axis, and the shortest width is the minor axis. These measurements help us understand just how stretched out the ellipse is. By using simple math, we can draw and study ellipses in many different ways.

Definition as locus of points

An ellipse is a special shape in math. Imagine two fixed points called foci. For any point on the ellipse, the total distance to both foci stays the same. This makes a curved shape around the two points.

When the two foci are at the same spot, the shape becomes a circle, which is a type of ellipse. The ellipse can be stretched more or less, and this stretch is called its eccentricity.

In Cartesian coordinates

An ellipse is a special shape that looks like a stretched circle. It has two points called foci, and for every point on the ellipse, the total distance to both foci stays the same.

A circle is a special type of ellipse where both foci are in the same place. The stretch of an ellipse is measured by something called eccentricity, which is a number between 0 (a perfect circle) and 1 (a very stretched shape).

Parametric representation

Using trigonometric functions, a special way to show an ellipse is: ( x , y ) = ( a cos ⁡ t , b sin ⁡ t ) , where t is a number.

Another way to think about an ellipse is by changing a simple shape called a circle slightly. This helps us understand how ellipses are formed.

Standard parametric representation

Polar forms

Polar form relative to center

In polar coordinates, we can describe an ellipse using a special math rule. If we place the center of the ellipse at the starting point and measure angles from the longest part of the ellipse, we can write the ellipse's shape with a formula.

Polar form relative to focus

We can also describe the ellipse using polar coordinates if we place the starting point at one of the two special points inside the ellipse called foci. The angle we measure from the longest part of the ellipse helps us write another formula for the ellipse's shape.

The angle we use is called the true anomaly. The number in the formula, ℓ = a(1-e²), is called the semi-latus rectum.

Eccentricity and the directrix property

An ellipse has two special lines called directrices. For any point on the ellipse, the ratio of its distance to a focus and its distance to the nearest directrix equals the ellipse’s eccentricity — a number between 0 and 1 that measures how "stretched" the ellipse is.

Focus-to-focus reflection property

An ellipse has a special property related to its two focal points. Imagine a light beam traveling from one focus to a point on the ellipse and then to the other focus. The angle at which the light hits the ellipse will always match the angle at which it leaves, similar to how light reflects off a mirror.

This property helps explain why whispering galleries work — if two people stand at the foci of an elliptical room, they can hear each other clearly even if they're far apart, because sound waves follow the same reflection rule as light.

Orthogonal tangents

Main article: Orthoptic (geometry)

For a special shape called an ellipse, there is a special circle connected to it. This circle is called the orthoptic or director circle. It shows where the lines that cross each other at right angles touch the ellipse.

Drawing ellipses

Ellipses appear in descriptive geometry as images of circles. There are various tools to draw an ellipse. Computers provide the fastest and most accurate method for drawing an ellipse. However, technical tools to draw an ellipse without a computer exist. The principle was known to the 5th century mathematician Proclus, and the tool now known as an elliptical trammel was invented by Leonardo da Vinci.

If there is no ellipsograph available, one can draw an ellipse using an approximation by the four osculating circles at the vertices.

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.

de La Hire's point construction

The following construction of single points of an ellipse is due to de La Hire. It is based on the standard parametric representation of an ellipse:

Draw the two circles centered at the center of the ellipse with radii a, b and the axes of the ellipse.
Draw a line through the center, which intersects the two circles at point A and B, respectively.
Draw a line through A that is parallel to the minor axis and a line through B that is parallel to the major axis. These lines meet at an ellipse point P (see diagram).
Repeat steps (2) and (3) with different lines through the center.

Pins-and-string method

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. In this method, pins are pushed into the paper at two points, which become the ellipse's foci. A string is tied at each end to the two pins; its length after tying is 2a. The tip of the pencil then traces an ellipse if it is moved while keeping the string taut. Using two pegs and a rope, gardeners use this procedure to outline an elliptical flower bed—thus it is called the gardener's ellipse. The Byzantine architect Anthemius of Tralles (c. 600) described how this method could be used to construct an elliptical reflector, and it was elaborated in a now-lost 9th-century treatise by Al-Ḥasan ibn Mūsā.

A similar method for drawing confocal ellipses with a closed string is due to the Irish bishop Charles Graves.

Paper strip methods

The two following methods rely on the parametric representation:

Method 1

The first method starts with

a strip of paper of length a + b.

The point, where the semi axes meet is marked by P. If the strip slides with both ends on the axes of the desired ellipse, then point P traces the ellipse. For the proof one shows that point P has the parametric representation , where parameter t is the angle of the slope of the paper strip.

A technical realization of the motion of the paper strip can be achieved by a Tusi couple (see animation). The device is able to draw any ellipse with a fixed sum a + b, which is the radius of the large circle. This restriction may be a disadvantage in real life. More flexible is the second paper strip method.

Method 2

The second method starts with

a strip of paper of length a.

One marks the point, which divides the strip into two substrips of length b and a − b. The strip is positioned onto the axes as described in the diagram. Then the free end of the strip traces an ellipse, while the strip is moved. For the proof, one recognizes that the tracing point can be described parametrically by , where parameter t is the angle of slope of the paper strip.

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.

Approximation by osculating circles

From Metric properties below, one obtains:

The radius of curvature at the vertices V1, V2 is:
The radius of curvature at the co-vertices V3, V4 is:

The diagram shows an easy way to find the centers of curvature C1 = (a − , 0), C3 = (0, b − ) at vertex V1 and co-vertex V3, respectively:

mark the auxiliary point H = (a, b) and draw the line segment V1V3,
draw the line through H, which is perpendicular to the line V1V3,
the intersection points of this line with the axes are the centers of the osculating circles.

(proof: simple calculation.)

The centers for the remaining vertices are found by symmetry.

With help of a French curve one draws a curve, which has smooth contact to the osculating circles.

Steiner generation

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

Given two pencils of lines at two points U, V and a projective but not perspective mapping π of B(U) onto B(V), then the intersection points of corresponding lines form a non-degenerate projective conic section.

For the generation of points of the ellipse one uses the pencils at the vertices V1, V2. Let P = (0, b) be an upper co-vertex of the ellipse and A = (−a, 2b), B = (a, 2b).

P is the center of the rectangle V1, V2, B, A. The side AB of the rectangle is divided into n equal spaced line segments and this division is projected parallel with the diagonal AV2 as direction onto the line segment V1B and assign the division as shown in the diagram. The parallel projection together with the reverse of the orientation is part of the projective mapping between the pencils at V1 and V2 needed. The intersection points of any two related lines V1Bi and V2Ai are points of the uniquely defined ellipse. With help of the points C1, … the points of the second quarter of the ellipse can be determined. Analogously one obtains the points of the lower half of the ellipse.

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.

As hypotrochoid

The ellipse is a special case of the hypotrochoid when R = 2r, as shown in the adjacent image. The special case of a moving circle with radius r inside a circle with radius R = 2r is called a Tusi couple.

Inscribed angles and three-point form

Circles

A circle is a special type of shape where all points are the same distance from the center. Three points that are not in a straight line can define a unique circle. To find the center and radius of this circle, we use a special idea called the inscribed angle theorem. This theorem helps us understand how angles formed by points on the circle relate to each other.

Ellipses

Ellipses are shapes that look like stretched circles. They are defined by a special rule involving two points called foci. Just like circles, an ellipse can also be determined by three points that are not in a straight line. This helps us understand how these special shapes can be drawn and measured.

The way angles are measured changes a little bit for ellipses compared to circles, but the basic idea of using points to define the shape still works.

Pole-polar relation

An ellipse can be shown with a special math rule. This rule connects points to lines in a certain way.

For a point on the ellipse, the rule gives the line that just touches the ellipse at that point. For a point outside the ellipse, the rule gives a line that connects two points where lines from the outside point just touch the ellipse. For a point inside the ellipse, the rule gives a line that does not touch the ellipse at all.

There are also special points and lines connected to the ellipse that follow this rule. These include points called foci and lines called directrices. These special points and lines are linked in pairs by this rule.

This idea of connecting points to lines also works for other shapes like hyperbolas and parabolas.

Metric properties

The area enclosed by a tilted ellipse is π y int x max {\displaystyle \pi \;y_{\text{int}}\,x_{\text{max}}} .

An ellipse is a shape that looks like a stretched circle. It has two points called foci (focus points), and for any point on the ellipse, the total distance to both foci is always the same.

The shape of an ellipse can be described by a number called its eccentricity, which tells us how "stretched" it is. An eccentricity of 0 means the shape is a perfect circle, while an eccentricity close to 1 means the ellipse is very stretched out.

x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}

A ellipse = π a b {\displaystyle A_{\text{ellipse}}=\pi ab}

A ellipse = π y int x max = π x int y max {\displaystyle A_{\text{ellipse}}=\pi \;y_{\text{int}}\,x_{\text{max}}=\pi \;x_{\text{int}}\,y_{\text{max}}}

In triangle geometry

Ellipses can be found in triangle geometry in a few special ways. One type is the Steiner ellipse, which passes through the points of a triangle and has its center at a special point called the centroid. Another group is called inellipses, which are ellipses that just touch the sides of a triangle. Two famous examples are the Steiner inellipse and the Mandart inellipse.

As plane sections of quadrics

Ellipses can be found when you cut certain 3D shapes with a flat surface. These shapes include:

Applications

Physics

Elliptical reflectors and acoustics

Planetary orbits

Main article: Elliptic orbit

In the 1600s, Johannes Kepler discovered that planets travel around the Sun in paths shaped like ellipses, with the Sun at one focus. Later, Isaac Newton explained this using his theory of gravity.

In space, when two objects are pulled toward each other by gravity, they move in elliptical paths. The point where they are closest together and the farthest apart are special points on these paths.

Harmonic oscillators

The path of certain moving objects, like a pendulum swinging in two directions or a weight on a spring, also forms an ellipse. Unlike planet orbits, the center of attraction for these objects is at the center of the ellipse.

Phase visualization

In electronics, two signals can be compared using a special display. If the display shows an ellipse instead of a straight line, it means the signals are not in perfect sync.

Elliptical gears

Gears shaped like ellipses can turn smoothly while staying in contact. These gears can help change the speed or force of moving parts in machines. For example, in bicycles, elliptical gears can make pedaling easier by changing the force needed at different points in the pedal stroke.

Optics

In certain materials, how light bends depends on its direction. This can be described using shapes similar to ellipses. Elliptical mirrors are also used in some lasers and light sources for microchip making.

Statistics and finance

In statistics, certain groups of numbers follow patterns that form ellipses. This idea helps in finance to understand how different investments change together.