Ellipse

An ellipse is a special 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.

Ellipses are 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 helps scientists understand how objects move in space. Ellipses are also used in engineering and physics.

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 how stretched out the ellipse is. We can use simple math to draw and study ellipses in many ways.

Definition as locus of points

An ellipse is a special shape in math. Picture two fixed points called foci. For any point on the ellipse, the total distance to both foci is always the same. This creates 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. 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. This is a number between 0 (a perfect circle) and 1 (a very stretched shape).

Parametric representation

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

We can also think of an ellipse by slightly changing a simple shape called a circle. This helps us see how ellipses are made.

Standard parametric representation

Polar forms

Polar form relative to center

In polar coordinates, we can describe an ellipse with a special math rule. If we put 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 put 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. This is a number between 0 and 1 that tells us how "stretched" the ellipse is.

Focus-to-focus reflection property

An ellipse has a special property with its two focal points. Imagine a light beam going 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, like 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 lines that cross each other at right angles touch the ellipse.

Drawing ellipses

Ellipses appear in descriptive geometry as images of circles. There are many tools to draw an ellipse. Computers are the fastest and most accurate way. But you can also draw an ellipse without a computer. The idea was known to the mathematician Proclus in the 5th century, and a tool called an elliptical trammel was invented by Leonardo da Vinci.

If you don’t have a special tool, you can draw an ellipse using four circles at the corners.

For any way to draw an ellipse below, you need to know the axes and the semi-axes. Or, you need to know the foci and the semi-major axis. If you don’t know these, you can find them using Rytz's construction.

de La Hire's point construction

This way to draw an ellipse was made by de La Hire. It uses a special math rule for ellipses:

Draw two circles at the center of the ellipse. One circle has radius a and the other has radius b. Also draw the axes of the ellipse.
Draw a line through the center. This line will touch the two circles at points A and B.
From point A, draw a line parallel to the minor axis. From point B, draw a line parallel to the major axis. These lines will meet at a point P on the ellipse (see diagram).
Repeat step 2 and 3 with different lines through the center.

Pins-and-string method

An ellipse is the place of points where the total distance to two points (the foci) is always the same. This idea helps us draw an ellipse using two drawing pins, a piece of string, and a pencil. Place the pins on the paper at the two foci. Tie the string to each pin so the total length is 2a. Then, move the pencil while keeping the string tight. It will trace an ellipse. Gardeners use this same idea with pegs and rope to mark flower beds, so it is called the gardener’s ellipse. The architect Anthemius of Tralles described this method long ago.

Paper strip methods

These next ways to draw an ellipse use paper strips and math rules.

Method 1

You will need a strip of paper that is a + b long.

Mark the point where the semi-axes meet as P. Slide the strip so its ends are on the axes of the ellipse. Point P will trace the ellipse. This works because of the math rule for ellipses.

A machine can copy this paper strip motion using a Tusi couple (see animation). It can draw any ellipse where a + b is fixed. But this can be a problem sometimes. The next method is more flexible.

Method 2

For this method, you need a strip of paper that is a long.

Mark the point that splits the strip into two parts of length b and a − b. Place the strip on the axes as shown in the diagram. As you move the strip, the free end will trace an ellipse. This works because of the math rule for ellipses.

This method is used in some ellipsographs.

Similar to method 1, you can change method 2 by cutting the part between the axes into halves.

Approximation by osculating circles

From math facts about ellipses, we get these:

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

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

Mark a helper point H = (a, b) and draw the line V1V3.
Draw a line through H that is perpendicular to V1V3.
Where this line meets the axes are the centers of the osculating circles.

(Proof: simple math.)

The centers for the other vertices are found by symmetry.

Using a French curve, you can draw a curve that touches these circles smoothly.

Steiner generation

This way to draw points on an ellipse uses a special math idea.

Given two sets of lines at points U and V, and a certain kind of mapping between them, the points where matching lines cross will make a conic section.

To make points on an ellipse, use the sets of lines at the vertices V1, V2. Let P = (0, b) be a top co-vertex and A = (−a, 2b), B = (a, 2b).

P is the center of the rectangle V1, V2, B, A. Divide the side AB into n even parts and project this division onto the side V1B using parallel lines. The points where lines from V1 and V2 match are points on the ellipse. Using these points, you can find more points on the ellipse. The same idea works for the bottom half.

Steiner generation can also be used for hyperbolas and parabolas. It is sometimes called a parallelogram method because you can start with a parallelogram instead of a rectangle.

As hypotrochoid

An ellipse is a special case of a hypotrochoid when R = 2r. This special case, where a small circle of radius r rolls inside a larger circle of radius R = 2r, is called a Tusi couple.

Inscribed angles and three-point form

Circles

A circle is a special shape where every point is the same distance from the center. Three points not in a straight line can define one circle. We use an idea called the inscribed angle theorem to find the center and size of the circle. This theorem shows how angles made by points on the circle are related.

Ellipses

Ellipses look like stretched circles. They follow a rule that uses two points called foci. Like circles, an ellipse can be defined by three points not in a straight line. This helps us learn how to draw and measure these shapes.

The way we measure angles is a little different for ellipses, but the idea of using points to define the shape stays the same.

Pole-polar relation

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

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.

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 special points called foci, 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. This number tells us how "stretched" the ellipse 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 appear 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 made when you cut some 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 found that planets move around the Sun in paths shaped like ellipses, with the Sun at one end. Later, Isaac Newton explained this using his ideas about gravity.

In space, when two objects are pulled together by gravity, they move in paths shaped like ellipses. The points where they are closest and farthest apart are special points on these paths.

Harmonic oscillators

The path of some moving objects, like a pendulum swinging in two directions or a weight on a spring, also makes an ellipse. Unlike planets, the point they move around is in the center of the ellipse.

Phase visualization

In electronics, two signals can be shown on a special screen. If the screen shows an ellipse instead of a straight line, it means the signals are not exactly in sync.

Elliptical gears

Gears shaped like ellipses can turn smoothly while touching each other. 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 some materials, how light bends depends on which way it is moving. This can be described using shapes like ellipses. Elliptical mirrors are also used in some lasers and light sources for making microchips.

Statistics and finance

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