Circles and PiConic Sections

The circle is one of four different shapes which can be created using “slices” through a cone. This can be demonstrated using the light cone of a torch:

Circle

If you point the torch vertically downwards, you see a of light. If you tilt the cone, you get an ellipse. If you tilt it even further, you get a parabola or a hyperbola.

Collectively, these four shapes are called conic sections. Even though they all look very different, they are closely related: in fact, they can all be generated using the same equation!

Conic sections were first studied by the ancient Greek mathematician Apollonius of Perga, who also gave them their unusual names.

In later courses, you’ll learn much more about parabolas and hyperbolas. For now, let’s have a closer look at the ellipse.

Ellipses

An ellipse just looks almost like an “elongated circle”. In fact, you could think about it as a circle with two centers – these are called focal points. Just like every point on a circle has the same distance from its center, every point on an ellipse has the same sum of distances to its two focal points.

If you have a long string connected to two fixed points, you can draw a perfect ellipse by tracing the maximum reach of the strings:

Coming soon: Ellipses drawing interactive

There are many other physical representations of how you could draw an ellipse:

Gears

Trammel

Disk

Swing

Planetary Orbits

You might remember from the very beginning of this course, that ancient Greek astronomers believed that the Earth is at the centre of the universe and that the sun, moon and planets move around Earth on circular orbits.

Unfortunately, astronomical observation of the sky didn’t quite support this. For example, the sun appeared larger during some parts of the year and smaller during others. On a circle, every point should have distance from its center.

Greek astronomer Hipparchus of Nicaea

To fix this, astronomers added Epicycles to their model of the solar system: planets move on a large circle around Earth, while simultaneously rotating on a smaller circle. While very complicated, this was the most widely accepted model of our universe for more than 1000 years:

This planet makes ${n} rotations around the small circle (the epicycle) during one rotation around the large circle (the deferent).

A 16-century drawing of epicycles in the Geocentric model. The Greek word “planetes” means “wanderers”.

Over time, people realised that Earth was just one of many planets orbiting the sun (the Heliocentric model), but it wasn’t until 1609, that the astronomer Johannes Kepler discovered that planets actually move on elliptical orbits.

The sun is in one of the two focal points of these ellipses. The planets speed up as they get closer to the sun, and slow down as they move further away.

A few decades later, Isaac Newton was able to prove Kepler’s observations, using his newly developed laws of gravity. Newton realised that there is a force between any two masses in the universe – similar to the attraction between two magnets.

Gravity is what makes everything fall to the ground and gravity is also what makes the planets move around the sun. It is only the great speed at which planets move, that prevents them from falling directly into the sun.

Frits Ahlefeldt

Using Newton’s laws, you can derive the path that objects take when moving under the force of gravity. It turns out that planets move on ellipses, but other objects like comets can travel on parabolic or hyperbolic paths: they fly close to the sun before turning around and shooting off into the universe, never to come back.

According to legend, a falling apple inspired Newton to think about gravity. He was one of the most influential scientists of all time, and his ideas shaped our understanding of the world for nearly 300 years – until Albert Einstein discovered relativity in 1905.

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Circles and PiConic Sections

Ellipses

Planetary Orbits