![](/content/sites/plus.maths.org/files/styles/small_square/public/homtopy_frontpage.png?itok=NhJbxNAv)
The concept of homotopy captures the idea that, sometimes, two shapes can be deformed into each other without doing anything drastic, like cutting them up or lifting them out of their surroundings. As an example, look at the yellow and the red curves in the GeoGebra applet below. They form two different types of S-shapes, but it's easy to imagine deforming one into the other. If you don't believe it, use the slider.
This applet illustrates how two curves in the plane can be continuously deformed into each other.
How can you make this intuition precise? First note that a curve can be thought of as a piece of straight line that has been arranged in the plane in a particular way, by bending it, or perhaps stretching or squeezing it. This means that there's a mathematical function which assigns to each point on the piece of straight line the corresponding point in the resulting curve. The function is continuous, which means that there are no gaps in the curve.
Given our two curves in our example above, let's write
![two functions](/content/sites/plus.maths.org/files/articles/2023/homotpy/both_funtions.png)
You can think of each curve as coming from taking the straight line shown on the left and arranging it in the plane in a particular way. This gives continuous functions from the piece of straight line to the plane. The functions take the end points A and B of the straight line segment to the end points of the two curves. A point x inside the straight line segment is taken by the functions to a point along each of the two curves.
Now deform the yellow curve in the applet below into the red one by using the slider. If you stop this process at some point you will see a third, dashed curved, which is somewhere along the way between the original two. It corresponds to a particular value of the variable
Using the slider you can deform the curves. If you stop at some point you see a third, dashed curve.
This third in-between curve can also be defined in terms of a continuous function from our straight line segment to the plane. Since the curve is associated to the variable
If you fix a point x on the line segment on which the functions Ht are defined and vary t, you see that the points Ht(x) trace out a continuous journey without jumps or gaps.
If the family
A homotopy between the two functions
- For all points
in , if we treat as a function of , then is continuous.
The concept of homotopy divides the set of continuous functions from
To see why this difference is important, see Maths in a minute: Topology and Maths in a minute: The fundamental group.
About this article
This article was produced as part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI) – you can find all the content from the collaboration here.
The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.
![INI logo](/content/sites/plus.maths.org/files/articles/2021/representation/ini_logo_green.jpg)