Maths GIFs: parabolas

One of the fantastic features of dynamic graphing software is that you can animate how a graph changes as some aspect of it is varied. I'm particularly keen on making short GIFs of animated graphs that demonstrate interesting properties. 

One advantage of animated GIFs is that they are really easy to distribute via Twitter. There are loads of them if you have a look at the hashtag #mathgif. 

Parabolas

I've been posting animated Maths GIFs to Twitter for quite a while now. Here are some of my favourites featuring parabolas:

The midpoint of the points of intersection of y = x² and y = mx + c lies on the line x = m/2.

The tangent at point P to a parabola with vertex Q can be constructed by finding R, the point of intersection of the vertical through P and the horizontal through Q, finding the midpoint, M, of QR and then drawing the line through P and M.

If the tangents at two points A and B on a parabola are perpendicular then the line AB goes through the focus and the tangents intersect on the directrix.

The tangent to y = x² + bx + c when x = 0 has equation y = bx + c.

The tangent to y = x² when x = a passes through the y-axis at y = –a² .

For two points on a parabola A and B the position of the point C on the parabola (between A and B) such that the triangle ABC has the maximum area lies on the tangent to the parabola that is perpendicular to AB.

If a pair of perpendicular parabolas have 4 points of intersection these will all lie on the same circle.

Tracing the parabolas y = x² + bx + 4 as b varies is aesthetically pleasing. 

Creating Maths GIFs

The best tool that I've found for creating Maths GIFs is GeoGebra Classic 5. This can be downloaded from https://www.geogebra.org/download 

To create one I first construct a file that displays the animation I want based on a single slider. This can be exported as a GIF using use File > Export > Graphics View as Animated GIF.

Some tips that I think are useful:

• Setting the slider to have about 50 increments and increasing only is usually a good compromise between sufficient for a smooth animation and getting a smallish file size.
• Using a smaller tiled window gives a smaller file size. A ratio of 2:1 displays well on Twitter.
• I like a time between frames of 5ms and the GIFs set to loop.