How I use GeoGebra to make animations in videos (Part II)
I said I use GeoGebra to make animations in videos, but how do I do this? Here is part 2 of 2 of the basics.
Please view this webpage on a PC, because the GeoGebra embedding does not work well on mobile.
Parametric Plots
If you want to create any lines / curves / surface with a certain formula in mind, you should use a parametric plot. For curves, the command would simply be
Curve (x(t), y(t), z(t), t, t_start, t_end)
x(t), y(t), z(t): the expression of x, y, z coordinates, with t as a variable
t_start, t_end: the start and end values of t in your curve.
In what follows, we will demonstrate a helix. The parametric equation for the helix would have
x(t) = r cos(t), y(t) = r sin(t), z(t) = vt
You will need to use the 3D view to view the helix if you are using the desktop app.
If you want to plot surfaces instead, then there is a similar command. The only difference is that we have two variables to parametrize a surface rather than 1 to parametrize a curve:
Surface (x(u,v), y(u,v), z(u,v), u, ustart, uend, v, vstart, v_end)
x(u,v), y(u,v), z(u,v): the expression of x, y, z coordinates, with u,v as variables
u_start, u_end, v_start, v_end: the start and end values of u, v in your surface.
I don’t think I have used this command for surfaces before in my videos. For simple surfaces where you can parametrize into z = f(x,y), you can simply input the formula into GeoGebra and it will handle it. GeoGebra also has a built-in sphere tool, so for my purposes, I haven’t used this command yet. But perhaps when it comes to things like a torus, I will need it!
Limitations (Parametric Plots)
The only limitation that I came across (that I really want GeoGebra to be able to make it much more convenient to remedy) is when it comes to customization. The whole line / curve / surface is a single object, and it can only be of the same color. If you want to have different colors at different points, you will have to go through a rather painful route, and as a result, I used Mathematica when I wanted to color different points of a surface a different color for complex analysis.
Sequence
This is the command to use if you want to construct multiple lines / curves / any object at the same time. This is the kind of command you want for simulating linear transformations. The command is:
Sequence (<object(t)>,t, t_start, t_end, t_int)
object(t): the usual command you use to draw curves / surfaces, but parametrized by t, i.e. drawing a curve / surface for each t
t_start, t_end: the start and end values of t in your curve.
t_int: the interval between consecutive t’s. Optional, but the default is 1. GeoGebra will first draw the object(t_start), then object(t_start + t_int) and so on until object(t_end)
Or, if the object being drawn is parametrized by 2 variables,
Sequence (Sequence (<object(u,v)>,u, u_start, u_end, u_int), v, v_start, v_end,v_int)
object(u,v): the usual command you use to draw curves / surfaces, but parametrized by u,v , i.e. drawing a curve / surface for each u and each v.
u_start, u_end, v_start, v_end: the start and end values of u, v in your surface.
u_int, v_int: the interval between consecutive u’s and v’s. Optional, but the default is 1.
For example, if you want to construct a bunch of parallel horizontal lines, then the corresponding command will be
Sequence (Curve (s,t,s,-10,10), t, -10, 10, 1)
This will construct a line “Curve (s,t,s,-10,10)” for each t from -10 to 10, with an interval of 1, i.e. t ranges from -10, -9, -8, … , 10. The result will be 21 parallel horizontal lines.
The example below is taken from one of my recent videos about matrix transpose. It demonstrates the left shear.
Limitations
The main limitation is that once you created the sequence object, the full set of lines / curves become one single object. It would not be possible to individually style each line / curve. If you want to individually color each line / curve, then you either have to use some other applications, or manually construct each of those objects and customize them individually.
Conclusion
Hopefully these two blog posts help you learn the basics of GeoGebra for creating animations. There are of course many things I omitted, but the slider, parametric plots and the sequence command are the main tools that I use for the animations in my videos.
