The second day (in Stuttgart) was more math than anything else, obviously. And it was mostly about industrial mathematics. First of all, I need to say that I was never attracted by these things, that seemed a little boring, but after this day I am convinced to give it a try and find out more about it. The day was divided into morning-presentations at the university and a visit to Daimler research department, which was one of the most interesting part of all the week.
The first presentation was about Motion Planning, and we discussed about ways of programming the ‘robot’ so that it moves from a start position to a goal position, while also considering some obstacles. A good example is the nail puzzle and we worked with this almost simple game all the presentation. First of all I need to remind you that to solve it one works with rotations (in our case α∈[0,2π]) and translations ( t∈[0,1]). So, in our case to solve the nail puzzle we are working in two different spaces: the working space, which is a R³ space, and the configuration space. We first solve the puzzle in the configuration space and then translate everything into the working space. We start with a graph in the configuration space, where the x-axis is represented by the translation and the y-axis by the rotation and so we can ‘easily’ graph the motion and put obstacles (considering the fact that we are not allowed collisions, obviously). For this we discussed 2 solutions: rapidly-exploring random tree and probabilistic road map. I will let you think a little about this problem and I will make a separate post explaining what I understood from these 2 solutions.
The second presentation was more like a visual thing. We had a brief introduction on how we see in 3D and after that we saw 3D shapes and discuss all their properties. We started with some basic fractal geometry (and you know that I like it a lot, check my fractal art album) and talked about the volume of those shapes ( if we have a fractal shape and we transform it infinitely, we get a volume of 0 though the feeling tells you that this is kind of impossible, but remember that we don’t completely understand infinity either).
We had a small presentation of a hypercube, but in 3D (he explained us how we can visualize it in 3D).
And then came the B-spline surfaces, which are very important in our digital art. Here are some examples:
These 2 were more like really applied mathematics and I enjoyed everything and it made me think of different ways in which mathematics exists in our society. they were refreshing and interesting presentations.