Monday, November 29, 2010

Bigfoot gets physical

Very busy with Killzone3, I had very little time to do some animation work,
but I still managed to add a couple of things to 'Bigfoot' in preparation for some motion retargeting experiments.

It now can analyze skeletons loaded from mocap files and split them into branches.
It can automatically assign masses to joints using a heuristic based on skeleton connectivity.
It also now contains a physics analyzer that can compute the position of the center of mass (C.O.M in video) it's velocity and acceleration for each frame, and the same for all other joints.

The results are actually pretty fun to watch, but the video came out with bad quality,

The blue trails and velocities correspond the center of mass.

Monday, November 15, 2010

Uniqueness of inverses for Groups, by contradiction.

In mathematical groups, the uniqueness of inverses is almost a direct consequence of the definition of a Group. Yet I could not prove it in 10 minutes, was too lazy and looked it up here: http://planetmath.org/encyclopedia/UniquenessOfInverseForGroups.html.

It bothered me that I could not come up with this proof, I tried to discover why. One of the attempts I tried was proving it by contradiction which also failed, but looking at the proof, I can see I gave up too early.
After peeking, I wanted to at least try to still prove it by contradiction (unlike the proposed proof) which intuitively seemed like the way to go for me, because the proposed proof 'constructs statements from nothing', starting with an expression, expanding it for no very obvious reason at that point (but of course obvious later), then collapsing it.

It is interesting how doing it by contradiction shows a slightly clearer path through the infinite 'ocean of statements' and the infinite number of 'valid paths' between each 2 statements, because you always have the 2 sides of the inequality to visually look at.

Here is my proof by contradiction:

Given other proofs me and Tom have attempted in the last months, I am sure I would have solved it if I gave it a bit more time, but I knew it had to be short so I wanted to solve it fast, but I failed. All in all this was a good insight into my problem solving technique.
This one falls into the category 'so obvious that it blocks you'.

Wednesday, November 10, 2010

Differentiable is 'stronger' than continuous...

As a reference for the calculus part of our course, we are using the excellent book 'Calculus Lifesaver' from Adrian Banner. This book is accompanied by extremely useful free downloadable video lectures with very clear explanations. Sometimes, it is not as deep as we would like it to be, but there is plenty of sources to remedy that when it happens.

In section 5.2.11, one proof of 'Differentiability implies continuity' is laid down, There is a step (step 4) in the proof that involves using a not so obvious trick.

I wanted to try to prove it myself in a more straightforward manner.
I will let Tom review it and tell me what he thinks, hopefully it convinces him :)

Sunday, November 7, 2010

What is natural

At project-perelman, We have inevitably started delving into Cauchy sequences and real analysis because if you really want to believe calculus proofs and have enough genuine interest and analytical mind, you will discover very fast that there are missing foundations and hand-waving proofs all over the place if you do not go deeper. This is best expressed by Karl Hahn in his excellent website.

As an example, my (failed) proof (http://jadnohra.net/release/math/evt.pdf) of the Extreme Value Theorem does not perfectly work because I could not prove that my induction would cover the whole space of inputs. This can only be done by digging deeper than calculus.

"

There is a natural way to "add" or "multiply" two points in the Euclidean plane. By "natural" I mean that the definitions have turned out to be useful for many applications, and that the definitions are fairly simple"

It puts in words what we many times feel but cannot express, when something mathematical feels 'natural'. Worth memorizing!

To all of this, I will add this unrelated and brilliant quote that my lovely wife just sent me:
"...höre nie auf zu zweifeln. Wenn Du keine Zweifel mehr hast, dann nur , weil du auf deinem Weg stehen geblieben bist. .... Aber achte auf eines: Lass nie zu, dass Zweifel dein Handeln lähmen. Treffe auch dann immer die notwendigen Entscheidungen, wenn du nicht sicher bist, ob deine Entscheidung richtig ist. .... Paolo Coelho, Brida"

Saturday, November 6, 2010

Geometric proof of sum(1 to n)

Ah, geometric proofs, so graphic, so intuitive and clear, so fitting to the way our brain works. I remember that last year I spent some time doubting Pythagoras' proof of c2 = a2 x b2, it sent me on a survey for all existing proofs, many of the more graphical ones did not 'really' convince me, specially when they involved rotating or moving shapes and then made claims about them fitting somehow, Euclid's proof on wikipedia did a better job, one of it's 'less convincing' points being the triangle area lemma. When I look at it I am pretty sure it came out of intuition (like most good proofs) and then the 'formal' proof got put into pieces after having 'devised a plan for the solution' as George Polya would put it in his classic book: HOW to solve it.

Still, it made me like geometric proofs more than I used to -- and the reason I liked algebraic ones more, in retrospect is because of the history of my Math teachers -- and so I started scribbling... I came up with a couple of tiny nice geometric proofs.

The first one is proving that and that . The key idea is (like the start of Pythagoras' theorem) is to use area when multiplying numbers by each other, and then figuring out a relationship between the resulting shapes (in this case all rectangles) and writing it down:

Another, nicer proof is the one for the sum of 1 to n being equal to n(n+1)/2 :
There are many ways to proove this one, by induction per example, or geometrically as demonstrated in one of the discrete math video lectures I am using. However, the geometric proof used there involves having to rotate shapes and comparing them, I did not really like that, so I tried to come up with a clearer proof and I found out that it is possible, I have not seen a proof that did it exactly this way, so here it is:

The idea is that we know that the square in the figure has n x n 'dots' in it, we therefore know that cutting in half by the diagonal, we have dots. This 'almost' covers the total number of dots (the answer we are looking for) and might not even be an integer.
We can see that all the last dots are cut by the diagonal in half, and those halves are all what is missing to account for the missing rest, therefore, we add them.
We have n halves since we have n rows, so adding to , we get the final answer: . We did it without having to move shapes, rotate them or compare their lengths. And that is nice!