Stupid physics and it's curvature models

[K.I.L.E.R]

I've been as usual, revising my mathematics in the last hour. I've derived the typical projection matrices in new ways that have never been defined before. Anyway the one thing that gets me is that I'm usually experimenting with getting curvature in my projections. If I am looking at an object in real life, there is some level of curvature that effects what I see. I'm thinking about approximating curvature by creating a projection that instead of hitting a plane, hits a sphere and rolling off the intersection point towards the COP in an evenly manner. Perhaps a parabola would better suit this purpose.

 

[Rolf N]

What's the order of a parabola function?
What's the order of a dot product?

That's the reason why it's called "linear algebra".

 

[K.I.L.E.R]

Exactly why can't I place the result of my calculations into a matrix? Why can I not approximate my result in a linear fashion?

 

[K.I.L.E.R]

I did it.
I missed the point of what you were saying but I still did it.

 

[ShootMyMonkey]

Exactly why can't I place the result of my calculations into a matrix? Why can I not approximate my result in a linear fashion?

I missed the point of what you were saying but I still did it.

The point he was trying to make is that even though you can put in results of non-linear curves into a matrix, a matrix multiplication, by itself, can never do anything other than a linear transformation. All structure is preserved across the transformation between the vector spaces (aka homomorphism) -- that which is curved to begin with remains curved, and that which is linear to begin with remains linear.

What you CAN do is do something that is overall non-linear, but involves a matrix or matrices at some point(s) along the way. Matrix transformations with extra ancillary operations preceding (e.g. changing what goes into the matrix and/or its operands in a non-linear fashion) or following them (e.g. using the results of the transformation in some non-linear operation) is the only way to get something non-linear. Which is precisely what you had to do in order to get the result you supposedly wanted.

http://en.wikipedia.org/wiki/Linear_transformation

BTW, I would also add that it's not true that the end goals that you mentioned in your original post have *never* been done before. It's just that nobody really talks about it much in the context of basic rasterization.

 

[Fred]

I absolutely guarentee everything there is to be done with projection matrices in 3d, has been done.

Ultimately the form of the matrices follows the representation theory of the various projective Lie groups involved (for instance SO(3) etc).

So while you might think something you wrote is novel and new, ultimately its just a homomorphism away from a textbook made 80+ years ago.

Those pesky mathematicians really quite have an extensive domain

 

[K.I.L.E.R]

Thanks for you inputs. I'm doing all these things myself so I can learn and get better. I don't really know what has or hasn't been done. I just get ideas into my head left and right and figure I'd try them out, thinking maybe possibly I might be the first person to do it. I didn't get what I'd imagined but I did get the result I was aiming for. All the matrix operations I had done was only a simple plane projection, which was then followed by mapping a point on a plane to a point on a curve which had to be done through the vertex shader manually.

 

[Basic]

Well, it's pretty natural to put the coefficients for a quadratic form in a matrix. And it is transformed by matrix multiplications (you have to multiply it on both sides though). But defining geometry with implict equations doesn't fit GPUs very well.