Math Is Fun Forum

hmmm... you're right! Mind you, thats exactly how the matrix equation was displayed in the book.

but regardless, that doesn't seem right. OpenGL performs the transformations in the reverse order of when you specify them.

Say you start with the model View matrix as the Identity. Then you specify a rotation matrix R, a translation matrix T, ands a scaler matrix S, in that order. Each transformation right multiplies the new matrix by the old one, so we get: (R*T*S) as our modelview matrix.

If we had to left multiply a 4vector V (or rather, a 1x4 matrix) by this matrix, V(R*T*S), that would result in the transformations occurring in the order in which they were specified, and not in reverse, which is not how its supposed to work. Unless perhaps, the reverse order process is not used for the Projection Matrix.

am I going mad, or does my openGL book not understand matrix multiplication?

by wrapping the cosine in absolute value, you can make the graph continuous!

ln(abs((cos(x))/(e^(-x))))

well I recognized the first as being the maclauren expansion of e^x where x = 1.

I seem to remember there being an easy convergency test to show that 1/n! converges though. Just can't remember which.

and how did you derive the second relation? thats kinda cool!

(edit) wait I get it! Every factorial after 2 has to be even! What a nifty trick!

well... I assume that integrator just gives you the antiderivative, so you could just plug (a) into that expression, and subtract that from the result of plugging (b) into that expression.
In otherwords,

then the evaluation is (F(b) - F(a))

But somehow i think you'd know that!

I think they are using inverse hyperbolic trigonometric substitution to do that:

let sinh(u) = 2x

then:

4x^2 = sinh^2(u) and dx = (1/2) cosh(u) du

so we have:

Note I've replaced a and b with a* and b* to represent the values of u where x = a and x = b respectively. Now using the identity, sinh^2 + 1 = cosh^2, we get

now I don't really remember how you handle even powers of hyper-trig functions, but I'm pretty sure there's an identity somewhere that will simplify this.

Anyway, when you're done finding the antiderivative in terms of u, you resubstitute for u using the relation sinh(u) = 2x, or equivalently, sinh^-1(2x) = u, and then simplify. Once you replace the expression with one involving only x, then your limits of evaluation move back to a, and b (rather than a* and b*). I'm assuming what you were given by the calculator is what you get as a result.

wow! I didn't even see this until just now.

argh! Row major format, of course! *face slap* silly me!

but I've never heard the term 'onto' to mean surjective. Thats weird!

pooossibly! But one thing I like making is a robot model that can be positioned based on a set of angle parameters. When updating its position, I would basically assemble each piece in a series of rotations and translations. For instance, you would rotate the forearm by say, leftElbowAngle, and then move it onto the bottom of the upper arm, and then the two of them would be rotated by leftArmShoulderAngle. The entire upper body could be rotated about its waste, and finally the entire model could be rotated and shifted to its appropriate location.

Now according to that, if different parts of the body are rotated at all these different angles, then calling glTranslate to shift its location would send them off in all different directions.

now i MAY be able to reorder it so that every piece is rotated before its translated, I just suspect that in certain scenarios, you might need to shift after rotating, and it would be a nightmare.

Hey guys! I was going to make a new thread about this but I remembered we had a global warming thread. So here's what I had to say.

Despite what appears to be growing concern for global warming, and carbon dioxide levels in the media and politics, all I ever see in my urban Philadelphia neighborhood is more and more trees being cut down because they are an annoyance. Back when I first moved to this neighborhood, about 15 years back, large oak and maple trees lined the avenues, and where plentiful in neighborhood streets, and backyards. By now, trees have all but completely disappeared. I once asked a neighbor why he was cuttting his tree down in his yard. He said "Because they're a mess. Thats all trees are, is a mess!"

aren't trees a little more than that? Aren't they a crucial component of our environment necessary for our own existence?

I understand that trees do make messes, and they can cause damage to plumbing systems, sewer pipes, and whatnot. But seriously, how much could global warming be reduced if we  spent time planting trees in places where they are scarce (such as urban neighborhoods and cities) instead of cutting them down?

well one way its easy to prove. But showing the implication works the other way puts a restriction on the solution sets. I had a proof that showed how this could be used for reduction, but i can't remember exactly what it was, or how i proved it. argh!

I'll keep thinking about it.

part 2.

In otherwords, where the coefficient of one of the variables is equal and opposite in sign, they may be added together, and the resulting inequality has a solution if and only if the first two have solutions.

To show this, suppose the first two inequalities have a solution (x, y, z), then:

we therefore, must have that for these solution values,

which implies the sum of the inequalities also has a solution.

Now suppose there is no point (x, y, z) such that

are both satisfied, and there is a point (x, y, z) such that

rearranging the latter inequality, for the solution point

now suppose we take a value that lies between these two by letting u equal the average of the values on either side. we therefore have:

or

rearranging we get

now suppose we chose the value x such that u = ax, (x = a/u)

we therefore have

a contradiction. Thus we are finished with the proof.

like I said, my solution deals primarily with, does a solution exist? I don't exactly remember to what extent it shows you the actual solutions, I think it does. But I don't remember how far it goes.

My method is complicated, and I'll have to remember as I go along. I'll start working on the post now.