Math: Equation / Formula questions

[vhelp]

Ok. I'm having a little bit of trouble here trying to understand a math syntax and in this case google is not so friendly because of the size of the search criteria I'm using -- you'll understand in a moment.

I used to see what looks like a notation or whatever the term is called when writing formulas. It looks something like this: U'= xyz. I see this a lot in color space conversion and also in other areas beyond video and images. But this ' notation prob means one kind of operation or function, I hope.

The question is, what does the { ' } mean or do ?? Its the pinky finger guy on the keyboard.

Thanks.

-vhelp 4784

[Midzuki]

[ very incorrect guess --- DELETED ]

[jagabo]

U' = f(U)

Means the new value of U, "U prime", after applying the function f to the old value of U.

For example, in a 3D rotation you start with a point X,Y,Z after the rotaion you have new coordinates X',Y',Z'.

Or in a color transformation of R,G,B you get a new color R',G',B'.

This is often used in iterative functions where you apply a function to a value, then apply that function to the new value, then apply the function again to the newest value, etc.

For example, with the function

N' = N+2

Starting with N=0 You generate the series 0, 2, 4, 6, 8...

[vhelp]

U' = f(U)

Means the new value of U after applying the function f to the old value of U.

Ok. So let me see if I get this straight. From a programing perspective, if I..

function Add(x: integer): integer;
begin
result := x+x;
end;

Where U=1

let us say f(U) [or f(1)] = my function's, Add(U) [or Add(1)] .. which holds the value of 1
and if I perform the operation, U':=Add(U) .. well, we take out the ' char of course.
then my new value for U will hold a value of 2 ??

Another example run..

Where U=25

Add(U) [or Add(25)] now holds the value of 50

. . .

Ok, one more..

function Mult(x: integer): integer;
begin
result := x*x;
end;

Where U=2
Mult(U) now holds the value of 4
Where U=15
Mult(U) now holds the value of 225

. . .

Am I getting close to the understanding ??

I wish our admin would extend on the forum boards feature for CODE to include PROGRAMMING syntax so that we can see the program code more effecient in block form. Oh well, maybe some day..

-vhelp 4785

EDIT: jagabo revised is prev response while I was still composing mine.

[jagabo]

You got it.

[JohnnyMalaria]

For colorspace stuff, the prime also indicates that a different set of coefficients are used for the transformation to/from a different space.

e.g., R'G'B' is different than RGB. The latter is traditional computer bitmap but the former is "studio RGB" that uses different ranges for the allowed pixel values. So, Y'U'V' goes with R'G'B' and YUV goes with RGB. Actually, "YUV" is often used incorrectly - it should be YCrCb (likewise with Y'Cr'Cb')

In a completely different arena, the prime also means the first differential (or derivative). For example, consider position and velocity. Velocity is how much position changes with time. If x = position, t is time and v = velocity, then:

v = change in x / change in t

written usually as:

v = dx/dt (the d's mean "change in")

more correctly, since v depends on x and t, it should read:

v(x, t) = dx/dt

Very often, a different notation is used for the differential:

v(x, t) = x'(t)

The ' indicates the differential of x with respect to t (i.e., dx/dt)

You can go one step further in this example and relate acceleration to position or velocity. Acceleration, a, is the rate of change of velocity w.r.t. time:

a(v, t) = v'(t)

which is the same as:

a(x, t) = x''(t) (note the double primes).

The velocity example is important for motion compensation. All of them are relevant for gaming and for fancy video transitions (like a rotating cube of mulitple video clips).

[deadrats]

I used to see what looks like a notation or whatever the term is called when writing formulas. It looks something like this: U'= xyz....But this '....

that's the formula for the anti-life equation.

ok, sorry, bad joke, i couldn't help myself.

in math this symbol ' is a calculus notation used to represent the derivative of a function, i.e how the function changes with respect to time:

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

http://mathworld.wolfram.com/Derivative.html

an simple example:

assume you have the equation:

y = x ^ 2

y' = 2x (first derivative)

as an interesting side note, the integral of the equation :

U' = xyz

is:

U = 1/2 x ^ 2 y z

which means that U' = xyz represents the instantaneous rate of change of the above equation or equivalently:

U = 1/2 x ^ 2 y z

represents the area under the curve if:

U' = xyz

was plotted on a graph.

as to what this has to do with color space, i couldn't even begin to tell you.

[vhelp]

..much appreciated guys.

But guys, don't get me wrong. I've got my color space equations working fine, and it had nothing to do with the question, sort of.

Its just that I came across it over the years and up till now, while researching on BLUR methods for a delphi project I'm working on, I came across this google find where Didée was describing something and wrote down the following equation: x' = x^2, and that got me on the investigative trail and I ultimated came here to ask the quesiton. Color space was another example of where I saw it and thought maybe some of you might be more familiar with it when used in that context and giveing me the understanding through an actual practice run.

--> variable blur 0.4: a gaussian/binomial/average variable radius blur -- by tsp;(807 ) Jan 19, 2005

So, in using Didée's example, I wrote another function to mimic this functionality:

function _Power(x,n: integer): integer;
begin
result := power(x,n);
end;

Given: x' = x^2, where x=2

and x := _Power(x,2)

then x would have the value of 25

-vhelp 4786