Recent content by da_nang

Easiest way to understand it is to look at a unit circle, where the sine is the y-coordinate. From there it should be easy to see that \sin(x) = \sin(\pi - x) not just for angles less than 90 degrees, but for all angles. Even better when you note that \sin(\pi - x) is just a reflection across...

Specifically, P(A | B^{C}) := P(A) - P(A \cap B) would still include B in the sample space. Given that B hasn't happen, you don't want B in the sample space.

Would you say the problem is discrete-time? If so, a Z-transform might help.

\vec x (t) = \vec{x}_1 + (\vec{x}_2 - \vec{x}_1)f(t) where 0 \leq f(t) \leq 1. If say f(t) = \frac{1}{\pi}\arctan(t) + \frac{1}{2}, then you've covered the entire real line. As for an equation, \vec{x}_1 + (\vec{x}_2 - \vec{x}_1)(\frac{1}{\pi}\arctan(t) + \frac{1}{2}) - \vec x = 0 works, yes?

Say you have two normal vectors, \vec{n}_1 = (a_1, b_1, c_1), \vec{n}_2 = (a_2, b_2, c_2) as well as two position vectors \vec{p}_1 = (x_1, y_1, z_1), \vec{p}_2 = (x_2, y_2, z_2). The intersection of two planes defined by these vectors is all position vectors \vec{x} = (x, y, z) that satisfy...

This is of course assuming the surface is a hyperbolic paraboloid, which is not necessarily the case. But it works as an approximation. An alternative approximation is to use two diagonally opposite points at heights z_1, z_2. This allows you to calculate the volume as V = \frac{z_1 + z_2}{2}...

Okay, I see what you mean. What you need to check then is that the parameters are multiplied by the right coefficients.

Alright, first problem I spotted is that your matrix is going to be rank deficient. Shouldn't cause too many problems since lstsq at least in the documentation should take care of it. Most likely some coefficients will end up being zero. Secondly, your matrix is highly ill-conditioned. A...

If c_3 is the total amount of memory in the system then dimensional analysis dictates that z has the unit of \frac{\text{Time}}{\text{Memory Unit}}. If that's not the case then your model is dimensionally inconsistent. For your second question, you may have made an error somewhere but I'd need...

If t(x,y) = c_1 x + c_2 y is your model, then it is presumed that the coefficients will be of appropriate units \frac{\text{Time}}{\text{Percent}} and so on. Whether or not this is a reasonable model is a different question with no easy answer. However, if the changes in each variable is small...

This will come in handy.

Well if you don't have any information on the interior then the best I think you can do without further information is to fit a plane to those coordinates and hope it doesn't vary too much from reality. Least-squares might do. After that, you use this approximation in the integral given by Mark44.

It's a surface, but it's not a plane. For any two points in a plane defined by the equation a x + b y + c z + d = 0, the equation a \Delta x + b \Delta y + c\Delta z = 0 must apply, where \Delta x represents the difference in x-coordinates and so on. This doesn't apply to the top face in...

I've attached an example image of what I mean. The model is the same as the one in your image. Notice the drastic change of color on the top face? That's because the quadrilateral polygon is made up of two triangles. If the polygon was planar, the triangles would be in the same plane and there'd...

By shape, I meant how the z-value of the top face changes wrt the domain bounded by the top face.