Math Is Fun Forum

Ok I'm tripping again. This happend in trig, a small lack of understanding of a concept occurs and it snowballs as I continue untill I don't know what the devil the book is talking about, and I can no longer move forward. I'm forced to go back, and review and think about for a few days. Darn.

Ok, it seems the problem solving patterns are relatively straightforward but I might as well be learning card tricks if I don't understand what I'm doing. I'm having trouble understanding all this dy dx buisiness. I think i'm having more trouble understanding what the notations stand for.

For I a while I thought dy/dx was just a notation for derivative, but it seems this fraction can be manipulted by multiplying by dx or dividing by dy. Now when we find the differential dy we essentially find the value of the rise, and the rise alone. (rather then the rise over the run) we find this by multiplying the slope (which is the rise over the run) by the run, to cancel out the run and leave the rise. But what is the run? What are we multiplying by? It seems the run could be anything! This is difficult because the slope of the function depends entirely on which point on the line we are talking about, and is dependant on that value of x. So when we multiply by dx or divide by dy, it seems we are multiplying or diving by undefined variables. Very confusing.

Ok look at this problem:

u = (x^2)y    find du

Ok I really don't know what they mean by "du" distance u? From what?

I know how to solve the problem but I really have no idea what I'm doing. :-(

lol!

Well some of this may be above my level as I'm having a bit of trouble understanding. I'll reread it a few times to see if it sinks in. I wasn't necessarily interested in velociy, (Though I might be able to use that)  I was more interested in rotating by a specified degree about a specified axis. What I can't figure out is what is that angle defined from and how do we specify which direction we want to rotate it. If you want to rotate 15 degrees, then 15 degrees in want direction? Depending on the axis and the location of the point that may end up rotating in the opposite direction you had desired.

Btw, one thing I did note that if a point is rotating about an axis, the point will always remain the distances from two points on the axis, even if those distances are not equal to eachother, the won't change.

Suppose we want to rotate a point (who's coordinates are x,y, and z) about the axis defined by the line connecting point1 and point2 (who's coordinates will be x1, y1, z1, x2, y2, z2 respectively)

We know the 3d distance from point1 and the point will not change, and that the 3d distance from point2 and the point will not change.

So distance1 = sqrt ((x1 - x)^2 + (y1 - y)^2 + (z1 - z)^2)
and distance2 = sqrt ((x2 - x)^2 + (y2-y)^2 + (z2 - z)^2)

Where distance1 and distance2 are the 3d distances from the point, to point1 and point2 respectively, before the rotation is made and the points coordinates are known.

That gives us two equations about the points. So I suppose if we know at least one of the coordinates of the point we should be able to calculate the remaining two coordinates to define its new location. I suppose this one coordinate must somehow be determined by the desired angle of rotation.

I heard in 3d engines they look for matrices that either never change or change only once in a while and the program is designed to skip reevaluatlion when not necessary and use the result of the previous calculation.

Hmmmm to tell you the truth I'm not 100% sure there is something wrong. I just color coded the points and things seem to look better, perhaps I was just confusing the points of the front of the box and the back of the box.

Anyway, like I said, looking at a screen is differant in many ways from looking through a window. For instance, in a window, as you look though differant area's of the glass, your angle of view is changing. On a screen, your angle of view stays the same no matter where you look. Also when you look att a tv from a reasonably mild angle, your mind tends to correct it as if you were looking at it straight on. Which is why we can watch a movie without being normal to the screen, and it won't look messed up. So this is another differance. How your mind percieves the illusion of 3d on a 2d plane is much differant then where you see objects through a sheet of glass. So the mind may be trying to correct the image on a 2d plane and is actually distorting it. I often find that wierd distortions onscreen tend to not look so weird if I move my face closer to the screen.

Perhaps, my 60 degree viewing cone is simply too wide and requires to to be too close to the screen. If that be the case maybe I just need to shrink it a bit.

Ok heres a review of how the rotation and projection works. In this example we will only be finding the y coordinate of the point. Finding the x coordinate is the same only we work on the XZ plane rather then the YZ plane.

A. The location of the box on the YZ plane, Y being verticle, and Z horizontal.

B. The direct distance to each point from the viewers location (on the YZ plane) is measured. Using the distance formula sqrt((view.z - z)^2 + (view.y - y)^2) then the angle to the object on the YZ plane is measured in relation to the viewer using my polarangle function. We now have a distance and an angle of each point so we now know their polarform.

C. To rotate our viewpoint, we add the negative of the desired view rotation on the YZ plane to the angle of each points polar form. If a point was at sqrt (104) @ 45 degree's, and we wished to rotate our view 15 degree's down, we subtract 15 from the polarform. sqrt(104) @ 30 degrees. This will rotate every point down by 15 degree's as if we had rotated our viewpoint UP by 15 degrees. We then convert back to rectangular form for the new temporary coordinates of the object. This should work because when you rotate your view, the DIRECT distance from you to every point does not change, but the angle at which you see them does.

D. Now we use the simple projection scaler to scale every point down to the picture plane which represents the computer screen. These distances give us the y coordinates of where the point is to be displayed on the screen.

The same process is done on the XY plane to find the x coordinate.

Whats important is that both the horizontal and verticle rotations must be done BEFORE the projection is made. Because both rotations effect the z distance to the viewer, even though the DIRECT distance to the point is unchanging when you rotate your view.

It all seems correct, so why are some funny shapes showing up with certain angles of rotation and certain distances? I don't know, but I intend to find out. Is it a bug in my program? Or is the mathematical formula flawed in some way? One more possibility exists, viewing an object on a screen is slightly differant then looking at the world through a sheet of glass. I'll get into that in a second.

One thing I did to test the rotation engine, I used the 3d distance formula sqrt(x1-x2)^2 + (y1 - y2)^2 + (z1 - z2)^2) and checked the direct distance from the viewer to a point and had the value constantly being reevaluated and printed to the screen. As I rotated my view the directed distance did not change, as I moved the viewers location the distance changed. Everything working as it should there.

Ok while typing this I just found a new flaw. The object is first rotated on the YZplane and then on the XZplane. This works but they are done one at a time. After one rotation is complete, the next rotation is done horizontally, rotating the points as they were after the verticle rotation. This is like (for instance) looking downward, and then rotating your eyes to the left. If you rotate your eyes left after you rotated them down, then your eyes will begin to look towards your left and upwards, thanks to the vertical rotation. This is hard to explain in words so maybe I'll draw a picture. Anyway, this would result in slight rotations on the XY plane, as if the ground wasn't flat. With that in mind, some of the wierd shapes may actually make sense.  I'll have to think for a bit on how to correct this.

Well I found out Game Maker actually has built in trigonometric functions and you can define your own scripts which work pretty much like functions. Game maker's language is pretty primitive as you can't define your own member data, there is only one type of object with preset member data. It has most of what you need but sometimes not being able to define your own can be a real pain. Also you can't pass objects into functions (or scripts) to acess their member data. Each new step has presented new difficulties and I've been forced to copy and recopy large amounts of the same code into each object. Its been a pain in the neck but its working.

Trying to work out some glitches now. The image is completely flawless with zero rotation. You can move around and the dots behave perfectly given their distance and location in relation to the viewer. But rotation seems to be iffy. At times, the rotation appears to be working fine. Other times, the objects get twisted into obviously wrong positions. I set restraints so the object is only visible if its with a 60 degree viewing cone. (if the view is too wide angled barelling is inevitable) Anyway I'm going over my design looking for possible errors, and checking the program for bugs.

Gimme a sec I need to draw up some pictures.

I go through this same episode every day. lol.

I see. Well my idiot mathbook never taught me the concept of a second differance. Perhaps there was a revised edition of a previous book in the series that I didn't have. If Saxon were still alive this would not have happened. >:-(