I am using a 3D engine called Electro which is programmed using Lua. It's not a very good 3D engine, but I don't have any choice in the matter.
Anyway, I'm trying to take a flat quadrilateral and transform it to be in a specific location and orientation. I know exactly where it is supposed to go (i.e. I know the exact vertices where the corners should end up), but I'm hitting a snag in getting it rotated to the right place.
Electro does not allow you to apply transformation matrices. Instead, you must transform models by using built-in scale, position (that is, translate), and rotation functions. The rotation function takes an object and 3 angles (in degrees):
E.set_entity_rotation(entity, xangle, yangle, zangle)
The documentation does not speficy this, but after looking through Electro's source, I'm reasonably certain that the rotation is applied in order of X rotation -> Y rotation -> Z rotation.
My question is this: If my starting object is a flat quadrilateral lying on the X-Z plane centered at the origin, and the destination position is in a different location and orientation where the destination vertices are known, how could I use Electro's rotation function to rotate it into the correct orientation before I move it to the correct place?
I've been racking my brain for two days trying to figure this out, looking at math that I don't understand dealing with Euler angles and such, but I'm still lost. Can anyone help me out?
Can you tell us more about the problem? It sounds odd phrased in this way. What else do you know about the final orientation you have to hit? Is it completely arbitrary or user-specified or can you use more knowledge to help solve the problem? Is there any other Electro API you could use to help?
If you really must solve this general problem, then too bad, it's hard, and underspecified. Here's some guy's code that may work, from euclideanspace.com.
First do the translation to bring one corner of the quadrilateral to the point you'd like it to be, then apply the three rotational transformations in succession:
If you know where the quad is, and you know exactly where it needs to go, and you're certain that there are no distortions of the quad to fit it into the place where it needs to go, then you should be able to figure out the angles using the vector scalar product.
If you have two vectors, the angle between them can be calculated by taking the dot product.
Related
I have a 3D point in space, and I need to know how to pitch/yaw/roll my current heading (in the form of a 3d unit vector) to face a point. I am familiar with quaternions and rotation matrices, and I know how to represent the total rotation necessary to get my desired answer.
However, I only have control over pitch, yaw, and roll velocities (I can 'instantaneously' set their respective angular velocities), and only occasional updates on my new orientation (once every second or so). The end goal is to have some sort of PID controller (or three separates ones, but I suspect it won't work like that) controlling my current orientation. the end effect would be a slow (and hopefully convergent) wobble towards a steady state in the direction of my destination.
I have no idea how to convert the current desired quaternion/rotation matrix into a set of pitch-yaw-roll angular velocities (some sort of quaternion derivative or something?). I'm not even sure what to search for. I'm also uncertain how to apply a PID controller to this system, because I suspect there will need to be one controller for the trio as opposed to treating them each independently (although intuitively I feel this should be possible). Can anyone offer any guidance?
As a side note, if there is a solution that just involves a duo (pitch/yaw, roll/pitch, etc), then that works just fine too. I should only need 2 rotational degrees of freedom for this, but that is further from a realm that I am familiar with so I was less confident forming the question around it.
First take a look if your problem can be solved using quaternion SLERP [1], which can let you specify a scalar between 0 and 1 as the control to move from q1-->q2.
If you still need to control using the angular rotations then you can calculate the error quaternion as Nico Schertler suggested.
From that error quaternion you can use the derivative property of the quaternion (Section 4 of http://www.ecsutton.ece.ufl.edu/ens/handouts/quaternions.pdf [2]) to work out the angular rates required.
I'm pretty sure that will work, but if it does not you can also look at using the SLERP derivative (eq. 23 of http://www.geometrictools.com/Documentation/Quaternions.pdf [3]) and equating that to the Right-Hand-Side of the equation in source [2] to again get angular rates. The disadvantage to this is that you need code implementations for the quaternion exponentiation and logarithm operations.
I am currently teaching myself linear algebra in games and I almost feel ready to use my new-found knowledge in a simple 2D space. I plan on using a math library, with vectors/matrices etc. to represent positions and direction unlike my last game, which was simple enough not to need it.
I just want some clarification on this issue. First, is it valid to express a position in 2D space in 4x4 homogeneous coordinates, like this:
[400, 300, 0, 1]
Here, I am assuming, for simplicity that we are working in a fixed resolution (and in screen space) of 800 x 600, so this should be a point in the middle of the screen.
Is this valid?
Suppose that this position represents the position of the player, if I used a vector, I could represent the direction the player is facing:
[400, 400, 0, 0]
So this vector would represent that the player is facing the bottom of the screen (if we are working in screen space.
Is this valid?
Lastly, if I wanted to rotate the player by 90 degrees, I know I would multiply the vector by a matrix/quarternion, but this is where I get confused. I know that quarternions are more efficient, but I'm not exactly sure how I would go about rotating the direction my player is facing.
Could someone explain the math behind constructing a quarternion and multiplying it by my face vector?
I also heard that OpenGL and D3D represent vectors in a different manner, how does that work? I don't exactly understand it.
I am trying to start getting a handle on basic linear algebra in games before I step into a 3D space in several months.
You can represent your position as a 4D coordinate, however, I would recommend using only the dimensions that are needed (i.e. a 2D vector).
The direction is mostly expressed as a vector that starts at the player's position and points in the according direction. So a direction vector of (0,1) would be much easier to handle.
Given that vector you can use a rotation matrix. Quaternions are not really necessary in that case because you don't want to rotate about arbitrary axes. You just want to rotate about the z-axis. You helper library should provide methods to create such matrix and transform the vector with it (transform as a normal).
I am not sure about the difference between the OpenGL's and D3D's representation of the vectors. But I think, it is all about memory usage which should be a thing you don't want to worry about.
I can not answer all of your questions, but in terms of what is 'valid' or not it all completely depends on if it contains all of the information that you need and it makes sense to you.
Furthermore it is a little strange to have the direction that an object is facing be a non-unit vector. Basically you do not need the information of how long the vector is to figure out the direction they are facing, You simply need to be able to figure out the radians or degrees that they have rotated from 0 degrees or radians. Therefore people usually simply encode the radians or degrees directly as many linear algebra libraries will allow you to do vector math using them.
I am quite new to this, and iv'e heard that i need to get my inversed projection matrix and so on to create a ray from a 2D point to a 3D world point, however since im using OpenglES and there are not as many methods as there would be regulary to help me with this. (And i simply don't know how to do it) im using a trigenomeric formula for this insted.
For each time i iterate one step down the negative Z-axis i multiply the Y-position on the screen (-1 to 1) with
(-z / (cot(myAngle / 2))
And the X position likewise but with a koefficent equally to the aspect ratio.
myAngle is the frustum perspective angle.
This works really good for me and i get very accurate values, so what i wonder is: Why should i use the inverse of the projection matrix and multiply it with some stuff instead of using this?
Most of the time you have a matrix lying around for your OpenGl camera. Using an inverse matrix is simple when you already have a camera matrix on hand. It is also (oh so very slightly at computer speeds) faster to do a matrix multiply. And in cases where you are doing a bajillion of these calculations per frame, it can matter.
Here is some good info on getting started on a camera class if you are interested:
Camera Class
And some matrix resources
Depending on what you are working on, I wouldn't worry too much about the 'best way to do it.' You just want to make sure you understand what your code is doing then keep improving it.
I use slerp to interpolate between two quaternions representing rotations. The resulting rotation is then extracted as Euler angles to be fed into a graphics lib. This kind of works, but I have the following problem; when rotating around two (one works just fine) axes in the direction of the green arrow as shown in the left frame
here
the rotation soon jumps around to rotate from the opposite site to the opposite visual direction, as indicated by the red arrow in the right frame.
This may be logical from a mathematical perspective (although not to me), but it is undesired. How could I achieve an interpolation with no visual flipping and changing of directions when rotating around more than one axis, following the green arrow at all times until the interpolation is complete?
Thanks in advance.
Your description of the problem is a little hard to follow, quite frankly. But it sounds like you need to negate one of your quaternions.
Remember, each rotation can actually be represented by two quaternions, q and -q. But the Slerp path from q to w will be different from the path from (-q) to w: one will go the long away around, the other the short away around. It sounds like you're getting the long way when you want the short way.
Try taking the dot product of your two quaternions (i.e., the 4-D dot product), and if the dot product is negative, replace your quaterions q1 and q2 with -q1 and q2 before performing Slerp.
How far is the total rotation? You may be asking for an interpolation for two orientation too far apart in angle. The math, quaternions or not, has trouble deciding which way to go, in a sense. Like not having enough keyframes in animation.
Determine a good intermediate orientation about halfway along, and make separate interpolations from the initial orientation to that intermediate one, and from the intermediate to the final.
So I have written a Quaternion based 3D Camera oriented toward new programmers so it is ultra easy for them to integrate and begin using.
While I was developing it, at first I would take user input as Euler angles, then generate a Quaternion based off of the input for that frame. I would then take the Camera's Quaternion and multiply it by the one we generated for the input, and in theory that should simply add the input rotation to the current state of the camera's rotation, and things would be all fat and happy. Lets call this: Accumulating Quaternions, because we are storing and adding Quaternions only.
But I noticed that there was a problem with this method. The more I used it, even if I was only rotating on one Euler angle, say Yaw, it would, over some iterations, begin bleeding over into another, say Pitch. It was slight, but fairly unacceptable.
So I did some more research and found an article stating it was better to accumulate Euler angles, so the camera stores it's current rotation as Euler angles, and input is simply added to them each frame. Then I generate a Quaternion from them each frame, which is in turn used to generate my rotation matrix. And this fixed the issue of rotation bleeding into improper axes.
So do any Stackoverflow members have any insight into this problem? Is that a proper way of doing things?
Multiplying quaternions is going to suffer from accumulation of floating-point roundoff issues (even simple angles like 45 degrees won't be exact). It's a great way to composite rotations, but the precision of each of your quaternion components is going to drop-off over time. The bleed-through is one side-effect, a visually worse one though is your quaternion could start incorporating a scale factor - to recover that, you'd have to renormalize back to Euler angles in any case. A fixed-point Euler angle isn't going to accumulate roundoff.
Recalculating the quaternion per-frame is minimal. I wouldn't bother trying to optimize it out. You could probably allow a few quaternions to accumulate before you renormalized to get the accuracy back, but it really isn't worth the effort.
Accumulation is an inexact process. Accumulating lots of incremental rotations will accumulate roundoff error whether you do it with quaternions or matrices.
I imagine something like this: you got your code up and running, but noticed that after a certain amount of navigation your camera was heeling over annoyingly -- violating an invariant you hadn't thought of in advance. Effectively, you've realized you don't want to accumulate rotations; instead you want to do something else.
You can look at this as more of an interface design issue than a numerical accuracy issue. Basically, people expect a camera to navigate according to pitch, yaw, and roll, so choosing to control and represent the angles directly can avoid a lot of problems.
The bummer here is that the quaterions seem to have become redundant (for this particular usage, at least). You still want the quaternions, though -- interpolating with the raw pitch/yaw/roll angles can be ugly. Again, it's an interface design question: you need to figure out where you'll need the quaternions, and how to get them in and out...
I've seen both argued for. I think the real question you'll have to deal with is flexibility in your camera system down the line; IMO yaw is generally more interesting in a third-person view (because you're going to rotate about the character's vertical axis). While you can arguably "yaw" around the vertical in first-person view as well, I'm not sure it's really the same thing.
However, I do think it's kind of a waste to recalculate your quaternions per-frame. Perhaps it would be better to store the latest quaternions and mark them dirty if your frame receives input?