Vectors and Orientation
Date: 05/29/2003 at 01:15:18 From: John Subject: Rotating a vector Hey, I am trying to write a game involving spaceships, and I have been coming across trouble regarding how to find relative positions. My ships are rectangular prisms and to check for the impact of a projectile (modeled as a point) I want to normalize a vector from the center of the ship to the projectile so that new vector's components give the x,y,z coordinates of the projectile relative to the ship (which can be rotated about all three axes) with the ship sitting along the axes (i.e. the sides of the ships are all normal or perpendicular to the axes). This will allow for easy hit detection. In addition, to find weapon facings I was hoping I could use a similar method to get a unit vector in the absolute direction of a weapon that is offset from the center of the ship, where the weapon's current direction is measured against the ship. And to make things easier for me to think about, I made the y-axis forward and the z-axis up, with theta measured against the y, clockwise positive, phi measured from y, but going to the z-axis, and a third angle I called eta to model the rotation of the ship about its y-axis. The example: A ship that naturally faces in the +y direction is rotated 45 degrees (clockwise in the XY plane), 45 degrees up (from the XY plane) and rotated about its central axis (i.e. about the vector representing the ship). A projectile is at (1, 2, 3). What is the position of the projectile relative to an unrotated ship? Also, if a gun is on the ship at (1,1,0) relative to the unrotated ship, and facing at -45 degrees theta and 45 degrees phi, what is its facing in the absolute system?
Date: 06/03/2003 at 10:44:27 From: Doctor Tom Subject: Re: Rotating a vector Hi John, Your question can have many answers, and the one that you use depends on how you're keeping track of the data in the program. Let me tell you how I would do it. If this won't work for you, write back and we'll try to come up with a better way. Assume you have a world coordinate system and that you know where everything is in that system with an x, y and z coordinate. You also need to know the velocity in this system that will be a vector (vx, vy, vz). If your spaceship is at (x, y, z) and has velocity (vx, vy, vz) then after traveling for a short time t, its new position will be (x + t(vx), y + t(vy), z + t(vz)). Obviously, as the game goes on, the velocity can change in size and direction, but for each time step, you can assume that the velocity is constant through that tiny time step. It may be that you need to take multiple small steps between each redisplay - it depends on the relative speeds of things. For example, if the projectiles are really fast, at one display time they might be on one side of a ship and on the next display time they may have completely passed through, so just checking for a collision before and after may miss the collision. This just depends on relative speeds, but if there's any chance of this happening, be sure to write your code so that you can do the calculations for multiple steps per display cycle. For projectiles, a position and velocity is sufficient to describe their state. For ships, you also have to keep orientation in mind. Usually the vector along the axis of the ship will be aligned with the velocity, but even if that were always the case, you still need to keep track of the "up" direction. In other words, if the pilot is facing in the direction of the velocity vector, his "up" direction could be at any angle perpendicular to this. The usual way to keep track of orientation is with a little reference frame: three perpendicular normalized vectors. One vector points "forward," another perpendicular to that points "up" and there is a third, perpendicular to both, that points sideways. As the spaceship flies along, the directions of the reference frame change. For example, if the pilot turns right, the "up" vector continues to point in the same direction, but the "forward" and "sideways" vectors change. If the pilot "pulls back on the stick" to climb, then the sideways vector stays the same, but the "forward" and "up" vectors change, et cetera. To calculate these rotations of the reference frame, all you need to do is rotations about the three axes with standard rotation matrices. These can be applied to each of the three vectors "forward," "up," and "sideways" to get new versions of all three. There will be roundoff error when you do this a whole bunch of times, so from time to time you'll need to renormalize. Here's how to do that: Make all the forward vectors have length 1 by dividing each component by its length. Take the vector cross product of this vector with the sideways vector to get a new "up" vector guaranteed perpendicular to the "forward" vector. Normalize this new "up" vector. Finally, calculate a new sideways vector by taking the negative cross product of the new forward and up vectors. Now you will have a set of three vectors, all mutually perpendicular, and all of length one, so you'll have a new normalized frame. If you put the three vectors "up," "forward," and "right" together to form a 3x3 matrix, this is exactly the matrix that will convert the reference frame of a ship in "standard position" with forward in the x direction, up in the y direction and sideways in the z direction to the current orientation. To get the current orientation back to the standard position, you simply need to invert this matrix and multiply your points by the inverted version. Luckily, inversion of these normalized matrices is trivial - just transpose the matrix (exchange rows and columns). So, suppose you have a ship at (x, y, z) and it has a frame of (fx, fy, fz) (forward), (ux, uy, uz) (up), and (sx, sy, sz) (sideways) all normalized and mutually perpendicular. The matrix: ( fx fy fz ) ( ux uy uz) ( sx sy sz ) will transform points to the ship's coordinate system, so the transpose: ( fx ux sx ) ( fy uy sy ) ( fz uz sz ) will do the inverse operation. Now, suppose you have a projectile at (px, py, pz). First, translate the center of the ship and the projectile so that the center of the ship is at (0, 0, 0) by subtracting (x, y, z) from the projectile's coordinates: (px-x, py-y, pz-z). Multiply that translated projectile's coordinates by the transposed matrix above. This will give the location of the projectile in ship's coordinates. Now see if the projectile is inside the ship: to be inside, its x-coordinate must be between the ship's minimum and maximum x-coordinates, et cetera. This may seem messy, but it's the easiest way I know. There may be minor errors in what I have just typed. As you code it up, test it with easy-to-check examples, say with the ship aligned with the axes, with the ship turned 90 degrees in some direction, with the ship rotated 45 degrees, and maybe one or two more. You can find additional details in books on computer graphics. Finally, I have written a couple of papers on matrices and transformations aimed at bright high-school kids that you might find helpful. Go to: Mathematical Circles Topics http://www.geometer.org/mathcircles and look at the PDF files: Homogeneous Coordinates for Computer Graphics http://www.geometer.org/mathcircles/cghomogen.pdf Introduction to Matrices http://www.geometer.org/mathcircles/Matrices.pdf Good luck! I'm sure you'll learn a lot. - Doctor Tom, The Math Forum http://mathforum.org/dr.math/
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