```- -- - ------------------------------------------------------------------------ 3D Rotations in Mathematics & Demos --------------------------------------------------------------------- ---- -- - Strange title, isn't it? :-) I just finished my exploration of matrices & 3D (yeah, after 3/4 year of using it :-)) so why not to write some lines about it... Everyone knows it ================= We want to rotate a point in 2D space (at first) : new x := x*cos(phi) - y*sin(phi) new y := x*sin(phi) + y*cos(phi) That means if we have a point in Carthesian Coordinate System (CCS), our point with coordinates (x,y) will be anticlockwise rotated around (0,0) in this system. Graphically: ^ +y | | | . <- rotated point with new (x,y) | / | / _. <- point (x,y) |/-- --------------> +x |(0,0) | | | | | | Clear. But! As you probably know, computer screen doesn't look like this. We can do two things if we want correct rotations: "convert" carthesian coordinates to computer ones or derive new rotation formulas. 1. Carthesian vs Computer Coordinates ===================================== Our situation looks like this: (0,0) -----------> +x |\--_. <- point (x,y) | \ | \ | \. <- rotated point with new (x,y) | | | | v +y We see two differences: y-direction and anti/clock- wise rotation. In 2D, both differences can be solved by a simple equation y := (yres-1) - y In 3D, it isn't so easy. Remember we're rotating around 3 axes, firstly around Z (gama angle), then Y (beta) and finally X (alpha) ( +cosG +sinG 0 ) ( +cosB 0 +sinB ) ( 1 +cosA +sinA ) MZ = ( -sinG +cosG 0 ) MY = ( 0 1 0 ) MX = ( 0 -sinA +cosA ) ( 0 0 1 ) ( -sinB 0 +cosB ) ( 0 0 0 ) So, we derive our new point by new [P] = [x y z] * MZ*MY*MX, right? In my short coder's life I've seen as many rotation matrices as sources released by coders =) After multiplying you will get: ( +cosBcosC +cosAsinC-sinAsinBcosC +sinAsinC+cosAsinBcosC ) ( -cosBsinC +cosAcosC+sinAsinBsinC +sinAcosC-cosAsinBsinC ) ( -sinB -sinAcosB +cosAcosB ) Believe me, this is the one and only correct solution :-) BUT. This matrix works in CCS, but not in computer space. So, what's wrong? Let's look at our two problems: a) y-direction: here there aren't any changes, we use y := (yres-1) - y b) if you run your 3D engine now, you'll get the feeling everything is alright (as I did for about 3/4 year ;-)) But there's that clockwise problem - if our engine has to be correct (that means it respects a lefthand rule since we're in lefthanded CCS = z-axis goes away from us) a rotation around y-axis has to switch to another direction: ^ +z | | | / | / . <- out rotated point (we're breaking the lefthand rule) | / |/___--. <- our point (x,y) ------*-------> +x /| / | / | / | /+y | So, we need to change from anticlockwise to clockwise rotation.. How do we do it? Simply ;) Just imagine how we define x & y coordinates on circle in 2D. If we're going "up" with our point, the y-value is increased. This is an anticlockwise rotation. And we want a clockwise one. And its as simple as negating the y coordinate! And y coordinate = r*sin(phi) -> every sin(beta) becomes -sin(beta). Remember "beta" is an angle we're rotating around y-axis. Our matrix becomes: ( +cosBcosC +cosAsinC+sinAsinBcosC +sinAsinC-cosAsinBcosC ) ( -cosBsinC +cosAcosC-sinAsinBsinC +sinAcosC+cosAsinBsinC ) ( +sinB -sinAcosB +cosAcosB ) And our rotation will look like: [rotate point using matrix above] [calculate projected coordinates] [center on screen: x := x + xres/2; y := (yres/2 - 1) -y ] [write pixel to screen] Now you've got a mathematically correct rotation in 2D and 3D... but there's one stupid thing and that is a need of one more instruction in the projection loop - you can't do for example "119 - y" in one instruction since you need the value 119 for the next use => at least two instructions are needed, that's bad if you want to rotate about 1000 points... ok, DSP has a +/- option in multiplying, but the solution above is a bit stupid ;-) A better solution is : 2. Deriving New Rotation Formulas =================================== What does this mean? When I began with 3D stuff I thought programs like Neon or 3DS use CCS instead of computer screen for calculating their y-coordinates. But this isn't true of course - why give coders some additional work? :-) So we have some 3d objects saved in a computer space system and we want to rotate it in mathematically correct way (ie. not breaking the lefhand-rule) When you try to use original matrix with: x := x + xres/2 y := y + yres/2 you'll get quite strange result ;-) Where's the problem? The original matrix is "built" for CCS. And... the difference between CCS and computer screen is in the y coordinate. In the sign of y coordinate. And what affects the y- coordinate in 3D? Rotation around x & z axis. So the only thing you have to do is negate the sin(alpha) and sin (gama) values: ( +cosBcosC -cosAsinC+sinAsinBcosC +sinAsinC+cosAsinBcosC ) ( +cosBsinC +cosAcosC+sinAsinBsinC -sinAcosC+cosAsinBsinC ) ( -sinB +sinAcosB +cosAcosB ) So people, this is the one and only matrix for correct rotation in a lefthanded coordinate system ! For sure, we're in the system which looks like this: | ^ +x ^ +z ^ +y | | | | | /|+z | | | | / | /| +z | /| +y | /| +x | / | / | / | / | / | / | / | / |/ |/ |/ |/ -------*-------> +x *-----> +y *-----> +x *-----> +z /| / | anticlockwise anticlockwise anticlockwise / | rotation around rotation around rotation around / | z-axis y-axis x-axis / | v +y And here it is... you see all rotations are correct, the lefthand rule stays unbroken, simply... this is what I like ;-) But beware! If you use this way, your vertices will be swapped in comparison with classic "conversion"! (clockwise->anticlockwise and vice versa) So you need to modify your backface culling routine (ble -> bge) and maybe some other things (for example, if you use realtime interpolation, right and left side will be swapped since the next point in anticlockwise order = the previous point in clockwise one) So, don't be stupid and use this way from the start and you'll save yourself a lot of work ;-) MiKRO ------------------------------------------------------------------------------- MiKRO XE/XL/MegaSTE/Falcon/CT60 mikro.atari.org ------------------------------------------------------------------------------- ```