[tutorial] 3D matrix (rotation,translation,scaling, rotation around an arbitrary axis)

[Homer_simpson]

Matrix mathematics can be very useful, for now i've only shown you how to use them for rotation,scaling and translation, i'll update this with rotation around an arbitrary axis. I'll also show you how to multiply matrices.

Quote:

type dot :
record
x, y, z : real
end record



type cam :
record
axy, axz, ayz, cx, cy, cz, zoom : real
end record
var camera : cam
camera.cx := -1
camera.cy := -1
camera.cz := -10
camera.axy := 0
camera.axz := 0
camera.ayz := 0
camera.zoom := 1000


proc draw3dline (d1, d2 : dot)
if d1.z + 2 not= 0 and d2.z + 2 not= 0 then
var xx : array 1 .. 3 of int
var yy : array 1 .. 3 of int
xx (1) := round ((d1.x - camera.cx) / ((d1.z + 2 - camera.cz) / camera.zoom)) + 320
xx (2) := round ((d2.x - camera.cx) / ((d2.z + 2 - camera.cz) / camera.zoom)) + 320
yy (1) := round ((d1.y - camera.cy) / ((d1.z + 2 - camera.cz) / camera.zoom)) + 200
yy (2) := round ((d2.y - camera.cy) / ((d2.z + 2 - camera.cz) / camera.zoom)) + 200
Draw.FillOval (xx (1), yy (1), 2, 2, 9)
Draw.Line (xx (1), yy (1), xx (2), yy (2), 12)
Draw.Dot (xx (2), yy (2), white)
end if
end draw3dline

proc draw3ddot (d1 : dot)
if d1.z + 2 not= 0 then
var xx : array 1 .. 3 of int
var yy : array 1 .. 3 of int
xx (1) := round ((d1.x - camera.cx) / ((d1.z + 2 - camera.cz) / camera.zoom)) + 320
yy (1) := round ((d1.y - camera.cy) / ((d1.z + 2 - camera.cz) / camera.zoom)) + 200
Draw.Dot (xx (1), yy (1), white)
end if
end draw3ddot

colorback (black)
cls
color (white)
View.Set ("offscreenonly")

type matrix : array 1 .. 3, 1 .. 4 of real
var rotationmatrix:matrix

procedure translate(x,y,z:real,var m:matrix)
m(1,1):=1 m(1,2):=0 m(1,3):=0 m(1,4):=x
m(2,1):=0 m(2,2):=1 m(2,3):=0 m(2,4):=y
m(3,1):=0 m(3,2):=0 m(3,3):=1 m(3,4):=z
end translate

procedure scale(x,y,z:real,var m:matrix)
m(1,1):=x m(1,2):=0 m(1,3):=0 m(1,4):=0
m(2,1):=0 m(2,2):=y m(2,3):=0 m(2,4):=0
m(3,1):=0 m(3,2):=0 m(3,3):=z m(3,4):=0
end scale


procedure rotateXY(a:real,var m:matrix)
m(1,1):=cosd(a) m(1,2):=-sind(a) m(1,3):=0 m(1,4):=0
m(2,1):=sind(a) m(2,2):=cosd(a) m(2,3):=0 m(2,4):=0
m(3,1):=0 m(3,2):=0 m(3,3):=1 m(3,4):=0
end rotateXY



procedure rotateYZ(a:real,var m:matrix)
m(1,1):=1 m(1,2):=0 m(1,3):=0 m(1,4):=0
m(2,1):=0 m(2,2):=cosd(a) m(2,3):=-sind(a) m(2,4):=0
m(3,1):=0 m(3,2):=sind(a) m(3,3):=cosd(a) m(3,4):=0
end rotateYZ


procedure rotateXZ(a:real,var m:matrix)
m(1,1):=cosd(a) m(1,2):=0 m(1,3):=sind(a) m(1,4):=0
m(2,1):=0 m(2,2):=1 m(2,3):=0 m(2,4):=0
m(3,1):=-sind(a) m(3,2):=0 m(3,3):=cosd(a) m(3,4):=0

end rotateXZ

procedure applymatrix(var x,y,z:real,m:matrix)

var tx := (x * m(1,1)) + (y * m(1,2)) + (z * m(1,3)) + m(1,4)

var ty := (x * m(2,1)) + (y * m(2,2)) + (z * m(2,3)) + m(2,4)

var tz := (x * m(3,1)) + (y * m(3,2)) + (z * m(3,3)) + m(3,4)

x:=tx
y:=ty
z:=tz
end applymatrix



var xy, yz, xz : real := 0
var p1, p2, vp : dot
p1.x := 0
p1.y := 0
p1.z := 0

p2.x := 0
p2.y := 0
p2.z := -1
var chars : array char of boolean

loop

Input.KeyDown (chars)
if chars (KEY_LEFT_ARROW) then
rotateXZ(1,rotationmatrix)
applymatrix(p2.x,p2.y,p2.z,rotationmatrix)
end if
if chars (KEY_RIGHT_ARROW) then
rotateXZ(-1,rotationmatrix)
applymatrix(p2.x,p2.y,p2.z,rotationmatrix)
end if
if chars (KEY_UP_ARROW) then
rotateYZ(-1,rotationmatrix)
applymatrix(p2.x,p2.y,p2.z,rotationmatrix)
end if
if chars (KEY_DOWN_ARROW) then
rotateYZ(1,rotationmatrix)
applymatrix(p2.x,p2.y,p2.z,rotationmatrix)
end if

for i : 1 .. 360
vp.x := sind (i)
vp.y := cosd (i)
vp.z := 0
draw3ddot (vp)
vp.x := sind (i)
vp.z := cosd (i)
vp.y := 0
draw3ddot (vp)
vp.y := sind (i)
vp.z := cosd (i)
vp.x := 0
draw3ddot (vp)


end for

draw3dline (p1, p2)
View.Update
delay (5)
cls
end loop

[saltpro15]

Have I mentioned I love you. I've learned more in your last 2 posts than I have in 3 months of Computer Science classes. I want to do an fps with raycasting, I'm not sure how to go about making the target appear on the screen and respond to mouse clicks, any ideas? no wait, if you do have an idea pm me, that topic isn't really covered by this thread.

[Tyr_God_Of_War]

Here, I added some camera movement to the numpad. You cam.axy(..ect.) does nothing at the moment.
I also simplified the matrices, and removed some unused variables. I might have shuffled it around a bit to.

Turing:

type dot :     record         x, y, z : real     end record type cam :     record         axy, axz, ayz, cx, cy, cz, zoom : real     end record type matrix : array 1 .. 3, 1 .. 4 of real var camera : cam camera.cx := -1 camera.cy := -1 camera.cz := -10 camera.axy := 0 camera.axz := 0 camera.ayz := 0 camera.zoom := 1000 proc draw3dline (d1, d2 : dot)     if d1.z + 2 not= 0 and d2.z + 2 not= 0 then         var xx : array 1 .. 3 of int         var yy : array 1 .. 3 of int         xx (1) := round ((d1.x - camera.cx) / ((d1.z + 2 - camera.cz) / camera.zoom)) + 320         xx (2) := round ((d2.x - camera.cx) / ((d2.z + 2 - camera.cz) / camera.zoom)) + 320         yy (1) := round ((d1.y - camera.cy) / ((d1.z + 2 - camera.cz) / camera.zoom)) + 200         yy (2) := round ((d2.y - camera.cy) / ((d2.z + 2 - camera.cz) / camera.zoom)) + 200         Draw.FillOval (xx (1), yy (1), 2, 2, 9)         Draw.Line (xx (1), yy (1), xx (2), yy (2), 12)         Draw.Dot (xx (2), yy (2), white)     end if end draw3dline proc draw3ddot (d1 : dot)     if d1.z + 2 not= 0 then         var xx : array 1 .. 3 of int         var yy : array 1 .. 3 of int         xx (1) := round ((d1.x - camera.cx) / ((d1.z + 2 - camera.cz) / camera.zoom)) + 320         yy (1) := round ((d1.y - camera.cy) / ((d1.z + 2 - camera.cz) / camera.zoom)) + 200         Draw.Dot (xx (1), yy (1), white)     end if end draw3ddot fcn translate (x, y, z : real) : matrix     var m : matrix     m (1, 1) := 1     m (1, 2) := 0     m (1, 3) := 0     m (1, 4) := x     m (2, 1) := 0     m (2, 2) := 1     m (2, 3) := 0     m (2, 4) := y     m (3, 1) := 0     m (3, 2) := 0     m (3, 3) := 1     m (3, 4) := z     result m end translate fcn scale (x, y, z : real) : matrix     var m : matrix     m (1, 1) := x     m (1, 2) := 0     m (1, 3) := 0     m (1, 4) := 0     m (2, 1) := 0     m (2, 2) := y     m (2, 3) := 0     m (2, 4) := 0     m (3, 1) := 0     m (3, 2) := 0     m (3, 3) := z     m (3, 4) := 0     result m end scale fcn rotateXY (a : real) : matrix     var m : matrix     m (1, 1) := cosd (a)     m (1, 2) := -sind (a)     m (1, 3) := 0     m (1, 4) := 0     m (2, 1) := sind (a)     m (2, 2) := cosd (a)     m (2, 3) := 0     m (2, 4) := 0     m (3, 1) := 0     m (3, 2) := 0     m (3, 3) := 1     m (3, 4) := 0     result m end rotateXY fcn rotateYZ (a : real) : matrix     var m : matrix     m (1, 1) := 1     m (1, 2) := 0     m (1, 3) := 0     m (1, 4) := 0     m (2, 1) := 0     m (2, 2) := cosd (a)     m (2, 3) := -sind (a)     m (2, 4) := 0     m (3, 1) := 0     m (3, 2) := sind (a)     m (3, 3) := cosd (a)     m (3, 4) := 0     result m end rotateYZ fcn rotateXZ (a : real) : matrix     var m : matrix     m (1, 1) := cosd (a)     m (1, 2) := 0     m (1, 3) := sind (a)     m (1, 4) := 0     m (2, 1) := 0     m (2, 2) := 1     m (2, 3) := 0     m (2, 4) := 0     m (3, 1) := -sind (a)     m (3, 2) := 0     m (3, 3) := cosd (a)     m (3, 4) := 0     result m end rotateXZ procedure applymatrix (var x, y, z : real, m : matrix)     const tx := (x * m (1, 1)) + (y * m (1, 2)) + (z * m (1, 3)) + m (1, 4)     const ty := (x * m (2, 1)) + (y * m (2, 2)) + (z * m (2, 3)) + m (2, 4)     const tz := (x * m (3, 1)) + (y * m (3, 2)) + (z * m (3, 3)) + m (3, 4)     x := tx     y := ty     z := tz end applymatrix View.Set ("offscreenonly") colorback (black) cls color (white) var p1, p2, vp : dot p1.x := 0 p1.y := 0 p1.z := 0 p2.x := 0 p2.y := 0 p2.z := -1 var chars : array char of boolean loop     Input.KeyDown (chars)     if chars (KEY_LEFT_ARROW) xor chars (KEY_RIGHT_ARROW) then         if chars (KEY_LEFT_ARROW) then             applymatrix (p2.x, p2.y, p2.z, rotateXZ (1))         else % right arrow key             applymatrix (p2.x, p2.y, p2.z, rotateXZ (-1))         end if     end if     if chars (KEY_UP_ARROW) xor chars (KEY_DOWN_ARROW) then         if chars (KEY_UP_ARROW) then             applymatrix (p2.x, p2.y, p2.z, rotateYZ (-1))         else % if chars (KEY_DOWN_ARROW) then             applymatrix (p2.x, p2.y, p2.z, rotateYZ (1))         end if     end if     if chars ('4') xor chars ('6') then         if chars ('4') then             camera.cx -= 0.05         else             camera.cx += 0.05         end if     end if     if chars ('8') xor chars ('2') then         if chars ('8') then             camera.cy += 0.05         else             camera.cy -= 0.05         end if     end if         if chars ('/') xor chars ('*') then         if chars ('/') then             camera.cz -= 0.05         else             camera.cz += 0.05         end if     end if         % if chars ('7') xor chars ('9') then  % axy does nothing?     %     if chars ('7') then     %         camera.axy += 100.05     %     else     %         camera.axy -= 100.05     %     end if     % end if     for i : 1 .. 360         vp.x := sind (i)         vp.y := cosd (i)         vp.z := 0         draw3ddot (vp)         vp.x := sind (i)         vp.y := 0         vp.z := cosd (i)         draw3ddot (vp)         vp.x := 0         vp.y := sind (i)         vp.z := cosd (i)         draw3ddot (vp)     end for     draw3dline (p1, p2)     View.Update     delay (5)     cls end loop