**Vectors are
cool!!!**

It's almost impossible to do graphics
programming without using vectors. Almost all math concerning 3d coding use
vectors. If you hate vectors, read on and you'll probably

love them more than
your girlfriend after you've finished reading this article. ;*)

**What are
vectors? **

First off, let me define 2 quantities: The *SCALAR* and *VECTOR*
quantities. Okay, scalar quantities are just values. One example of it is
Temperature. You say, "It's 40 degrees Celsius here", and that's it. No sense of
direction. But to define a vector you need a direction or sense. Like when the
pilot say's, "We are 40 kilometers north of Midway". So a scalar quantity is
just a value while a vector is a value + direction.

Look at the figure
below: The *arrow(Ray)* represents a vector. The
"Head" is its *"Sense"*(direction is not
applicable here) and the "Tail" is its starting point. The distance from head to
tail is called its "magnitude".

In this vector there
are 2 components, the X and Y component. X is the horizontal and Y is the
vertical component. Remember that **"ALL VECTOR OPERATIONS ARE DONE WITH
ITS COMPONENTS."**

I like to setup my vectors in this TYPE:

x as single

y as single

End TYPE

either way on that line.

**Definitions:**

*Orthogonal vectors are vectors perpendicular to each other. It's sticks up 90 degrees.

To get a vector between 2
points:

2d:

**v = (x2 - x1) + (y2 - y1)**
3d:

vx = x2 - x1

vy = y2 - y1

vz = z2 - z1

where: (x2-x1) is the horizontal component and so on.

Vectors are not limited to the cartesian coordinate system.

In polar form:

**Resolving a vector
by its components**

Suppose a vector v has a magnitude 5 and direction given by Theta = 30 degrees. Where theta is the angle the vector makes with the positive x-axis. How do we resolve this vectors' components?

Remember the Polar to Cartesian
conversion?** v.x = cos(theta) v.y
= sin(theta)**

Let vx = horizontal component

Let vy = horizontal component

Let Theta = Be the angle

So...

v.x = 5 * cos(30)

v.x = 4.33

v.y = |v| * sin(theta)

v.y = 5 * sin(30)

v.y = 2.50

What I've been showing you is a 2d vector. Making a 3d vector is just adding another component, the Z component.

x as single

y as single

z as single

End TYPE

**1. Scaling a
vector(Scalar multiplication)**

Purpose:

This is used to scale a vector by a
scalar value. Needed in the scaling of models and changing the velocity of
projectiles.

In equation:** v = v *
scale**In
qbcode:

v.y = v.y * Scale

v.z = v.z * Scale

**2. Getting the
Magnitude(Length) of a vector**

Purpose:

Used in *"Normalizing"*

Equation:** |V| = Sqr(v.x^2 + v.y^2 +
v.z^2)**

QBcode:

**3. Normalizing a
vector**

Purpose:

Used in light sourcing, camera transforms, etc. Makes the vector a
** "unit-vector"** that is a vector having a magnitude of 1. Divides
the vector by its length.

Equ:

-------

|v|

QBCode:

v.x = v.x / mag!

v.y = v.y / mag!

v.z = v.z / mag!

Purpose:

Used in many things like lightsourcing and vector
projection. Returns the cosine of the angle between any two vectors ( Assuming
the vectors are Normalized) . A Scalar.

The dot product is also called the
"Scalar" product.

Equ:** v.w = v.x* w.x + v.y* w.y + v.z*
w.z **

QBCode:

Fun fact:

Proof: "What is the cosine of 90?"

Purpose:

Used in lightsourcing, camera transformation, back-face culling, etc. The cross
product of 2 vectors returns another vector that is orthogonal to the plane that
has the first 2

vectors. Let's say we have vectors U and
F.

Equ:** U x F = R**

QB code:

R.y = U.z * F.x - F.z * U.x

R.z = U.x * F.y - F.x * U.y

Fun facts:** C is the vector
orthogonal to A and B.* C is the NORMAL to the plane that includes A and
B. The cross-product of any two vectors can best be remembered by the
CRAMERS RULE on DETERMINANTS. Thought of it while
taking a bath. I'll tell you when I finish my matrix chapter.*
The cross product is exclusive to 3d and its also called the "Vector"
product.*

**6. Vector
Projection**

Purpose:

Used in resolving the second vector of the camera matrix (Thanks Toshi!). For vectors A and B...

Equ:** U.Z *
Z**

*QB
code:*

Let N = vector projection of U to Z. The vector parallel to Z.

** T! = Vector.Dot(U,
Z) N.x = T! * Z.x N.y = T! *
Z.y N.z = T! * Z.z**

Purpose:

Used in camera and
object movements. Anything that you'd want to move relative to your
camera. Adding vectors is just the same as adding their components. Let A
and B be vectors in 3d, and C is the sum:

Equ:** C = A + B C = (ax + bx) +
(ay + by) + (az + bz)**

QBcode:** c.x = a.x +
b.x c.y = a.y + b.y c.z = a.z +
b.z **

Now that Most of the Math is out of the way....

**I. WireFraming and BackFace culling**

** **I
like to make use of types with my 3d engines.

For Polygons:

** Type
Poly p1 as
integer p2 as
integer p3 as
integer end type**

P1 is the first
vertex, p2 second and p3 third. Let's say you have a nice rotating cube composed
of points, looks spiffy but you want it to be composed of polygons(Triangles) in
this case). If we have a cube with vertices:** Vtx1 50, 50, 50
:x,y,z Vtx2 -50, 50,
50 Vtx3 -50,-50,
50 Vtx4 50,-50,
50 Vtx5 50,
50,-50 Vtx6 -50,
50,-50 Vtx7
-50,-50,-50 Vtx8
50,-50,-50**

What we need are connection points that define a face. The one below is a Quadrilateral face(4 points)

** Face1 1, 2, 3,
4 Face2 2, 6, 7,
3 Face3 6, 5, 8,
7 Face4 5, 1, 4,
8 Face5 5, 6, 2,
1 Face6 4, 3, 7,
8**

Face1 would have vertex 1, 2, and 3 as its connection vertices.

Now since we want triangles instead of quads, we divide each quad into 2 triangles, which would make 12 faces. It' also imperative to arrange your points in counter-clockwise or

clockwise order so that backface culling would work. In this case I'm using counter-clockwise.

The following code divide the quads into 2 triangles with vertices arranged in counter-clockise order. Tri(j).idx will be used for sorting.

j = 1

FOR i = 1 TO 6

READ p1, p2, p3, p4 'Reads the face(Quad)

Tri(j).p1 = p1

Tri(j).p2 = p2

Tri(j).p3 = p4

Tri(j).idx = j

j = j + 1

Tri(j).p1 = p2

Tri(j).p2 = p3

Tri(j).p3 = p4

Tri(j).idx = j

j = j + 1

NEXT i

To render the cube without backface culling, here's the pseudocode:

2. Rotatepoints

3. Project points

4. Sort(Not needed for cubes and other simple polyhedrons)

5. Get Triangles' projected coords

ie.

x1 = Model(Tri(i).P1).ScreenX

y1 = Model(Tri(i).P1).ScreenY

x2 = Model(Tri(i).P2).ScreenX

y2 = Model(Tri(i).P2).ScreenY

x3 = Model(Tri(i).P3).ScreenX

y3 = Model(Tri(i).P3).ScreenY

6. Draw

Tri x1,y1,x2,y2,x3,y3,color

Backface
culling is also called *"Hidden face removal"*.
In essense, it's a way to speed up your routines by NOT showing a polygon if
it's not facing towards you. But how do we know what face of the polygon is the
"right" face? Let's take a CD as an example, there are 2 sides to a particular
CD. One side that the data is to be written and the other side where the label
is printed. What if we decide that the Label-side should be the right side? How
do we do it? Well it turns out that the answer is our well loved NORMAL. :*) But
for that to work, we should *sequentially* arrange our vertices in counter or
clockwise order.

If you arranged your
polys' vertices in counter- clockwise order as most 3d modelers do, you just get
the projected z-normal of the

poly and check if its greater than(>)0. If
it is, then draw triangle. Of course if you arranged the vertices in
clockwise

order, then the poly is facing us when the Z-normal is
<0.

** Counter-Clockwise arrangement of
vertices**:

**Clockwise Arrangement of
vertices:**

Since we only need the
z component of the normal to the poly, we could even use the "projected"
coords(2d) to get the z component!*QBcode:*** Znormal = (x2 - x1) * (y1 - Y3) -
(y2 - y1) * (x1 - X3) IF (Znormal > 0) THEN '>0 so
vector facing us Drawpoly
x1,y1,x2,y2,x3,y3 end if**

Here's the example file:

There are numerous sorting techniques that I use in my 3d renders here are the most common:

**1. Bubble sort
(modified) 2. Shell sort 3. Quick
sort 4. Blitz sort (PS1 uses this according to
Blitz) **

I won't go about
explaining how the sorting algorithms work. I'm here to discuss how to implement
it in your engine. It may not be apparent to you (since you are rotating
a

simple cube) but you need to sort your polys to make your renders look
right. The idea is to draw the farthest polys first and the nearest last. Before
we could go about sorting

our polys we need a new element in our
polytype.** Type
Poly p1 as
integer p2 as
integer p3 as
integer idx as
integer zcenter as
integer end type**

IDX, not by subscript.

*Zcenter is the theoretical center of the polygon. It's a 3d coord (x,y,z)

To get the center of any polygon or polyhedra(model),you add all the 3 coordinates and divide it by the number of vertices(In this case 3).

Since we only want to get the z center:

** Zcenter=
Model(Poly(i).p1)).z + Model(Poly(i).p2)).z +
Model(Poly(i).p3)).z Zcenter =
Zcenter/3**

Optimization trick:

We don't really need to find the *real* Zcenter since all the z values that were added are going to be still sorted

right. Which means... No divide!!!

Now you sort the polys like this:

**FOR i% = Lbound(Poly) TO
UBOUND(Poly) Poly(i%).zcenter = Model(Poly(i%).p1).Zr +
Model(Poly(i%).p2).Zr +
Model(Poly(i%).p3).Zr Poly(i%).idx
= i%NEXT i%Shellsort Poly(), Lbound(Poly),
UBOUND(Poly)**

To Draw the model, you use the
index(Poly.idx)*QBcode:*

j = Poly(i).idx

x1 = Model(Poly(j).p1).scrx 'Get triangles from "projected"

x2 = Model(Poly(j).p2).scrx 'X and Y coords since Znormal

x3 = Model(Poly(j).p3).scrx 'Does not require a Z coord

y1 = Model(Poly(j).p1).scry

y2 = Model(Poly(j).p2).scry

y3 = Model(Poly(j).p3).scry

'Use the Znormal,the Ray perpendicular(Orthogonal) to the

'Screen defined by the Triangle (X1,Y1,X2,Y2,X3,Y3)

'if Less(>) 0 then its facing in the opposite direction so

'don't plot. If <0 then its facing towards you so Plot.

Znormal = (x2 - x1) * (y1 - y3) - (y2 - y1) * (x1 - x3)

IF Znormal < 0 THEN

DrawTri x1,y1,x2,y2,x3,y3

END IF

NEXT i

Here's a working example:

These 2 systems are extentions of the polar coordinate system. Where polar is 2d these 2 are 3d. :*)

The cylindrical coodinate system is useful if you want to generate models mathematically. Some examples are Helixis, Cylinders(of course), tunnels or any tube-like model. This

system works much like 2d, but with an added z component that doesn't need and angle. Here's the equations to convert cylindrical to rectangular coordinate system.

Here's the Cylindrical to rectangular coordinate conversion equations. Almost like 2d. Of course this cylinder will coil on the z axis. To test yourself, why dont you change

the equations to coil it on the y axis?

y = SIN(theta)

z = z

To generate a cylinder:

z! = zdist * Slices / 2

FOR Slice = 0 TO Slices - 1

FOR Band = 0 TO Bands - 1

Theta! = (2 * PI / Bands) * Band

Model(i).x = radius * COS(Theta!)

Model(i).y = radius * SIN(Theta!)

Model(i).z = -z!

i = i + 1

NEXT Band

z! = z! - zdist

NEXT Slice

Here's a 9 liner I made using that equation.

**9Liner.Bas**

This is another useful system. It can be used for Torus and Sphere generation. Here's the conversion:

**x = SIN(Phi)* COS(theta) y
= SIN(Phi)* SIN(theta) z =
COS(Phi)**

Where: Theta = Azimuth ; Phi = Elevation

To generate a sphere:

**QBcode:**

** i =
0 FOR SliceLoop = 0 TO Slices -
1 Phi! = PI / Slices *
SliceLoop FOR BandLoop = 0 TO
Bands - 1
Theta! = 2 * -PI / Bands *
BandLoop
Model(i).x = -INT(radius * SIN(Phi!) *
COS(Theta!))
Model(i).y = -INT(radius * SIN(Phi!) *
SIN(Theta!))
Model(i).z = -INT(radius *
COS(Phi!))
i = i + 1 NEXT
BandLoop NEXT SliceLoop**

Here's a little particle engine using the spherical coordinate system.

Here's an example file to generate models using those equations:

**III. Different Polygon fillers**

Now how do we make a flat filler? Let me introduce you first to the idea of

screen at location (x1,y1) to (x2,y2) in 10 steps?

Let A = (x1,y1)

B = (x2,y2)

Steps = 10

f(x) = (B-A)/Steps

So....

*QBcode:*

dx! = (x2-x1)/steps

dy! =
(y2-y1)/Steps

x! = x1

y! =
y1

For a = 0 to steps -
1

Pset(x,y),
15

x! = x!
+ dx!

y! =
y! + dy!

next
a

That's all to there is to interpolation.
:*)

Now that we have an idea of what linear
interpolation is we could make a flat triangle filler.**The 3 types of triangle**

*A. Flat
Filled*

**1. Flat Bottom**

**2. Flat Top
**

3. Generic
Triangle

In both the Flat Top and
Flat bottom cases, it's easy to do both triangles as we only need to interpolate
A to B and A to

C in Y steps. We draw a horizontal line in between (x1,y) and
(x2,y).

The problem lies when we want to draw a generic
triangle since we don't know if it's a flat top or flat bottom. But it turns out
that there is an all too easy way to get around with this. Analyzing the generic
triangle, we could just divide the triangle into 2 triangles. One Flat Bottom
and One Flat Top!

We draw it with 2 loops.
The first loop is to draw the Flat Bottom and the second loop is for the Flat
Top.

PseudoCode:

TOP PART ONLY!!!!(FLAT BOTTOM)

1. Interpolate **a.x** and draw each scanline from
**a.x** to **b.x** in **(b.y-a.y)** steps.

** ie.
a.x = x3 - x1**

** b.x = y3 - y1**

** Xstep1! = a.x /
b.x**

2. Interpolate **a.x** and draw each
scanline from **a.x** to **c**.**x** in **(c.y-a.y)**
steps.

** ie.
a.x = x1 - x3**

** c.x = y1 - y3**

** Xstep3! = a.x /
c.x**

3. Draw each scanline(Horizontal line) from
**a.y** to **b.y **incrementing **y** with one in each step,
interpolating **LeftX** with** Xstep1!** and **RightX** with
**Xstep3!**. You've just finished drawing the TOP part of the
triangle!!!

4. Do the same with the bottom-half
interpolating from **b.x** to **c.x** in **b.y** steps.

*PseudoCode:*

**1. Sort VerticesIF y2 < y1
THEN SWAP y1, y2 SWAP x1,
x2END IFIF y3 < y1 THEN SWAP y3,
y1 SWAP x3, x1END IFIF y3 < y2
THEN SWAP y3, y2 SWAP x3, x2END
IF2.
Interpolate A to Bdx1 =
x2 - x1 dy1 = y2 - y1 IF dy1 <> 0
THEN Xstep1! = dx1 /
dy1 ELSE Xstep1! = 0END
IF3. Interpolate B
to Cdx2 = x3 -
x2 dy2 = y3 - y2IF dy2 <> 0 THEN
Xstep2! = dx2 / dy2ELSE Xstep2! = 0END
IF4. InterPolate A
to Cdx3 = x1 -
x3dy3 = y1 - y3IF dy3 <> 0 THEN Xstep3! =
dx3 / dy3ELSE Xstep3! = 0END
IF5. Draw Top
PartLx! = x1 'Starting
coordsRx! = x1FOR y = y1 TO y2 - 1 LINE (Lx!,
y)-(Rx!, y), clr Lx! = Lx! + Xstep1! 'increment
derivatives Rx! = Rx! + Xstep3!NEXT
y6. Draw Lower
PartLx! = x2FOR
y = y2 TO y3 LINE (Lx!, y)-(Rx!, y),
clr Lx! = Lx! + delta2! Rx! = Rx! +
delta3!NEXT y**

Here's an example file:

**B. Gouraud
Filled**

There is
not that much difference between the flat triangle and the gouraud
triangle. In the calling sub, instead of just the 3 coodinates, there are
3 paramenters more. Namely: **c1,c2,c3**. They are the colors we could
want to interpolate between vertices. And since you know how to interpolate
already, it would not be a problem. :*)

First we need a horizontal line routine that draws with interpolated colors. Here's the code. It's self explanatory.

*dc! is the ColorStep(Like the Xsteps)

*QBcode:*

** HlineG
(x1,x2,y,c1,c2) dc! = (c2 - c1)/ (x2 -
x1) c! = c1 For x = x1 to
x2 Pset(x , y) ,
int(c!) c! = c! +
dc! next x**

Now that we have a horizontal gouraud line, we will modify some code into our flat filler to make it a gouraud filler. I won't give you the whole code, but some important snippets.

**1. In the sorting stuff:
(You have to do this to all the IF's.**

**IF y2 < y1
THEN SWAP y1, y2 SWAP x1,
x2 SWAP c1, c2END IF**

**dx1 = x2 - x1 dy1 = y2 -
y1 dc1 = c2 - c1IF dy1 <> 0 THEN
Xstep1! = dx1 / dy1 Cstep1! = dc1 /
dy1ELSE Xstep1! = 0 Cstep1! =
0END IF**

**5. Draw Top
Part**

Lx! = x1 'Starting
coords

Rx! = x1

Lc! = c1 'Starting colors

Rc! = c1

FOR y = y1 TO
y2 - 1

HlineG Lx!, Rx!, y, Lc!, Rc!

Lx! = Lx! + Xstep1!

Rx! = Rx! +
Xstep3!

Lc! = Lc! + Cstep1! 'Colors

Rc! = Rc! + Cstep3!

NEXT y

It's that easy! You have to interpolate just 3 more values! Here's the complete example file:

C. Affine
Texture Mapped

Again, there is not much difference between the previous 2 triangle routines
from this. Affine texturemapping also involves the same algo as that of
the flat filler. That is, Linear interpolation. That's probably why
it doesn't look good. :*( But it's fast. :*). If in the gouraud
filler you need to interpolate between **3 colors**, you need to interpolate
between **3 U and 3 V **texture coordinates in the affine mapper.
That's **6** values in all. In fact, it's almost the same as
gouraud filler!

Now we have to modify our Gouraud Horizontal line routine to a textured line routine.

**This assumes that the
texture size is square and a power of 2. Ie. 4*4, 16*16, 128*128,etc. And is
used to prevent from reading pixels outside the texture.*The texture mapper
assumes a QB GET/PUT compatible image. Array(1) = width*8; Array(2) = Height;
Array(3) = 2 pixels.* HlineT also assumes that a DEF SEG = Varseg(Array(0))
has been issued prior to the call. TOFF is the Offset of the image in multiple
image arrays. ie: TOFF = VARPTR(Array(0))*TsizeMinus1 is Texturesize
-1.*

du! = (u2 - u1)/ (x2 - x1)

dv! = (v2 - v1)/ (x2 - x1)

u! = u1

v! = v1

TsizeMinus1 = Tsize - 1

For x = x1 to x2

'get pixel off the texture using

'direct memory read. The (+4 + TOFF)

'is used to compensate for image

'offsetting.

Tu=u! AND TsizeMinus1

Tv=v! AND TsizeMinus1

Texel = Peek(Tu*Tsize + Tv + 4 + TOFF)

Pset(x , y) , Texel

u! = u! + du!

v! = v! + dv!

next x

Now we have to modify the rasterrizer to support U and V coords. All we have to do is interpolate between all the coords and we're good to go.

*1. In the sorting stuff:
(You have to do this to all the IF's.*IF y2 < y1 THEN

SWAP y1, y2

SWAP x1, x2

SWAP u1, u2

SWAP v1, v2

END IF

dx1 = x2 - x1

dy1 = y2 - y1

du1 = u2 - u1

dv1 = v2 - v1

IF dy1 <> 0 THEN

Xstep1! = dx1 / dy1

Ustep1! = du1 / dy1

Vstep1! = dv1 / dy1

ELSE

Xstep1! = 0

Ustep1! = 0

Vstep1! = 0

END IF

Lx! = x1 'Starting coords

Rx! = x1

Lu! = u1 'Starting U

Ru! = u1

Lv! = v1 'Starting V

Rv! = v1

FOR y = y1 TO y2 - 1

HlineT Lx!, Rx!, y, Lu!, Ru!, Lv!, Rv!

Lx! = Lx! + Xstep1!

Rx! = Rx! + Xstep3!

Lu! = Lu! + Ustep1! 'U

Ru! = Ru! + Ustep3!

Lv! = Lv! + Vstep1! 'V

Rv! = Rv! + Vstep3!

NEXT y

Here's the example demo for you to learn from. Be sure to check the algo as it uses fixpoint math to speed things up quite a bit. :*)

**IV. Shading and Mapping Techniques**

So you want your cube filled and lightsourced, but don't know how to? The answer is Lambert Shading. And what does

Lambert shading use? The NORMAL. Yes, it's the cross-product thingy I was writing about. How do we use the normal you say. First, you have a filled cube composed of triangles (Polys), now we define a vector orthogonal to that plane(Yep, the Normal) sticking out.

How do we calculate normals? Easy, use the cross product!

2. get poly's x, y and z coords

3. define vectors from 3 coords

4. get the cross-product(our normal to a plane)

5. Normalize your normal

FOR i = 1 TO UBOUND(Poly)

P1 = Poly(i).P1 'get poly vertex

P2 = Poly(i).P2

P3 = Poly(i).P3

x1 = Model(P1).x 'get coords

x2 = Model(P2).x

x3 = Model(P3).x

y1 = Model(P1).y

y2 = Model(P2).y

y3 = Model(P3).y

Z1 = Model(P1).z

Z2 = Model(P2).z

Z3 = Model(P3).z

ax! = x2 - x1 'derive vectors

bx! = x3 - x2

ay! = y2 - y1

by! = y3 - y2

az! = Z2 - Z1

bz! = Z3 - Z2

'Cross product

xnormal! = ay! * bz! - az! * by!

ynormal! = az! * bx! - ax! * bz!

znormal! = ax! * by! - ay! * bx!

'Normalize

Mag! = SQR(xnormal! ^ 2 + ynormal! ^ 2 + znormal! ^ 2)

IF Mag! <> 0 THEN

xnormal! = xnormal! / Mag!

ynormal! = ynormal! / Mag!

znormal! = znormal! / Mag!

END IF

v(i).x = xnormal! 'this is our face normal

v(i).y = ynormal!

v(i).z = znormal!

NEXT i

root alone would make my renders slow as hell!!"

A: No. You only need to do this when setting up your renders.

ie. Only do this once, and at the top of your proggie.

Now that we have our normal, we define a light source. Your light source is also a vector. Be sure that both vectors

are normalized.

ie.

Light.y > The light vector

Light.z/

Polynormal.x\

Polynormal.y > The Plane normal

Polynormal.z/

The angle in
the pic is the ** incident angle** between the light and the plane
normal. the

product by the color range!!! In screen 13: Dot*255.

ny! = PolyNormal.y

nz! = PolyNormal.z

lx! = LightNormal.x

ly! = LightNormal.y

lz! = LightNormal.z

Dot! = (nx! * lx!) + (ny! * ly!) + (nz! * lz!)

IF Dot! < 0 then Dot! = 0

Clr = Dot! * 255

FlatTri x1, y1, x2, y2, x3, y3, Clr

end if

Here's an example file in action:

After the lambert shading, we progress into gouraud shading.Q: But how do we find a normal to a point? A: You can't. There is no normal to a point. The cross-product is exclusive to planes(3d) so you just can't. You don't have to worry though, as there are ways around this problem.

What we need
to do is to find adjacent faces that the vertex is located and average their
** face normals**. It's an approximation but it works!

Let: V()= Face normal; V2()
vertexnormal*QBcode:* FOR i = 1 TO
Numvertex

xnormal! = 0

ynormal! = 0

znormal! = 0

FaceFound = 0

FOR j = 0 TO UBOUND(Poly)

IF Poly(j).P1 = i OR Poly(j).P2 = i OR Poly(j).P3 = i THEN

xnormal! = xnormal! + v(j).x

ynormal! = ynormal! + v(j).y

znormal! = znormal! + v(j).z

FaceFound = FaceFound + 1 'Face adjacent

END IF

NEXT j

xnormal! = xnormal! / FaceFound

ynormal! = ynormal! / FaceFound

znormal! = znormal! / FaceFound

v2(i).x = xnormal! 'Final vertex normal

v2(i).y = ynormal!

v2(i).z = znormal!

NEXT i

Now that you have calculated the vertex normals, you only have to pass the rotated vertex normals into our gouraud filler!!! ie. Get the dot product between the rotated vertex normals and multiply it with the color range. The product is your color coordinates.

*QBcode:*

**IF znormal < 0
THEN nx1! = CubeVTXNormal2(Poly(i).P1).X
'Vertex1 ny1! =
CubeVTXNormal2(Poly(i).P1).Y nz1! =
CubeVTXNormal2(Poly(i).P1).Z nx2! =
CubeVTXNormal2(Poly(i).P2).X 'Vertex2 ny2! =
CubeVTXNormal2(Poly(i).P2).Y nz2! =
CubeVTXNormal2(Poly(i).P2).Z nx3! =
CubeVTXNormal2(Poly(i).P3).X 'Vertex3 ny3! =
CubeVTXNormal2(Poly(i).P3).Y nz3! =
CubeVTXNormal2(Poly(i).P3).Z lx! =
LightNormal.X ly! = LightNormal.Y
lz! = LightNormal.Z 'Calculate dot-products of vertex
normals Dot1! = (nx1! * lx!) + (ny1! * ly!) + (nz1! *
lz!) IF Dot1! < 0 THEN
'Limit Dot1! =
0 ELSEIF Dot1! > 1 THEN Dot1! =
1 END IF Dot2! = (nx2! * lx!) +
(ny2! * ly!) + (nz2! * lz!) IF Dot2! < 0
THEN Dot2! =
0 ELSEIF Dot2! > 1
THEN Dot2! =
1 END IF Dot3! = (nx3! * lx!) +
(ny3! * ly!) + (nz3! * lz!) IF Dot3! < 0
THEN Dot3! =
0 ELSEIF Dot3! > 1
THEN Dot3! =
1 END IF 'multiply by color
range clr1 = Dot1! * 255 clr2 =
Dot2! * 255 clr3 = Dot3! * 255
GouraudTri x1, y1, clr1, x2, y2, clr2, x3, y3, clr3END IF**

Here's and example file:

**3. Phong Shading(Fake)
**

Phong shading is a shading technique which utilizes diffuse, ambient and specular lighting. The only way to do Real phong shading is on a per-pixel basis. Here's the equation:

**Intensity=Ambient + Diffuse
* (L • N) + Specular * (R • V)^Ns**

Where:

**Ambient** = This is the light
intensity that the objects reflect upon the environment. It reaches even
in shadows.

**Diffuse** = Light that scatters
in all direction

**Specular** = Light intensity that
is dependent on the angle between your eye vector and the reflection
vector. As the angle between them increases, the less intense it
is.

**L.N **= The dot product of the
Light(L) vector and the Surface Normal(N)

**R.V **= The dot product of the
Reflection(R) and the View(V) vector.

**Ns** = is the specular intensity
parameter, the greater the value, the more intense the specular light
is.

*L.N could be substututed to R.V which makes our equation:

**Intensity=Ambient + Diffuse
* (L • N) + Specular * (L • N)^Ns**

Technically, * this should be done for every pixel of the polygon.
*But since we are making real-time engines and using QB, this is almost
an impossibilty. :*(

Fortunately, there are some ways around this. Not as good looking but works nonetheless. One way is to make a phong texture and use environment mapping to simulate light. Another way is to modify your palette and use gouraud filler to do the job. How do we do it then? Simple! Apply the equation to the RGB values of your palette!!!

First we
need to calculate the angles for every, color index in our pal. We do this
by interpolating our ** Normals' angle(90 degrees)** and

**PseudoCode:**

**Range = 255 -
0 'screen 13**

**Angle! = PI /
2 '90 degrees**

**Anglestep! =
Angle!/Range 'interpolate**

**For Every color
index...**

**
Dot! = Cos(Angle!)**

**
'''Apply equation**

**
'RED**

**
Diffuse! = RedDiffuse * Dot! **

**
Specular! = RedSpecular + (Dot! ^Ns)**

**
Red% = RedAmbient! + Diffuse! + Specular!**

**
'GREEN**

**
Diffuse! = GreenDiffuse * Dot! **

**
Specular! = GreenSpecular + (Dot! ^Ns)**

**
Green% = GreenAmbient! + Diffuse! + Specular!**

**
'BLUE**

**
Diffuse! = BlueDiffuse * Dot! **

**
Specular! = BlueSpecular + (Dot! ^Ns)**

**
Red% = BlueAmbient! + Diffuse! + Specular!**

**
WriteRGB(Red%,Green%,Blue%,ColorIndex)**

** Angle! =
AngleStep!**

**Loop
until maxcolor **

** This idea came from a Cosmox
3d demo by Bobby 3999. Thanks a bunch!*

Here's an example file:

**4. Texture
Mapping**

** **Texture mapping is a type of fill that uses a
Texture(image) to fill a polygon. Unlike our previous fills, this one "plasters"
an image(the texture) on your cube. I'll start by explaining what are
those U and V coordinates in the Affine mapper part of the article. The U
and V coordinates are the Horizontal and vertical coordinates of the bitmap(our
texture). How do we calculate those coordinates? Fortunately, most
3d modelelers already does this for us automatically. :*).

However, if you like to make your models the math way, that is generating them mathematically, you have to calculate them by yourself. What I do is divide the quad into two triangles and blast the texture coordinates on loadup. Lookat the diagram to see what I mean.

*Textsize is the width or height of the bitmap

*QBcode:*

** FOR j = 1
TO UBOUND(Poly) u1 =
0 v1 =
0 u2 =
TextSize% v2 =
TextSize% u3 =
TextSize% v3 =
0 Poly(j).u1 =
u1 Poly(j).v1 =
v1 Poly(j).u2 =
u2 Poly(j).v2 =
v2 Poly(j).u3 =
u3 Poly(j).v3 =
v3 j = j +
1 u1 =
0 v1 =
0 u2 =
0 v2 =
TextSize% u3 =
TextSize% v3 =
TextSize% Poly(j).u1 =
u1 Poly(j).v1 =
v1 Poly(j).u2 =
u2 Poly(j).v2 =
v2 Poly(j).u3 =
u3 Poly(j).v3 =
v3 NEXT j**

After loading the textures, you just call the TextureTri sub passing the right parameters and it would texture your model for you. It's a good idea to make a 3d map editor that let's you pass texture coordinates, instead of calculating it on loadup. Here's a code snippet to draw a textured poly.

*QBcode:*

**u1 = Poly(i).u1 'Texture
Coordsv1 = Poly(i).v1u2 = Poly(i).u2v2 = Poly(i).v2u3 =
Poly(i).u3v3 = Poly(i).v3TextureTri x1, y1, u1, v1, x2, y2, u2, v2, x3,
y3, u3, v3, TSEG%, TOFF%END IF**

**Tseg% and Toff% are the
Segment and Offset of the Bitmap.*

Here's an example file:

**5. Environment
Mapping**

Environment mapping(also called Reflection Mapping) is a way to display a model as if it's reflecting a surface in front of it. Your model looks like a warped-up mirror! It looks so cool, I jumped off my chair when I first made one. :*) We texture our model using the texture mapper passing a vertex-normal modified texture coordinate. What does it mean? It means we calculate our texture coordinate using our vertex normals!

Here's the formula:

**TextureCoord =
Wid/2+Vertexnormal*Hie/2**

Where:

Wid = Width of the bitmap

Hei = Height of the bitmap

Now, assuming your texture has the same width and height:

*QBcode:*

**Tdiv2! = Textsize% /
2FOR i = 1 TO UBOUND(Poly) u1! = Tdiv2! +
v(Poly(i).P1).x * Tdiv2! 'Vertex1 v1! = Tdiv2! +
v(Poly(i).P1).y * Tdiv2! u2! = Tdiv2! + v(Poly(i).P2).x *
Tdiv2! 'Vertex2 v2! = Tdiv2! + v(Poly(i).P2).y *
Tdiv2! u3! = Tdiv2! + v(Poly(i).P3).x * Tdiv2!
'Vertex3 v3! = Tdiv2! + v(Poly(i).P3).y *
Tdiv2! Poly(i).u1 = u1! Poly(i).v1 =
v1! Poly(i).u2 = u2! Poly(i).v2 =
v2! Poly(i).u3 = u3! Poly(i).v3 =
v3!NEXT i**

After setting up the vertex normals and the texture coordinates, inside your rasterizing loop:

*1. Rotate Vertex
normals*

*2. Calculate texture
coordinates*

*3. Draw
model *

That's it! Your own environment mapped rotating object. ;*)

Here's a demo:

Another one that simulates textures with phong shading using a phongmapped texture.

*Playing with
colors!!!*

**6. Shading in
multicolor**

Our previous shading techniques, Lambert, gouraud, and phong, looks good but you are limited to a single gradient. Not a good fact if you want to use colors. But using colors in screen 13 limits you to flat shading. I bet you would want a gouraud or phong shaded colored polygons right? Well, lo and behold! There is a little way around this problem. :*)

We use a subdivided gradient palette! A subdivided gradient palette divides your whole palette into gradients of colors limited to its subdivision. Here's a little palette I made using data statements and the gradcolor sub.

If you look closely, each line starts with a dark color and progresses to an intense color. And if you understood how our fillers work, you'll get the idea of modifying the fillers to work with this pal. Okay, since I'm feeling good today, I'll just give it to you. After you calculated the Dot-product between the light and poly normals:

**This assumes a 16 color gradient
palette. You could make it 32 or 64 if you want. Of course if you make it
32, you should multiply by 32 instead of 16. :*)*

*QBcode: *

** Clr1 =
(Dot1! * 16) + Poly(j).Clr '16 color grad Clr2 = (Dot2! *
16) + Poly(j).Clr Clr3 = (Dot3! * 16) +
Poly(j).Clr GouraudTri x1, y1, Clr1, x2, y2, Clr2, x3,
y3, Clr3**

Here's an example:

**7. Translucency **

** **A lot of people have asked me about the algo behind my
translucent teapot in ** Mono and Disco**. It's not that hard
once you know how to make a translucent pixel. This is not really TRUE
translucency, It's a gradient-based blending algorithm. You make a 16
color gradient palette and apply it to the color range(Same grad above.
:*)).

**PseudoCode:**

**For Every pixel in the
poly...**

**TempC = PolyPixel and
15**

**BaseColor = PolyPixel -
TempC**

**DestC =
Color_Behind_Poly_Pixel and 15**

**C = (TempC +
DestC)/2**

**C = C +
Basecolor**

**Pset(x,y),C**

What this does for every pixel is to average the polygons color with the color behind it(the screen or buffer) and add it to the basecolor. The basecolor is the starting color for each gradient. Ie. (0-15): 0 is the base color; (16 to 31): 16 is the base color. Hence the AND 15. Of course, you can make it a 32 color gradient and AND it by 31. :*)

Here's a little demo of Box translucency I made for my Bro. Hex. ;*)

Here's the 3d translucency demo:

**Final Words:**

To make good models, use a 3d modeler and import it as an OBJ file as it's easy to read 3d Obj files. Lightwave3d and Milkshape3d can import their models in the OBJ format. In fact I made a loader myself. ;*) Note that some models does not have textures, notably, Ship.l3d, fighter.l3d, etc. The only ones with saved textures are Cubetext, Maze2, TriforcT, and Pacmaze2.

Zipped with OBJs:

Bas File:

This article is just a stepping stone for you into bigger things like Matrices, viewing systems and object handling. I hope you learned something from this article as this took me a while to write. Making the example files felt great though. :*) Any questions, errors in this doc, etc., you can post questions at http://forum.qbasicnews.com/. Chances are, I would see it there.

Next article, I will discuss Matrices and how to use them effectively on your 3d engine. I would also discuss polygon clipping and probably, if space permits, 3d viewing systems. So bye for now, Relsoft, signing off...

**Credits:**

*God for making me a little healthier.
;*)*

*Dr. Davidstien for all the 3d OBJs.
*

*Plasma for SetVideoSeg*

*Biskbart for the Torus*

*Bobby 3999 for the Phong sub*

*CGI Joe for the original
polyfillers*

*Blitz for the things he taught me.*

*Toshi for the occasional help*