Three Dimensional Shading In Computer Graphics

ﾖﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄｴ% VLA Proudly Presents %ﾃﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄｷ

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ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;

ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄ

ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄ

Three Dimensional Shading In Computer Graphics

ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄ

ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄ

ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;

By Lithium /VLA

Hopefully you have read the companion document 3DROTATE.DOC, as this one

will build apon the concepts presented in my attempt to teach some of the

math need to make 3D graphics a reality. This file will cover such important

topics as the Dot Product and how routines are best constructed for real-time

3D rotations and planar shading.

Our Friend, The Dot Product

The Dot Product is a neat relation that will allow you to quickly find

the angle between any two vectors. It's easiest to explain graphicly, so

I will exercise my extended-ASCII keys.

Two Vectors A & B

A (Xa, Ya, Za) A = (Xa) + (Ya) + (Za)

B (Xb, Yb, Zb) B = (Xb) + (Yb) + (Zb)

Where Xa, and the others coorispond to some value on their respective Axis's

A

/

/

/

/

\ <-- Angle Theta between vector A and B

\

\

\

B

Cos( = Xa * Xb + Ya * Yb + Za * Zb

ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄ

A * B

Example:

A (1,2,3) A = 1 14) = 3.7417

B (4,5,6) b = 4 77) = 8.7750

Cos( = 1 * 4 + 2 * 5 + 3 * 6 = 4 + 10 + 18 = 32 = 0.9746

ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2; ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2; ﾄﾄᦍ 858s1812i 2;ﾄﾄ

(3.7417)*(8.7750) 32.8334 32.8334

ArcCos (.9746) = 12.9

So, your wondering how this revolutionizes you code, huh? Well, remember

our other friend, the Normal vector? You use Normal vectors that define

the directions of everything in our 3D world. Let's say that vector A was

the Normal vector from my plane, and B is a vector that shows the direction

that the light in my scene is pointing. If I do the Dot Product of them,

you will get the angle between them, if that angle is >= 90 and <= 270
then no light falls on the visible surface and it doesn't need to be

displayed.

Also notice, the way the values of the Cosine orient themselves

90 Cos 000 = 1

Cos 090 = 0

Cos 180 = -1

Negative Positive Cos 270 = 0

180・ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾅﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄ 0 An angle between a light and a plane that

is less than 90 or greater than 270 will

be visible, so you can check if the Cos(

Negative Positive is greater than 0 to see if it is visible.

270

How Do You Implement The Code? Easy As

Examples in ASM structures

We will define our points like this

STRUC XYZs

Xpos dd ?

Ypos dd ?

Zpos dd ?

Dist dd ?

ENDS XYZs ;size is 16 bytes

The X,Y,Zpos define a point in 3D space, Dist is the distance from the origin

Dist = X + Y + Z

Precalculate these values and have them handy in your data area

Our planes should look something like this

STRUC PlaneSt

NumPts db ? ;3 or 4

NormIndex dw ?

PtsIndex dw ?

dw ?

dw ?

dw ?

ENDS PlaneSt

The number of points that in the plane depends on the number your fill

routines can handle you must have at least 3 and more than 6 is not suggested

Then we set up our data like this

MaxPoints = 100

MaxPlanes = 100

PointList XYZs MaxPoints DUP()

PlaneList PlaneSt MaxPlanes DUP()

NormalList XYZs <0,0,0, 10000h> , MaxPlanes DUP()

Non-ASM User Note:

I set up points in a structure that had an X,Y,Z and Distance

value. I set up a plane structure that had the number of points

the index number of the normal vector for that plane and the index

numbers for the points in the plane.

The next lines set up arrays of these points in PointList, and

the number of points was defined as MaxPoints. An array of planes

was created as PlaneList with MaxPlanes as the total number of

plane structures in the array. NormalList is an array of the vectors

that are normal to the planes, one is set up initally (I'll explain

that next) and then one for each possible plane is allocated.

You'll notice that I defined the first Normal and then created space for

the rest of the possible normals. I'll call this first normal, the

Zero Normal. It will have special properties for planes that don't shade

and are never hidden.

Well, before I start telling all the tricks to the writting code, let me

make sure a couple of points are clear.

- In the 3DROTATE.DOC I said that you could set your view point on the

Z-Axis and then figure out if planes were visible by the post-rotation

Normal vectors, if their Z was > 0 then display, if not, don't

That is an easy way to set up the data, and I didn't feel like going

into the Dot Product at the time, so I generalized. So, what if you

don't view your plane from the Z-Axis, the answer is you use the...

Dot Product!

that's right. The angle will be used now to figure wheither or not to

display the plane.

- I have been mentioning lights and view points as vectors that I can

use with the Normal vector from my plane. To work correctly, these

vectors for the lights and view should point in the direction that you

are looking or the direction that the light is pointing, *NOT* a vector

drawn from the origin to the viewer position or light position.

- True Normal vectors only state a direction, and should therefore have

a unit distance of 1. This will have the advantage of simplifying the

math involved to figure you values. Also, for God's sake, pre-compute

your normal, don't do this everytime. Just rotate them when you do your

points and that will update their direction.

If the Normal's have a length of 1 then A * B = 1 * 1 = 1

So:

Cos( = Xa * Xb + Ya * Yb + Za * Zb

ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄ

A * B

Is Reduced To:

Cos( = Xa * Xb + Ya * Yb + Za * Zb

We eliminated a multiply and a divide! Pat yourself on the back.

- You ASM users might be wondering why I defined my Zero Normal as:

<0,0,0,10000h> How does 10000h = a length of 1 ?

Well, this is a trick you can do in ASM, instead of using floating point

values that will be slow on computers without math co-processors, we can

use a double word to hold our value. The high word holds the integer

value, and the low word is our decimal. You do all of your computations

with the whole register, but only pull the high word when you go to

display the point. So, with that under consideration, 10000h = 1.00000

Not bad for integers.

- How does the Zero Normal work? Since the X,Y,and Z are all 0, the

Cos( = 0, so if you always display when Cos( = 0, then that plane

will always be seen.

So, Beyond The Babble... How To Set Up Your Code

Define Data Points, Normals, and Planes

Pre-Calculate as many values as possible

Rotate Points and Normals

Determin Visible Planes With Dot Product

(Save this value if you want to shade)

Sort Visible Planes Back to Front

(Determin Shade From Dot Product)

Clip Plane to fit scene

Draw to the screen

Change Angles

Goto Rotation

A quick way to figure out which color to shade your plane if you are

using the double word values like I described before is to take the

Dot Product result, it will lie between 10000h - 0h if you would like

say 16 shades over the angles, then take that value and shr ,12 that will

give you a value from 0h - 10h (0-16, or 17 colors) if you make 10h into

0fh, add that offset to a gradient in your palette, then you will have

the color to fill your polygon with.

Note also that the Cosine function is weighted toward the extremes.

If you want a smooth palette change as the angles change, your palette

should weight the gradient accordingly.

A useful little relation for depth sorting is to be able to find the

center of a triangle.

E The center C = (D + E + F)/3

^

/ \ Divide each cooridinate by (Xd + Xe + Xf)/3 = Xc

/ C \ and do the same for the Y's and Z's if you

/ \ choose to sort with this method. Then rotate

Dﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;ﾄﾄᦍ 858s1812i 2;F that point and use it to depth sort the planes

Phong and Goraud Shading

Recently, someone asked me about the practiblity of real-time phong and

goraud shading. The technique is common to ray-tracers and requires a great

deal of calculation when working with individual rays cast from each pixel,

but when only using this for each plane, it is possible. This type of shading

involves taking into account the reduced luminousity of light as distance

increases. For each light, you define a falloff value. This value should be

the distance a which the light will be at full intensity. Then at 2*FallOff

you will have 1/2 intensity, 3*FallOff will yeild 1/3 and so on. To implement

this type of shading, you will need to determin the distance from the light

to the center of the plane. If distance < FallOff, then use the normal

intensity. If it is greater, divide the FallOff value by the distance. This

will give you a scalar value that you can multiple by the shading color that

the plane should have. Use that offset and it will be darker since it is

further away from the light source.

However, to determin the distance form the light to each plane, you must

use a Square Root function, these are inherently slow unless you don't care

about accuracy. Also, it would be difficult to notice the use of this

technique unless you have a relatively small FallOff value and your objects

move about in the low intesity boundries.

Well, that's all that I feel like doing tonight, and besides, Star Trek is on!

So, see VLA.NFO for information about contacting myself or any of the other

members of VLA.

Happy Coding!