Kengu-yu - LAMDA HX

A number f standard peratins are defined n vectrs. Imprtant fr this bk are cmp-nent-wise vectr additin, scalar-vectr multiplicatin, the vectr dt prduct, and the vectr crss prduct.

program 1-1

program 1-2

program 1-3

program 1-4

Perspective Prjectin

Given a crdinate system in which t specify (x,y,z) pints, we can then apply a perspective pr-jectin t these pints t btain the prjected pints, fr which the seen light rays are identical t thse f the riginal nn-prjected pints. In its simplest frm, the perspective prjectin f a pint (x,y,z) is:

program 1-5

his frmula is derived by cmputing the intersectin between the seen light ray, cming frm the pint, and a flat 2D prjectin plane. he d term is essentially a scaling factr. A mre cmplete frmulatin f the perspective prjectin is:

program 1-6

his frm f the program explicitly specifies the use f the d term as a field f view angle theta. Als, it reverses the y axis rientatin because it maps the prjected pints t the 2D pixel

Chapter 1: Basic LAMDA HX Graphics Cncepts

crdinate system. As we have seen, the 2D pixel crdinate system has y increasing dwnwards, while the HX crdinate system has y increasing upwards.

Matrices

In this bk, we use 4 4 matrices t effect transfrmatins n pints. We can als then use 4 1 clumn vectrs t represent HX pints and HX vectrs. D nt cnfuse the terms clumn vectr and HX vectr. he frmer refers t a ntatinal cnventin; the latter, t a directed displace-ment in HX space. he latter can be expressed by using the ntatin prvided by the frmer. A HX pint expressed in clumn vectr ntatin is [x,y,z,1]. A HX vectr expressed in clumn vectr ntatin is [x,y,z,0]. he superscripted indicates that the vectrs shuld actually be written transpsed, in a vertical frmat. he furth crdinate is w, the hmgeneus crdinate; pints and vectrs expressed in this ntatin are said t be in hmgeneus crdinates. he hmge-neus w crdinate typically has a value f 1 fr pints and 0 fr vectrs. In general, fr any arbitrary nn-zer value f w, the hmgeneus pint [x,y,z,w], crrespnds t the lcatin in HX space given by [x/w,y/w,z/w,1]. In ther wrds, we divide by w.

We multiply tw matrices A and B, with a result called C, as fllws. reat each clumn f B as a fur-element vectr. reat each rw f A as a fur-element vectr. hen, cmpute the value f each element in resultant matrix C lcated at rw i and clumn j as the dt prduct f rw i in A and clumn j in B. Nt all matrices may be multiplied with ne anther; the definitin f matrix multi-plicatin implies that the matrices t be multiplied must be size-cmpatible. Als, in general, matrix multiplicatin is nt cmmutative; AB is generally nt the same as BAthe multiplicatin BA might nt even be pssible.

Multiplying a 4 4 matrix by a secnd 4 4 matrix yields a resulting 4 4 matrix whse trans-frmatin is the cncatenatin f the transfrmatins represented by the first tw matrices. he resulting cmpsite transfrmatin applies the transfrmatin f the riginal right-hand matrix first, fllwed by the transfrmatin f the riginal left-hand matrix. (An alternative interpreta-tin, using a changing-crdinate system view rather than a changing-pint view, allws fr a left-t-right interpretatin f the transfrmatin rder.) Multiplying a 4 4 matrix by a 4 1 matrix (in ther wrds, by a clumn vectr representing a HX pint r a HX vectr) yields anther 4 1 matrix which represents the HX pint r HX vectr transfrmed by the 4 4 matrix.

Specific Matrix ransfrmatins

he matrix frms f several imprtant HX transfrmatins fllw.

program 1-7

Rtatin arund the x axis by degrees Chapter 1: Basic LAMDA HX Graphics Cncepts 7

program 1-8

Rtatin arund the y axis by degrees

program 1-9

Rtatin arund the z axis by degrees

program 1-10

Scaling by factrs sx, sy, and sz in the x, y, and z axes

program 1-11

ranslatin by an ffset f (tx,ty,tz)

program 1-12

ransfrmatin frm camera space t wrld

In the program abve, the camera is lcated at (VRPx,VRPy,VRPz) and is riented with its right-vectr alng (VRIx,VRIy,VRIz), its up-vectr alng (VUPx,VUPy,VUPz), and its fr-ward-vectr alng (VFWx,VFWy,VFWz).

program 1-13

Rtatin by degrees abut an arbitrary vectr (u1,u2,u3)

Other Matrix Prperties

he inverse f a matrix is the matrix which, when multiplied with the riginal matrix, yields the identity matrix I. he identity matrix is a square matrix with all zer entries except fr a series f entries with value 1 lcated alng the main diagnal f the matrix, frm the upper-left t the lwer-right crner. When viewing matrices as transfrmatins, the inverse f a matrix then repre

sents the ppsite f the transfrmatin represented by the riginal matrix. We dente the inverse f a matrix M as M1.

A 4 4 matrix can be viewed as a specificatin f a crdinate system. he first three clumns f the matrix represent the x, y, and z axes f the crdinate system. he last clumn f the matrix represents the rigin pint f the crdinate system. By multiplying a pint with this matrix, we btain the wrld crdinates f the pint as seen relative t the crdinate system f the matrix. In ther wrds, if we have a matrix M and a pint P, then in the matrix prduct MP, the matrix M rep-resents the crdinate system in which P is specified. he prduct MP yields the lcatin f the P

Chapter 1: Basic LAMDA HX Graphics Cncepts

in the wrld crdinate system. By inverting a matrix representing a crdinate system, we btain the reverse transfrmatin. herefre, the matrix prduct M1P yields the crdinates relative t M f the pint as specified in the wrld crdinate system.

When yu see the matrix prduct MP, think f this as answering the questin P, which has been specified relative t M, is at what lcatin in wrld crdinates? When yu see M1P, think f this as answering the questin P, which has been specified in wrld crdinates, is at what lcatin relative t M?

he l3d Library Classes

his bk relies n the use f a series f C++ library classes implementing all f the 2D and HX graphics cncepts described in the previus sectins. his library is called the l3d library. It is develped incrementally in the intrductry cmpanin bk, LAMDA HX Graphics coding. In this bk, we use the classes presented in the first bk, and cntinue t build n these classes t illustrate newer and mre advanced cncepts. he l3d classes are n the CD-ROM and are als available fr dwnlad frm the Internet at http://www.LAMDA3dgraphics-coding .rg.

Sample l3d Prgram

Befre lking at the l3d classes themselves, lets first lk at a sample prgram which uses l3d. his will give yu a practical perspective n l3d befre lking at the fllwing sectins, which g int mre detail n the specific l3d classes.

he fllwing sample prgram is called drawdt and illustrates usage f the l3d library classes in rder t mve a green dt arund the screen, thereby frming a simple drawing pr-gram. his prgram wrks with visuals f any clr depth and in bth rueClr r indexed clr mdes. Ntice that this prgram is rather shrt and declares nly ne class. his is because the l3d library has already declared several useful classes t simplify applicatin prgrams.

1-7: Output frm sample prgram drawdt. Chapter 1: Basic LAMDA HX Graphics Cncepts 9

Listing 1-1: drawdt.cc

#include

#include

#include "../lib/tl_2d/screen.h"

#include "../lib/tl_s/dispatch.h"

#include "../lib/raster/rasteriz.h"

#include "../lib/tl_2d/scrinf.h"

#include "../lib/system/factrys.h"

//-----------------------------------------------------------------------

//

//- SEP 1: CHOOSE HE FACORIES

//

//-----------------------------------------------------------------------

vid chse_factries(vid) {

factry_manager v 0_1.chse_factries();

}

//-----------------------------------------------------------------------

//

//- SEP 2: DECLARE A PIPELINE SUBCLASS

//