is called the geometric transformation. In computers the world is represented by numbers; thus geometrical properties and transformations must

Transcription

1 Capter 5 TRANSFORMATIONS, CLIPPING AND PROJECTION 5.1 Geometric transformations Tree-dimensional grapics aims at producing an image of 3D objects. Tis means tat te geometrical representation of te image is generated from te geometrical data of te objects. Tis cange of geometrical description is called te geometric transformation. In computers te world is represented by numbers; tus geometrical properties and transformations must also be given by numbers in computer grapics. Cartesian coordinates provide tis algebraic establisment for te Euclidean geometry, wic dene a 3D point by tree component distances along tree, non-coplanar axes from te origin of te coordinate system. Te selection of te origin and te axes of tis coordinate system may ave a signicant eect on te complexity of te denition and various calculations. As mentioned earlier, te world coordinate system is usually not suitable for te denition of all objects, because ere we are not only concerned wit te geometry of te objects, but also wit teir relative position and orientation. A brick, for example, can be simplistically dened in a coordinate system aving axes parallel to its edges, but te description of te box is quite complicated if arbitrary orientation is required. Tis consid- 99

2 TRANSFORMATIONS, CLIPPING AND PROJECTION eration necessitated te application of local coordinate systems. Viewing and visibility calculations, on te oter and, ave special requirements from a coordinate system were te objects are represented, to facilitate simple operations. Tis means tat te denition and te potograping of te objects may involve several dierent coordinate systems suitable for te dierent specic operations. Te transportation of objects from one coordinate system to anoter also requires geometric transformations. Working in several coordinate systems can simplify te various pases of modeling and image syntesis, but it requires additional transformation steps. Tus, tis approac is advantageous only if te computation needed for geometric transformations is less tan te decrease of te computation of te various steps due to te specically selected coordinate systems. Representations invariant of te transformations are te primary candidates for metods working in several coordinate systems, since tey can easily be transformed by transforming te control or denition points. Polygon mes models, Bezier and B-spline surfaces are invariant for linear transformation, since teir transformation will also be polygon meses, Bezier or B-spline surfaces, and te vertices or te control points of te transformed surface will be tose coming from te transformation of te original vertices and control points. Oter representations, sustaining non-planar geometry, and containing, for example, speres, are not easily transformable, tus tey require all te calculations to be done in a single coordinate system. Since computer grapics generates 2D images of 3D objects, some kind of projection is always involved in image syntesis. Central projection, owever, creates problems (singularities) in Euclidean geometry, itistus wortwile considering anoter geometry, namely te projective geometry, to be used for some pases of image generation. Projective geometry is a classical branc of matematics wic cannot be discussed ere in detail. A sort introduction, owever, is given to igligt tose features tat are widely used in computer grapics. Beyond tis elementary introduction, te interested reader is referred to [Her91] [Cox74]. Projective geometry can be approaced from te analysis of central projection as sown in gure 5.1. For tose points to wic te projectors are parallel wit te image plane no projected image can be dened in Euclidean geometry. Intuitively speaking tese image points would be at \innity" wic is not part of te Eu-

3 5.1. GEOMETRIC TRANSFORMATIONS 101 "ideal points" vanising line center of projection affine lines projection plane Figure 5.1: Central projection of objects on a plane clidean space. Projective geometry lls tese oles by extending te Euclidean space by new points, called ideal points, tat can serve as te image of points causing singularities in Euclidean space. Tese ideal points can be regarded as \intersections" of parallel lines and planes, wic are at \innity". Tese ideal points form a plane of te projective space, wic is called te ideal plane. Since tere is a one-to-one correspondence between te points of Euclidean space and te coordinate triples of a Cartesian coordinate system, te new elements obviously cannot be represented in tis coordinate system, but a new algebraic establisment is needed for projective geometry. Tis establisment is based on omogeneous coordinates. For example, by te metod of omogeneous coordinates a point of space can be specied as te center of gravity of te structure containing mass X at reference point p 1, mass Y at point p 2, mass Z at point p 3 and mass w at point p 4. Weigts are not required to be positive, tus te center of gravity can really be any point of te space if te four reference points are not co-planar. Alternatively, if te total mass, tat is = X +Y +Z +w, is not zero and te reference points are in Euclidean space, ten te center of gravity will also be in te Euclidean space. Let us call te quadruple (X ;Y ;Z ;), were = X + Y + Z + w, te omogeneous coordinates of te center of gravity. Note tat if all weigts are multiplied by te same (non-zero) factor, te center of gravity, tat is te point dened by te omogeneous coordi-

4 TRANSFORMATIONS, CLIPPING AND PROJECTION nates, does not cange. Tus a point (X ;Y ;Z ;) is equivalent to points (X ;Y ;Z ;), were is a non-zero number. Te center of gravity analogy used to illustrate te omogeneous coordinates is not really important from a matematical point of view. Wat sould be remembered, owever, is tat a 3D point represented by omogeneous coordinates is a four-vector of real numbers and all scalar multiples of tese vectors are equivalent. Points of te projective space, tat is te points of te Euclidean space (also called ane points) plus te ideal points, can be represented by omogeneous coordinates. First te representation of ane points wic can establis a correspondence between te Cartesian and te omogeneous coordinate systems is discussed. Let us dene te four reference points of te omogeneous coordinate system in points [1,0,0], [0,1,0], [0,0,1] and in [0,0,0] respectively. If=X +Y +Z +wis not zero, ten te center of gravity in Cartesian coordinate system dened by axes i; j; k is: r(x ;Y ;Z ;)= 1 (X [1; 0; 0] + Y [0; 1; 0] + Z [0; 0; 1] + w [0; 0; 0]) = X i + Y j + Z k: (5:1) Tus wit te above selection of reference points te correspondence between te omogeneous coordinates (X ;Y ;Z ;) and Cartesian coordinates (x; y; z) of ane points ( 6= 0) is: x = X ; y = Y ; z = Z : (5:2) Homogeneous coordinates can also be used to caracterize planes. In te Cartesian system a plane is dened by te following equation: a x + b y + c z + d =0 (5:3) Applying te correspondence between te omogeneous and Cartesian coordinates, we get: a X + b Y + c Z + d =0 (5:4) Note tat te set of points tat satisfy tis plane equation remains te same if tis equation is multiplied by a scalar factor. Tus a quadruple [a; b; c; d]

5 5.1. GEOMETRIC TRANSFORMATIONS 103 of omogeneous coordinates can represent not only single points but planes as well. In fact all teorems valid for points can be formulated for planes as well in 3D projective space. Tis symmetry is often referred to as te duality principle. Te intersection of two planes (wic is a line) can be calculated as te solution of te linear system of equations. Suppose tat we avetwo parallel planes given by quadruples [a; b; c; d] and [a; b; c; d 0 ] (d 6= d 0 ) and let us calculate teir intersection. Formally all points satisfy te resulting equations for wic a X + b Y + c Z =0 and =0 (5:5) In Euclidean geometry parallel planes do not ave intersection, tus te points calculated in tis way cannot be in Euclidean space, but form a subset of te ideal points of te projective space. Tis means tat ideal points correspond to tose omogeneous quadruples were = 0. As mentioned, tese ideal points represent te innity, but tey make a clear distinction between te \innities" in dierent directions tat are represented by te rst tree coordinates of te omogeneous form. Returning to te equation of a projective plane or considering te equation of a projective line, we can realize tat ideal points may also satisfy tese equations. Terefore, projective planes and lines are a little bit more tan teir Euclidean counterparts. In addition to all Euclidean points, tey also include some ideal points. Tis may cause problems wen we want to return to Euclidean space because tese ideal points ave no counterparts. Homogeneous coordinates can be visualized by regarding tem as Cartesian coordinates of a iger dimensional space (note tat 3D points are dened by 4 omogeneous coordinates). Tis procedure is called te embedding of te 3D projective space into te 4D Euclidean space or te straigt model [Her91] (gure 5.2). Since it is impossible to create 4D drawings, tis visualization uses a trick of reducing te dimensionality and displays te 4D space as a 3D one, te real 3D subspace as a 2D plane and relies on te reader's imagination to interpret te resulting image. A omogeneous point is represented by a set of equivalent quadruples f(x ;Y ;Z ;) j 6=0g; tus a point is described as a 4D line crossing te origin, [0,0,0,0], in te straigt model. Ideal points are in te = 0 plane and ane points are represented by tose lines tat are not parallel to te = 0 plane.

6 TRANSFORMATIONS, CLIPPING AND PROJECTION affine points embedded Euclidean space =1 plane z ideal point =0 plane Figure 5.2: Embedding of projective space into a iger dimensional Euclidean space Since points are represented by a set of quadruples tat are equivalent in omogeneous terms, a point may be represented by any of tem. Still, it is wort selecting a single representative from tis set to identify points unambiguously. For ane points, tis representative quadruple is found by making te fourt () coordinate equal to 1, wic as a nice property tat te rst tree omogeneous coordinates are equal to te Cartesian coordinates of te same point taking equation 5.2 into account, tat is: ( X ; Y ; Z ; 1)=(x; y; z;1): (5:6) In te straigt model tus te representatives of ane points correspond to te =1yperplane (a 3D set of te 4D space), were tey can be identied by Cartesian coordinates. Tis can be interpreted as te 3D Euclidean space and for ane points te omogeneous to Cartesian conversion of coordinates can be accomplised by projecting te 4D point onto te = 1yperplane using te origin as te center of projection. Tis projection means te division of te rst tree coordinates by te fourt and is usually called omogeneous division. Using te algebraic establisment of Euclidean and projective geometries, tat is te system of Cartesian and omogeneous coordinates, geometric transformations can be regarded as functions tat map tuples of coordinates onto tuples of coordinates. In computer grapics linear functions are

7 5.1. GEOMETRIC TRANSFORMATIONS 105 preferred tat can conveniently be expressed as a vector-matrix multiplication and a vector addition. In Euclidean geometry tis linear function as te following general form: [x 0 ;y 0 ;z 0 ]=[x; y; z] A 33 +[p x ;p y ;p z ]: (5:7) Linear transformations of tis kind map ane points onto ane points, terefore tey are also ane transformations. Wen using omogeneous representation, owever, it must be taken into account tat equivalent quadruples diering only by a scalar multiplication must be transformed to equivalent quadruples, tus no additive constant is allowed: [X 0 ;Y0 ;Z0 ;0 ]=[X ;Y ;Z ;]T 44: (5:8) Matrix T 44 denes te transformation uniquely in omogeneous sense; tat is, matrices diering in a multiplicative factor are equivalent. Note tat in equations 5.7 and 5.8 row vectors are used to identify points unlike te usual matematical notation. Te preference for row vectors in computer grapics as partly istorical reasons, partly stems from te property tat in tis way te concatenation of transformations corresponds to matrix multiplication in \normal", tat is left to rigt, order. For column vectors, it would be te reverse order. Using te straigt model, equation 5.7 can be reformulated for omogeneous coordinates: p T 1 [x 0 ;y 0 ;z 0 ;1] = [x; y; z;1] 4 A : (5:9) Note tat te 33 matrix A is accommodated in T as its upper left minor matrix, wile p is placed in te last row and te fourt column vector of T is set to constant [0,0,0,1]. Tis means tat te linear transformations of Euclidean space form a subset of omogeneous linear transformations. Tis is a real subset since, as we sall see, by setting te fourt column to a vector dierent from [0,0,0,1] te resulting transformation does not ave an ane equivalent, tat is, it is not linear in te Euclidean space. Using te algebraic treatment of omogeneous (linear) transformations, wic identies tem by a 4 4 matrix multiplication, we can dene te concatenation of transformations as te product of transformation matrices

8 TRANSFORMATIONS, CLIPPING AND PROJECTION and te inverse of a omogeneous transformation as te inverse of its transformation matrix if it exists, i.e. its determinant is not zero. Taking into account te properties of matrix operations we can see tat te concatenation of omogeneous transformations is also a omogeneous transformation and te inverse of a omogeneous transformation is also a omogeneous transformation if te transformation matrix is invertible. Since matrix multiplication is an associative operation, consecutive transformations can always be replaced by a single transformation by computing te product of te matrices of dierent transformation steps. Tus, any number of linear transformations can be expressed by a single 4 4 matrix multiplication. Te transformation of a single point of te projective space requires 16 multiplications and 12 additions. If te point must be mapped back to te Cartesian coordinate system, ten 3 divisions by te fourt omogeneous coordinate may be necessary in addition to te matrix multiplication. Since linear transformations of Euclidean space ave a[0;0;0;1] fourt column in te transformation matrix, wic is preserved by multiplications wit matrices of te same property, any linear transformation can be calculated by 9multiplications and 9 additions. According to te teory of projective geometry, transformations dened by 44 matrix multiplication map points onto points, lines onto lines, planes onto planes and intersection points onto intersection points, and terefore are called collinearities [Her91]. Te reverse of tis statement is also true; eac collinearity corresponds to a omogeneous transformation matrix. Instead of proving tis statement in projective space, a special case tat as importance in computer grapics is investigated in detail. In computer grapics te geometry is given in 3D Euclidean space and aving applied some omogeneous transformation te results are also required in Euclidean space. From tis point of view, te omogeneous transformation of a 3D point involves: 1. A 4 4 matrix multiplication of te coordinates extended by a fourt coordinate of value A omogeneous division of all coordinates in te result by te fourt coordinate if it is dierent from 1, meaning tat te transformation forced te point out of 3D space. It is important to note tat a clear distinction must be made between te

9 5.1. GEOMETRIC TRANSFORMATIONS 107 central or parallel projection dened earlier wic maps 3D points onto 2D points on a plane and projective transformations wic map projective space onto projective space. Now let us start te discussion of te omogeneous transformation of a special set of geometric primitives. A Euclidean line can be dened by te following equation: ~r(t) =~r 0 +~vt; were t is a real parameter. (5:10) Assuming tat vectors ~v 1 and ~v 2 are not parallel, a Euclidean plane, on te oter and, can be dened as follows: ~r(t 1 ;t 2 )=~r 0 +~v 1 t 1 +~v 2 t 2 ; were t 1 ;t 2 are real parameters. (5:11) Generally, lines and planes are special cases of a wider range of geometric structures called linear sets. By denition, a linear set is dened by a position vector ~r 0 and some axes ~v 1 ;~v 2 ;:::;~v n by te following equation: ~r(t 1 ;:::;t n )=~r 0 + nx i=1 t i ~v i : (5:12) First of all, te above denition is converted to a dierent one tat uses omogeneous-like coordinates. Let us dene te so-called spanning vectors ~p 0 ;:::;~p n of te linear set as: ~p 0 = ~r 0 ; ~p 1 = ~r 0 + ~v 1 ;. ~p n = ~r 0 + ~v n : Te equation of te linear set is ten: ~r(t 1 ;:::;t n )=(1 t 1 ::: t n )~p 0 + Introducing te new coordinates as nx i=1 (5:13) t i ~p i : (5:14) 0 =1 t 1 ::: t n ; 1 =t 1 ; 2 = t 2 ; :::; n = t n ; (5:15) te linear set can be written in te following form: S = f~p j ~p = nx i=0 i ~p i ^ nx i=0 i =1g: (5:16)

10 TRANSFORMATIONS, CLIPPING AND PROJECTION Te weigts ( i ) are also called te baricentric coordinates of te point ~p wit respect to ~p 0, ~p 1,:::,~p n. Tis name reects te interpretation tat ~p would be te center of gravity of a structure of weigts ( 0 ; 1 ;:::; n )at points ~p 0 ;~p 1 ;:::;~p n. Te omogeneous transformation of suc a point ~p is: [~p; 1] T =[ ( nx i=0 nx i=0 i ~p i ; 1] T =[ i [~p i ; 1]) T = nx i=0 nx i=0 i ~p i ; since P n i=0 i = 1. Denoting [~p i ; 1] T by [ ~ P i ; i ]we get: [~p; 1] T = nx i=0 i [ ~ P i ; i ]=[ nx i=0 i ] T = i ([~p i ; 1] T) (5:17) nx i=0 i ~ P i ; nx i=0 i i ]: (5:18) If te resulting fourt coordinate P n i=0 i i is zero, ten te point ~p is mapped onto an ideal point, terefore it cannot be converted back to Euclidean space. Tese ideal points must be eliminated before te omogeneous division (see section 5.5 on clipping). After omogeneous division we are left wit: [ nx i=0 i i P n j=1 j j P ~ i ; 1]=[ i nx i=0 i ~p i ; 1] (5:19) were ~p i is te omogeneous transformation of ~p i. Te derivation of i guarantees tat P n i=0 i =1. Tus, te transformation of te linear set is also linear. Examining te expression of te weigts ( ), we can conclude i tat generally i 6= meaning te omogeneous transformation may destroy equal spacing. In oter words te division ratio is not projective i invariant. In te special case wen te transformation is ane, coordinates i will be 1, tus i = i, wic means tat equal spacing (or division ratio) is ane invariant. A special type of linear set is te convex ull. Te convex ull is dened by equation 5.16 wit te provision tat te baricentric coordinates must be non-negative.

11 5.1. GEOMETRIC TRANSFORMATIONS 109 To avoid te problems of mapping onto ideal points, let us assume te spanning vectors to be mapped onto te same side of te =0yperplane, meaning tat te i -s must ave te same sign. Tis, wit i 0, guarantees tat no points are mapped onto ideal points and i = n X i=0 i i P n i=0 i i 0 (5:20) Tus, baricentric coordinates of te image will also be non-negative, tat is, convex ulls are also mapped onto convex ulls by omogeneous transformations if teir transformed image does not contain ideal points. An arbitrary planar polygon can be broken down into triangles tat are convex ulls of tree spanning vectors. Te transformation of tis polygon will be te composition of te transformed triangles. Tis means tat a planar polygon will also be preserved by omogeneous transformations if its image does not intersect wit te = 0 plane. As mentioned earlier, in computer grapics te objects are dened in Euclidean space by Cartesian coordinates and te image is required in a 2D pixel space tat is also Euclidean wit its coordinates wic correspond to te pysical pixels of te frame buer. Projective geometry may be needed only for specic stages of te transformation from modeling to pixel space. Since projective space can be regarded as an extension of te Euclidean space, te teory of transformations could be discussed generally only in projective space. For pedagogical reasons, owever, we will use te more complicated omogeneous representations if tey are really necessary for computer grapics algoritms, and deal wit te Cartesian coordinates in simpler cases. Tis combined view of Euclidean and projective geometries may be questionable from a purely matematical point of view, but it is accepted by te computer grapics community because of its clarity and its elimination of unnecessary abstractions. We sall consider te transformation of points in tis section, wic will lead on to te transformation of planar polygons as well.

12 TRANSFORMATIONS, CLIPPING AND PROJECTION Elementary transformations Translation Translation is a very simple transformation tat adds te translation vector ~p to te position vector ~r of te point to be transformed: Scaling along te coordinate axes ~r 0 = ~r + ~p: (5:21) Scaling modies te distances and te size of te object independently along te tree coordinate axes. If a point originally as [x; y; z] coordinates, for example, after scaling te respective coordinates are: x 0 = S x x; y 0 = S y y; z 0 = S z z: (5:22) Tis transformation can also be expressed by a matrix multiplication: ~r 0 = ~r S x S y S z Rotation around te coordinate axes : (5:23) Rotating around te z axis by an angle, te x and y coordinates of a point are transformed according to gure 5.3, leaving coordinate z unaected. y (x,y ) φ (x,y) z x Figure 5.3: Rotation around te z axis By geometric considerations, te new x; y coordinates can be expressed as: x 0 = x cos y sin ; y 0 = x sin + y cos : (5:24)

13 5.1. GEOMETRIC TRANSFORMATIONS 111 Rotations around te y and x axes ave similar form, just te roles of x; y and z must be excanged. Tese formulae can also be expressed in matrix form: ~r 0 (x; ) =~r cos sin sin cos ~r 0 (y; ) =~r ~r 0 (z; ) =~r cos 0 sin sin 0 cos cos sin 0 sin cos : (5:25) Tese rotations can be used to express any orientation [Lan91]. Suppose tat K and K 000 are two Cartesian coordinate systems saring a common origin but aving dierent orientations. In order to determine tree special rotations around te coordinate axes wic transform K into K 000, let us dene a new Cartesian system K 0 suc tat its z 0 axis is coincident wit z and its y 0 axis is on te intersection line of planes [x; y] and [x 000 ;y 000 ]. To transform axis y onto axis y 0 a rotation is needed around z by angle. Ten a new rotation around y 0 by angle as to be applied tat transforms x 0 into x 000 resulting in a coordinate system K 00. Finally te coordinate system K 00 is rotated around axis x 00 = x 000 by an angle to transform y 00 into y 000. Te tree angles, dening te nal orientation, are called roll, pitc and yaw angles. If te roll, pitc and yaw angles are, and respectively, te transformation to te new orientation is: ~r 0 = ~r cos sin 0 sin cos Rotation around an arbitrary axis cos 0 sin sin 0 cos cos sin 0 sin cos : (5:26) Let us examine a linear transformation tat corresponds to a rotation by angle around an arbitrary unit axis ~t going troug te origin. Te original and te transformed points are denoted by vectors ~u and ~v respectively.

14 TRANSFORMATIONS, CLIPPING AND PROJECTION Let us decompose vectors ~u and ~v into perpendicular (~u? ;~v? ) and parallel (~u k ;~v k ) components wit respect to ~t. By geometrical considerations we can write: ~u k = ~t(~t ~u) ~u? = ~u ~u k = ~u ~t(~t ~u) (5.27) Since te rotation does not aect te parallel component, ~v k = ~u k. t u u =v v t x u u φ v Figure 5.4: Rotating around ~ t by angle Since vectors ~u? ;~v? and ~t ~u? = ~t ~u are in te plane perpendicular to ~t, and ~u? and ~t ~u? are perpendicular vectors (gure 5.4), ~v? can be expressed as: ~v? = ~u? cos + ~t ~u? sin : (5:28) Vector ~v, tat is te rotation of ~u, can ten be expressed as follows: ~v = ~v k + ~v? = ~u cos + ~t ~u sin + ~t(~t ~u)(1 cos ): (5:29) Tis equation, also called te Rodrigues formula, can also be expressed in matrix form. Denoting cos and sin by C and S respectively and assuming ~t to be a unit vector, we get: ~v = ~u C (1 t 2 x)+t 2 x t x t y (1 C )+S t z t x t z (1 C ) S t y t y t x (1 C ) S t z C (1 t 2 y)+t 2 y t x t z (1 C )+S t x t z t x (1 C )+S t y t z t y (1 C ) S t x C (1 t 2 z)+t 2 z 3 7 5: (5:30)

15 5.2. TRANSFORMATION TO CHANGE THE COORDINATE SYSTEM 113 It is important to note tat any orientation can also be expressed as a rotation around an appropriate axis. Tus, tere is a correspondence between roll-pitc-yaw angles and te axis and angle of nal rotation, wic can be given by making te two transformation matrices dened in equations 5.26 and 5.30 equal and solving te equation for unknown parameters. Searing Suppose a searing stress acts on a block xed on te xy face of gure 5.5, deforming te block to a parallepiped. Te transformation representing te distortion of te block leaves te z coordinate unaected, and modies te x and y coordinates proportionally to te z coordinate. z y x Figure 5.5: Searing of a block In matrix form te searing transformation is: ~r 0 = ~r a b Transformation to cange te coordinate system : (5:31) Objects dened in one coordinate system are often needed in anoter coordinate system. Wen we decide to work in several coordinate systems and to make every calculation in te coordinate system in wic it is te

16 TRANSFORMATIONS, CLIPPING AND PROJECTION simplest, te coordinate system must be canged for eac dierent pase of te calculation. Suppose unit coordinate vectors ~u, ~v and ~w and te origin ~o of te new coordinate system are dened in te original x; y; z coordinate system: ~u =[u x ;u y ;u z ]; ~v =[v x ;v y ;v z ]; ~w =[w x ;w y ;w z ]; ~o=[o x ;o y ;o z ]: (5:32) Let a point ~p ave x; y; z and ; ; coordinates in te x; y; z and in te u; v; w coordinate systems respectively. Since te coordinate vectors ~u; ~v; ~w as well as teir origin, ~o, are known in te x; y; z coordinate system, ~p can be expressed in two dierent forms: ~p = ~u + ~v + ~w + ~o =[x; y; z]: (5:33) Tis equation can also be written in omogeneous matrix form, aving introduced te matrix formed by te coordinates of te vectors dening te u; v; w coordinate system: T c = u x u y u z 0 v x v y v z 0 w x w y w z 0 o x o y o z ; (5:34) [x; y; z;1] = [;;;1] T c : (5:35) Since T c is always invertible, te coordinates of a point ofte x; y; z coordinate system can be expressed in te u; v; w coordinate system as well: [;;;1] = [x; y; z;1] T c 1 : (5:36) Note tat te inversion of matrix T c can be calculated quite eectively since its upper-left minor matrix is ortonormal, tat is, its inverse is given by mirroring te matrix elements onto te diagonal of te matrix, tus: T 1 c = o x o y o z u x v x w x 0 u y v y w y 0 u z v z w z : (5:37)

17 5.3. DEFINITION OF THE CAMERA Denition of te camera Having dened transformation matrices we can now look at teir use in image generation, but rst some basic denitions. In 3D image generation, a window rectangle is placed into te 3D space of te virtual world wit arbitrary orientation, a camera or eye is put beind te window, and a poto is taken by projecting te model onto te window plane, supposing te camera to be te center of projection, and ignoring te parts mapped outside te window rectangle or tose wic are not in te specied region in front of te camera. Te data, wic dene ow te virtual world is looked at, are called camera parameters, and include: front clipping plane fb bp vpn v w u eye z vrp window y x Figure 5.6: Denition of te camera Position and orientation of te window. Te center of te window, called te view reference point, is dened as a point, or a vector vrp, ~ in te world coordinate system. Te orientation is dened by au; v; w ortogonal coordinate system, wic is also called te window coordinate system, centered at te view reference point, wit ~u and ~v specifying te direction of te orizontal and vertical sides of te window rectangle, and ~w determining te normal of te plane of te window. Unit coordinate vectors ~u; ~v; ~w are obviously

18 TRANSFORMATIONS, CLIPPING AND PROJECTION not independent, because eac of tem is perpendicular to te oter two, tus tat dependence as also to be taken care of during te setting of camera parameters. To ease te parameter setting pase, instead of specifying te coordinate vector triple, two almost independent vectors are used for te denition of te orientation, wic are te normal vector to te plane of te window, called te view plane normal, or vpn ~ for sort, and a so-called view up vector, or vup, ~ wose component tat is perpendicular to te normal and is in te plane of vpn ~ and vup ~ denes te direction of te vertical edge of te window. Tere is a sligt dependence between tem, since tey sould not be parallel, tat is, it must always old tat vup ~ vpn ~ 6= 0. Te ~u; ~v; ~w coordinate vectors can easily be calculated from te view plane normal and te view up vectors: ~w = vpn ~ j vpnj ~ ; ~u = ~w vup ~ ; ~v = ~u ~w: (5:38) j~w vupj ~ Note tat unlike te x; y; z world coordinate system, te u; v; w system as been dened left anded to meet te user's expectations tat ~u points to te rigt, ~v points upwards and ~w points away from te camera located beind te window. Size of te window. Te lengt of te edges of te window rectangle are dened by two positive numbers, te widt by wwidt, te eigt by weigt. Potograpic operations, suc as zooming in and out, can be realized by proper control of te size of te window. To avoid distortions, te widt/eigt ratio as to be equal to widt/eigt ratio of te viewport on te screen. Type of projection. Te image is te projection of te virtual world onto te window. Two dierent types of projection are usually used in computer grapics, te parallel projection (if te projectors are parallel), and te perspective projection (if all te projectors go troug a given point, called te center of projection). Parallel projections are furter classied into ortograpic and oblique projections depending on weter or not te projectors are perpendicular to te plane of projection (window plane). Te attribute \oblique" may also refer to perspective projection if te projector from te center of

19 5.3. DEFINITION OF THE CAMERA 117 te window is not perpendicular to te plane of te window. Oblique projections may cause distortion of te image. Location of te camera or eye. Te camera is placed beind te window in our conceptual model. For perspective projection, te camera position is, in fact, te center of projection, wic can be dened by apoint eye ~ in te u; v; w coordinate system. For parallel projection, te direction of te projectors as to be given by te u; v; w coordinates of te direction vector. Bot in parallel and perspective projections te dept coordinate w is required to be negative in order to place te camera \beind" te window. It also makes sense to consider parallel projection as a special perspective projection, wen te camera is at an innite distance from te window. Front and back clipping planes. According to te conceptual model of taking potos of te virtual world, it is obvious tat only tose portions of te model wic lie in te innite pyramid dened by te camera as te apex, and te sides of te 3D window (for perspective projection), and in a alf-open, innite parallelepiped (for parallel projection) can aect te poto. Tese innite regions are usually limited to a nite frustum of a pyramid, or to a nite parallelepiped respectively, toavoid overows and also to ease te projection task by eliminating te parts located beind te camera, by dening two clipping planes called te front clipping plane and te back clipping plane. Tese planes are parallel wit te window and tus ave constant w coordinates appropriate for te denition. Tus te front plane is specied by anfp value, meaning te plane w = fp, and te back plane is dened by abp value. Considering te objectives of te clipping planes, teir w coordinates ave tobe greater tan te w coordinate of te eye, and fp < bp sould also old.

20 TRANSFORMATIONS, CLIPPING AND PROJECTION 5.4 Viewing transformation Image generation involves: 1. te projection of te virtual world onto te window rectangle, 2. te determination of te closest surface at eac point (visibility calculation) by dept comparisons if more tan one surface can be projected onto te same point in te window, and 3. te placement of te result in te viewport rectangle of te screen. Obviously, te visibility calculation as to be done prior to te projection of te 3D space onto te 2D window rectangle, since tis projection destroys te dept information. Tese calculations could also be done in te world coordinate system, but eac projection would require te evaluation of te intersection of an arbitrary line and rectangle (window), and te visibility problem would require te determination of te distance of te surface points along te projectors. Te large number of multiplications and divisions required by suc geometric computations makes te selection of te world coordinate system disadvantageous even if te required calculations can be reduced by te application of te incremental concept, and forces us to look for oter coordinate systems were tese computations are simple and eective to perform. In te optimal case te points sould be transformed to a coordinate system were X; Y coordinates would represent te pixel location troug wic te given point is visible, and a tird Z coordinate could be used to decide wic point is visible, i.e. closest to te eye, if several points could be transformed to te same X; Y pixel. Note tat Z is not necessarily proportional to te distance from te eye, it sould only be a monotonously increasing function of te distance. Te appropriate transformation is also expected to map lines onto lines and planes onto planes, allowing simple representations and linear interpolations during clipping and visibility calculations. Coordinate systems meeting all te above requirements are called screen coordinate systems. In a coordinate system of tis type, te visibility calculations are simple, since sould two or more points ave te same X; Y pixel coordinates, ten te visible one as te smallest Z coordinate.

21 5.4. VIEWING TRANSFORMATION 119 From a dierent perspective, if it as to be decided weter one point will ide anoter, two comparisons are needed to ceck weter tey project onto te same pixel, tat is, weter tey ave te same X; Y coordinates, and a tird comparison must be used to select te closest. Te projection is very simple, because te projected point as, in fact, X; Y coordinates due to te denition of te screen space. For pedagogical reasons, te complete transformation is dened troug several intermediate coordinate systems, altoug eventually it can be accomplised by a single matrix multiplication. For bot parallel and perspective cases, te rst step of te transformation is to cange te coordinate system to u; v; w from x; y; z, but after tat tere will be dierences depending on te projection type World to window coordinate system transformation First, te world is transformed to te u; v; w coordinate system xed to te center of te window. Since te coordinate vectors ~u, ~v, ~w and te origin vrp ~ are dened in te x; y; z coordinate system, te necessary transformation can be developed based on te results of section 5.2 of tis capter. Te matrix formed by te coordinates of te vectors dening te u; v; w coordinate system is: T uvw = u x u y u z 0 v x v y v z 0 w x w y w z 0 vrp x vrp y vrp z ; (5:39) [x; y; z;1] = [;;;1] T uvw : (5:40) Since ~u, ~v, ~w are perpendicular vectors, T uvw is always invertible. Tus, te coordinates of an arbitrary point of te world coordinate system can be expressed in te u; v; w coordinate system as well: [;;;1]=[x; y; z;1] T 1 uvw : (5:41)

22 TRANSFORMATIONS, CLIPPING AND PROJECTION Window to screen coordinate system transformation for parallel projection Searing transformation For oblique transformations, tat is wen eye u or eye v is not zero, te projectors are not perpendicular to te window plane, tus complicating visibility calculations and projection (gure 5.7). Tis problem can be solved by distortion of te object space, applying a searing transformation in suc a way tat te non-oblique projection of te distorted objects sould provide te same images as te oblique projection of te original scene, and te dept coordinate of te points sould not be aected. A general (0,0,eye ) w w window P=eye Figure 5.7: Searing searing transformation wic does not aect te w coordinate is: T sear = s u s v : (5:42) Te unknown elements, s u and s v, can be determined by examining te transformation of te projector P ~ =[eye u ;eye v ;eye w ;1]. Te transformed projector is expected to be perpendicular to te window and to ave dept coordinate eye w, tat is: ~P T sear =[0;0;eye w ;1]: (5:43)

23 5.4. VIEWING TRANSFORMATION 121 Using te denition of te searing transformation, we get: s u = eye u eye w ; s v = eye v eye w : (5:44) Normalizing transformation Having accomplised te searing transformation, te objects for parallel projection are in a space sown in gure 5.8. Te subspace wic can be projected onto te window is a rectangular box between te front and back clipping plane, aving side faces coincident to te edges of te window. To allow uniform treatment, a normalizing transformation can be applied, wic maps te box onto a normalized block, called te canonical view volume, moving te front clipping plane to 0, te back clipping plane to 1, te oter boundaries to x =1, y=1, x= 1 and y = 1 planes respectively. v 1 fp window bp w 1-1 Figure 5.8: Normalizing transformation for parallel projection Te normalizing transformation can also be expressed in matrix form: T norm = =wwidt =weigt =(bp fp) fp=(bp fp) : (5:45) Te projection in te canonical view volume is very simple, since te projection does not aect te (X; Y ) coordinates of an arbitrary point, but only its dept coordinate.

24 TRANSFORMATIONS, CLIPPING AND PROJECTION Viewport transformation Te space inside te clipping volume as been projected onto a 2 2 rectangle. Finally, te image as to be placed into te specied viewport of te screen, dened by te center point, (V x ;V y ) and by te orizontal and vertical sizes, V sx and V sy. For parallel projection, te necessary viewport transformation is: T viewport = V sx = V sy = V x V y : (5:46) Summarizing te results, te complete viewing transformation for parallel projection can be generated. Te screen space coordinates formed by te (X; Y ) pixel addresses and te Z dept value mapped into te range of [0::1] can be determined by te following transformation: T V = Tuvw 1 T sear T norm T viewport ; [X; Y; Z; 1] = [x; y; z;1] T V : (5.47) Matrix T V, called te viewing transformation, is te concatenation of te transformations representing te dierent steps towards te screen coordinate system. Since T V is ane, it obviously meets te requirements of preserving lines and planes, making bot te visibility calculation and te projection easy to accomplis Window to screen coordinate system transformation for perspective projection As in te case of parallel projection, objects are rst transformed from te world coordinate system to te window, tat is u; v; w, coordinate system by applying Tuvw. 1 View-eye transformation For perspective projection, te center of te u; v; w coordinate system is translated to te camera position witout altering te direction of te axes.

25 5.4. VIEWING TRANSFORMATION 123 Since te camera is dened in te u; v; w coordinate system by avector eye, ~ tis transformation is a translation by vector eye, ~ wic can also be expressed by a omogeneous matrix: T eye = Searing transformation eye u eye v eye w : (5:48) As for parallel projection, if eye u or eye v is not zero, te projector from te center of te window is not perpendicular to te window plane, requiring te distortion of te object space by asearing transformation in suc away tat te non-oblique projection of te distorted objects provides te same images as te oblique projection of te original scene and te dept coordinate of te points is not aected. Since te projector from te center of te window ( P=[eye ~ u ;eye v ;eye w ;1]) is te same as all te projectors for parallel transformation, te searing transformation matrix will ave te same form, independently of te projection type: T sear = Normalizing transformation eye u =eye w eye v =eye w : (5:49) After searing transformation te region wic can be projected onto te window is a symmetrical, nite frustum of te pyramid in gure 5.9. By normalizing tis pyramid, te back clipping plane is moved to 1, and te angle at its apex is set to 90 degrees. Tis is a simple scaling transformation, wit scales S u, S v and S w determined by te consideration tat te back clipping plane goes to w = 1, and te window goes to te position d wic is equal to alf te eigt and alf te widt of te normalized window: S u wwidt=2 =d; S v weigt=2 =d; eye w S w = d; S w bp =1 (5:50)

26 TRANSFORMATIONS, CLIPPING AND PROJECTION w fp d 1 w fp window bp window bp Figure 5.9: Normalizing transformation for perspective projection Solving tese equations and expressing te transformation in a omogeneous matrix form, we get: T norm = eye w =(wwidt bp) eye w =(weigt bp) =bp 0 5 : (5:51) In te canonical view volume, te central projection of a point X c ;Y c ;Z c onto te window plane is: 3 X p = d X c Z c ; Y p = d Y c Z c : (5:52) Perspective transformation Te projection and te visibility calculations are more dicult in te canonical view volume for central projection tan tey are for parallel projection because of te division required by te projection. Wen calculating visibility, it as to be decided if one point (X 1 c ;Y1 c ;Z1 ) ides anoter point c (X 2 c ;Y2 c ;Z2 ). Tis involves te ceck for relations c [X 1 c =Z 1 c ;Y1 c =Z1 c]=[x 2 c =Z2 c ;Y2 c =Z2 c] and Z 1 c <Z2 c wic requires division in a way tat te visibility ceck for parallel projection does not. To avoid division during te visibility calculation, a transformation is needed wic transforms te canonical view volume to meet te

27 5.4. VIEWING TRANSFORMATION 125 requirements of te screen coordinate systems, tat is, X and Y coordinates are te pixel addresses in wic te point is visible, and Z is a monotonous function of te original distance from te camera (see gure 5.10). eye 1 V x,vy 1 V sx,vsy canonical view volume screen coordinate system Figure 5.10: Canonical view volume to screen coordinate system transformation Considering te expectations for te X and Y coordinates: X = X c Z c V sx 2 + V x; Y = Y c Z c V sy 2 + V y: (5:53) Te unknown function Z(Z c ) can be determined by forcing te transformation to preserve planes and lines. Suppose a set of points of te canonical view volume are on a plane wit te equation: a X c + b Y c + c Z c + d =0 (5:54) Te transformation of tis set is also expected to lie in a plane, tat is, tere are parameters a 0 ;b; 0 c; 0 d 0 satisfying te equation of te plane for transformed points: a 0 X + b 0 Y + c 0 Z + d 0 =0 (5:55) Inserting formula 5.53 into tis plane equation and multiplying bot sides by Z c,we get: a 0 V sx 2 X c + b 0 V sy 2 Y c + c 0 Z(Z c ) Z c +(a 0 V x +b 0 V y +d 0 )Z c =0 (5:56)

28 TRANSFORMATIONS, CLIPPING AND PROJECTION Comparing tis wit equation 5.54, we can conclude tat bot Z(Z c ) Z c and Z c are linear functions of X c and Y c, requiring Z(Z c ) Z c to be a linear function of Z c also. Consequently: Z(Z c ) Z c = Z c + =) Z(Z c )=+ Z c : (5:57) Unknown parameters and are set to map te front clipping plane of te canonical view volume (fp 0 = fp=bp) to 0 and te back clipping plane (1) to 1: fp 0 + =0; 1+=1 + (5:58) =bp=(bp fp); = fp=(bp fp) Te complete transformation, called te perspective transformation, is: X = X c Z c V sx 2 + V x; Y = Y c Z c V sy 2 + V y; Z = Z c bp fp (bp fp) Z c : (5:59) Examining equation 5.59, we can see tat X Z c, Y Z c and Z Z c can be expressed as a linear transformation of X c ;Y c ;Z c, tat is, in omogeneous coordinates [X ;Y ;Z ;]=[XZ c ;Y Z c, ZZ c ;Z c ] can be calculated wit a single matrix product by T persp : T persp = V sx = V sy =2 0 0 V x V y bp=(bp fp) fp=(bp fp) : (5:60) Te complete perspective transformation, involving omogeneous division to get real 3D coordinates, is: [X ;Y ;Z ;]=[X c ;Y c ;Z c ;1] T persp ; [X; Y; Z; 1] = [ X ; Y ; Z ; 1]: (5:61) Te division by coordinate is meaningful only if 6= 0. Note tat te complete transformation is a omogeneous linear transformation wic consists of a matrix multiplication and a omogeneous division to convert te omogeneous coordinates back to Cartesian ones.

29 5.4. VIEWING TRANSFORMATION 127 Tis is not at all surprising, since one reason for te emergence of projective geometry as been te need to andle central projection someow by linear means. In fact, te result of equation 5.61 could ave been derived easily if it ad been realized rst tat a omogeneous linear transformation would solve te problem (gure 5.10). Tis transformation would transform te eye onto an ideal point and make te side faces of te viewing pyramid parallel. Using omogeneous coordinates tis transformation means tat: T : [0; 0; 0; 1] 7! 1 [0; 0; 1; 0]: (5:62) Multiplicative factor 1 indicates tat all omogeneous points diering by a scalar factor are equivalent. In addition, te corner points were te side faces and te back clipping plane meet sould be mapped onto te corner points of te viewport rectangle on te Z = 1 plane and te front clipping plane must be moved to te origin, tus: T : [1; 1; 1; 1] 7! 2 [V x + V sx =2;V y +V sy =2; 1; 1]; T : [1; 1; 1; 1] 7! 3 [V x + V sx =2;V y V sy =2; 1; 1]; T : [ 1; 1; 1; 1] 7! 4 [V x V sx =2;V y +V sy =2; 1; 1]; T : [0; 0;fp 0 ;1] 7! 5 [V x ;V y ;0;1]: (5:63) Transformation T is dened by a matrix multiplication wit T 44. Its unknown elements can be determined by solving te linear system of equations generated by equations 5.62 and Te problem is not determinant since te number of equations (20) is one less tan te number of variables (21). In fact, it is natural, since scalar multiples of omogeneous matrices are equivalent. By setting 2 to 1, owever, te problem will be determinant and te resulting matrix will be te same as derived in equation As as been proven, omogeneous transformation preserves linear sets suc as lines and planes, tus deriving tis transformation from te requirement tat it sould preserve planes also guaranteed te preservation of lines. However, wen working wit nite structures, suc as line segments, polygons, convex ulls, etc., omogeneous transformations can cause serious problems if te transformed objects intersect te = 0 yperplane. (Note tat te preservation of convex ulls could be proven for only tose cases wen te image of transformation as no suc intersection.) To demonstrate tis problem and ow perspective transformation works, consider an example wen V x = V y =0;V sx = V sy =2;fp =0:5;bp= 1 and

30 TRANSFORMATIONS, CLIPPING AND PROJECTION 1. Canonical view volume in 3D Euclidean space eye B A = 1 an Euclidean line segment z 2. After te perspective transformation x "points" =1 eye A z B x 3. After te omogenous division eye 8 A =1 B intersection wit =0 plane z x line segment wit wrap-around Figure 5.11: Steps of te perspective transformation and te wrap-around problem

31 5.4. VIEWING TRANSFORMATION 129 examine wat appens wit te clipping region and wit a line segment dened by endpoints [0.3,0,0.6] and [0.3,0,-0.6] in te Cartesian coordinate system (see gure 5.11). Tis line segment starts in front of te eye and goes beind it. Wen te omogeneous representation of tis line is transformed by multiplying te perspective transformation matrix, te line will intersect te = 0 plane, since originally it intersects te Z c = 0 plane (wic is parallel wit te window and contains te eye) and te matrix multiplication sets = Z c. Recall tat te = 0 plane corresponds to te ideal points in te straigt model, wic ave no equivalent in Euclidean geometry. Te conversion of te omogeneous coordinates to Cartesian ones by omogeneous division maps te upper part corresponding to positive values onto a Euclidean alf-line and maps te lower part corresponding to negative values onto anoter alf-line. Tis means tat te line segment falls into two alf-lines, a penomenon wic is usually referred to as te wrap-around problem. Line segments are identied by teir two endpoints in computer grapics. If wrap-around penomena may occur we do not know weter te transformation of te two endpoints really dene te new segment, or tese are te starting points of two alf-lines tat form te complement of te Euclidean segment. Tis is not surprising in projective geometry, since a projective version of a Euclidean line, for example, also includes an ideal point in addition to all ane points, wic glues te two \ends" of te line at innity. From tis point of view projective lines are similar (more precisely isomorpic) to circles. As two points on a circle cannot identify an arc unambiguously, two points on a projective line cannot dene a segment eiter witout furter information. By knowing, owever, tat te projective line segment does not contain ideal points, tis denition is unambiguous. Te elimination of ideal points from te omogeneous representation before omogeneous division obviously solves te problem. Before te omogeneous division, tis procedure cuts te objects represented by omogeneous coordinates into two parts corresponding to te positive and negative values respectively, ten projects tese parts back to te Cartesian coordinates separately and generates te nal representation as te union of te two cases. Recall tat a clipping tat removes object parts located outside of te viewing pyramid must be accomplised somewere in te viewing pipeline. Te cutting proposed above iswort combining wit tis clipping step, meaning tat te clipping (or at least te so-called dept clip-