Projections. Let us start with projections in 2D, because there are easier to visualize.

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1 Projetions Let us start ith projetions in D, beause there are easier to visualie. Projetion parallel to the -ais: Ever point in the -plane ith oordinates (, ) ill be transformed into the point ith oordinates (, ). he matri of the transformation is Projetion along a line ith slope m: he point (,) ill be transformed into the point (- /m, ). he point (, ) ill be transformed into ( p, ). Using similar triangles on the net figure, i.e the triangle (,), (-/m, ) and (,) and the triangle (,), ( p, ) and (,), e find that - p (/m) herefore the matri of the transformation is m or equivalentl m m

2 Perspetive projetion: Here e have a OP given b (,, ) Given an point (,, ) on the plane, e dra the line joining it ith the OP. he projetion point is the point here this line meets the -ais, at ( p, ). Using again similar triangles in the previous figure, i.e the triangle (, ), ( p,), (, ) and the triangle (,), ( p, ), (, ), e an rite p p Solving for p e find p. In other ords, using homogeneous oordinates, e obtain the projeted point to have the folloing oordinates: (,, ), or using the propert of homogeneous oordinates (,, ). his generates the matri of the transformation: hanged to: or if the enter is given b (,, ) then the matri ill be REMARK: he parallel projetion is a partiular ase of the perspetive projetion the a e have defined in the last matri.

3 If the enter of projetion is the diretion of the oblique line (, m, ), then b substitution e find the matri of the beginning of the page. B generaliation (using the same tehniques to obtain the values of the projetion matri), e an rite that the projetion onto the -plane, using the enter of projetion (,,, ) is the folloing one: 3-D Projetions Parallel Projetion. his is a projetion folloing a diretion. In the lassial vieing e have the folloing projetions: Orthographi projetion: he diretion of projetion is perpendiular to the projetion plane. In a multivie orthographi projetion, the projetion plane is parallel to one of the prinipal faes of the objet. Aonometri projetion: he DOP is perpendiular to the plane of projetion, but the objet ma have an orientation ith respet to the projetion plane. Isometri projetion is a tpe of aonometri projetion in hih the projetion plane is plaed smmetriall ith respet to three prinipal faes of the objet that meet at a orner. Dimetri projetion is another aonometri projetion, but in this one, the projetion plane is plaed smmetriall ith respet to to prinipal faes of the objet. Oblique projetions: he diretion of projetion is not orthogonal to the projetion plane. Among them the most famous are the avalier and the abinet projetions. In both of them one prinipal fae of the objet is parallel to the projetion plane, but the diretion of projetion is suh that in the ase of abinet projetion: he prinipal fae perpendiular to the projetion plane has half the sale that the fae parallel to the projetion plane.

4 avalier projetion: he prinipal fae perpendiular to the projetion plane has the same sale that the fae parallel to the projetion plane. PerspetiveProjetion. he enter of projetion is a point at a finite distane. In lassial vieing for the perspetive projetion, e have one-, to-, and three-point perspetives. he one-point perspetive projetion has a prinipal fae of the objet parallel to the projetion plane. he to-point one has a line of the objet parallel to the projetion plane. he three-point does not have an line parallel to the projetion plane. he one-point perspetive projetion has a vanishing point (point here the parallel lines no parallel to the projetion plane onverge) he to-point perspetive projetion has to vanishing points (that generate the line of the horion, b the vanishing points of the parallel lines not parallel to the projetion plane) he three-point perspetive projetion has three vanishing points. All these projetions are partiular ases of the projetion transformation that e ill generate. Projetion eample: Let s assume that the enter of projetion is P [,,, ]. e have the bo A [,,, ] ; B [5,,, ] ; [5,, -5, ] ; D [,, -5, ] ; E [,,, ] ; F [5,,, ] ; G [5,, -5, ] ; H [,, -5, ] ; e ant to find the oordinates of the projeted bo. First e find the projetion onto the -plane, using the OP given b P. But before omputing the projetion matri, e reall ompute the prearping matri.

5 P Note that e keep the pseudo-distane (for the -buffer) to distinguish the points that are farther aa. hen the onl thing left to obtain the projetion is to multipl b the orthogonal projetion matri, hih is equivalent to hanging the -oordinate value to ero. First e ould immediatel rite hat happens in the -plane, sine the OP is there, and so are the points A, B,, D. As a matter of fat, e an obtain P. A P. [,,, ] [,,, ]. Note that the pseudo-depth is. P. B P. [5,,, ] [5,,, ]. Note that the pseudo-depth is again ero. P. P. [5,, -5, ] [8,, -.6, ]. Note that the pseudo-depth is.6. P. D P. [,, -5, ] [6,, -.6, ]. Pseudo-depth is.6. P. E P. [,,, ] [,,, ]. Pseudo-depth is again ero. P. F P. [5,,, ] [5,,, ]. Pseudo-depth is again ero. P. G P. [5,, -5, ] [8, 4, -.6, ]. Pseudo-depth is.6. P. H P. [,, -5, ] [6, 4, -.6, ]. Pseudo-depth is.6.

6 If e represent these ne points, ithout the pseudo-depth, e obtain the folloing figure: Note that the vanishing point V has oordinates [,,, ]. Oblique projetion: It is a parallel projetion hose diretion of projetion is not perpendiular to the vieing plane. e need to find the matri of the transformation.

7 learl the diretion of projetion is given b [ p -, p,, ], but in order to determine the values of p, p, e ould need the diretion of projetion. So e need another approah. e ould determine p, p, if e kne the angle α, and the angle φ. he angle α is determined b the line going from the point [,,, ] to [ p, p,, ] and the line going from the same point [ p, p,, ] to the point [,,, ] ( See the previous figure). he angle φ is determined b the line going from [,,, ] to the point [ p, p,, ] and the -ais. If these to angles ere given, e ould determine the diretion of projetion. Let us see: Let L be the distane from the point ( p, p, ) to the point (,, ). he e an rite, p L os φ p L sin φ. But L tan - α herefore p tan - α os φ p tan - α sin φ and onsequentl the projetion matri is: ' osφ tanα sinφ tanα

8 omparing this result ith the general projetion matri e enounter before, e an establish that the enter of projetion for the oblique projetion is given b,, tan sin, tan os α φ α φ OP If tan α, then e have a abinet projetion. If it is equal to, then e have the avalier projetion. r some eamples to see the relationship beteen the distanes on the plane parallel to the projetion plane, and the distanes on a plane perpendiular to the projetion plane. Projetion Normaliation. Ever projetion an be transformed into the produt of to transformations. First e appl a distortion, and then an orthogonal projetion. he first transformation, hih makes the distortion of the objet, is sometimes referred as prearping. Let s ork on D, and then e an generalie. Imagine e have the projetion matri,, that an be deomposed into the produt of these to: if, or (note that and an not both be ero simultaneousl) Let us all the matri on the right and the projetion matri. Note that the other matri orresponds to an orthogonal projetion.

9 hen e appl e obtain a pseudo-distane to the -ais as a bprodut. his pseudodistane preserves the relative distane order of the points ith respet to the -ais. Eample: Assume (,, ) (,, ) hen the matri ill be given b. onsider the points ( - - α, -α, ) α α α α. So the ne points in artesian oordinates are α α α,, and the distanes to the -ais are given b. α α he distane to the -ais of the original points as α, and the ne distane is α. α You ma hek that the ne distane preserves the order of the points. he ones originall farther aa, go to a ne distane that is also further aa. For instane, if α.5, the ne distane is 3, if α as.75, the ne distane is 7 3, if α as, the ne distane is.5, if α as 3, then the ne distane is 4 3, et But before e even appl the distortion of the objet, e perform another transformation, to hange the given vieing volume to the ube ith enter at the origin and determined b the planes ±, ±, ±. his is alled the anonial vie volume. Note that e ould have used a unit ube, ith a orner at the origin, and the results ould be ver similar. Let s start ith an orthogonal projetion. e have to suppl the vieing volume, given b the folloing parallelepiped: ( L, R, B,, N, F ) - L stands for left, R stands for right, B stands for bottom, stands for top, N stands for near, and F stands for far. Other notations an be found, ith the folloing meaning: L min, R ma, B min, ma, N min, F ma.

10 In order to make the hange to the anonial vie, e have to make a translation first, hose matri is given b N F B L R hen e make a saling, given b the matri S: N F B L R S Multipling both matries e obtain the transformation matri N F N F N F B B B L R L R L R P Oblique and perspetive projetion. In this ase a good tehnique is to define the orld-vieing indo in the folloing a. First e define a vieing indo in the plane, i.e. e delare L, R, B and. hen e delare the to planes N, F. he vieing 3D indo is the region of the spae beteen these to planes, that projets into the retangle ( L, R, B, ). In this ase the prearping hanges the orld-vieing indo into a parallelepiped, and e are in the previous ase. Eample in D: onsider the enter of projetion (,, ), and the indo (in the -ais) determined b (,,) and (,,). For the orld-vieing volume e fi - (near) and - (far). his determines that the orld-vieing volume is the quadrilateral defined b the folloing points: (,-,), (3, -, ), (6, -,) and (4, -,). (See net figure)

11 he matri for the prearping is given b, hih transforms the quadrilateral into a retangle, hose verties are given b (, -.5,), (, ,), (, ,) and (,-.5,). No e an hange this retangle into the anonial vie, before e appl the orthographi projetion.

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