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2IV60 Computer graphics set 5: Viewing Jack van Wijk TU/e Viewing • Transformation worldscreen • Clipping: Removing parts outside screen • 2D (chapter 8) • 3D (chapter 10) 2D Viewing pipeline 1 World: Screen: Clipping window Viewport ywmax yvmax ywmin yvmin xwmin xwmax Clipping window: What do we want to see? xvmin xvmax Viewport: Where do we want to see it? H&B 8-1:258-259 2D Viewing pipeline 2 World: Screen: Clipping window Viewport ywmax yvmax ywmin yvmin xwmin xwmax xvmin xvmax Clipping window: Panning… H&B 8-2:259-261 2D Viewing pipeline 2 World: Screen: Clipping window Viewport ywmax yvmax ywmin yvmin xwmin xwmax xvmin xvmax Clipping window: Panning… H&B 8-2:259-261 2D Viewing pipeline 3 World: Screen: Viewport ywmax yvmax ywmin yvmin xwmin xwmax xvmin xvmax Clipping window: Zooming… H&B 8-2:259-261 2D Viewing pipeline 3 World: Screen: ywmax Viewport yvmax yvmin ywmin xwmin xwmax xvmin xvmax Clipping window: Zooming… H&B 8-2:259-261 2D Viewing pipeline 4 MC: Modeling Coordinates Apply model transformations WC: World Coordinates Determine visible parts VC: Viewing Coordinates To standard coordinates NC: Normalized Coordinates Clip and determine pixels DC: Device Coordinates H&B 8-2:259-261 Clipping window Clipping window ywmax ywmax ywmin ywmin xwmin xwmax xwmax xwmin - Clipping window usually an axis-aligned rectangle - Sometimes rotation - From world to view coordinates: T xwmin , ywmin possibly followed by rotation - More complex in 3D H&B 8-2:259-261 To normalized coordinates 1 ywmax -ywmin yvmax -yvmin xwmax-xwmin Scale with : xvmax-xvmin xv max xv min yv max yv min S , xwmax xwmin ywmax ywmin If the two scale factors are unequal, then the aspect - ratio changes : distortion! H&B 8-3:261-265 To normalized coordinates 2 1 Viewport yvmax yvmax -yvmin yvmin xvmax-xvmin xvmin xvmax 1 T ranslate with : T xv min , yv min H&B 8-3:261-265 To normalized coordinates 3 Clipping window 1 ywmax yvmax ywmin yvmin xwmin xwmax Viewport xvmin xvmax 1 All together : xvmax xvmin yvmax yvmin T( xvmin , yvmin )S , xwmax xwmin ywmax ywmin T( xwmin , ywmin ) H&B 8-3:261-265 OpenGL 2D Viewing 1 Specification of 2D Viewing in OpenGL: - Standard pattern, follows terminology. First, this is about projection. Hence, select and the Projection Matrix (instead of the ModelView matrix) with: glMatrixMode(GL_PROJECTION); H&B 8-4:265-267 OpenGL 2D Viewing 2 Next, specify the 2D clipping window: gluOrtho2D(xwmin, xwmax, ywmin, ywmax); xwmin, xwmax: horizontal range, world coordinates ywmin, ywmax: vertical range, world coordinates ywmax ywmin xwmin xwmax H&B 8-4:265-267 OpenGL 2D Viewing 3 Finally, specify the viewport: glViewport(xvmin, yvmin, vpWidth, vpHeight); xvmin, yvmin: coordinates lower left corner (in pixel coordinates); vpWidth, vpHeight: width and height (in pixel coordinates); vpHeight vpWidth (xvmin, yvmin) H&B 8-4:265-267 OpenGL 2D Viewing 4 In short: glMatrixMode(GL_PROJECTION); gluOrtho2D(xwmin, xwmax, ywmin, ywmax); glViewport(xvmin, yvmin, vpWidth, vpHeight); To prevent distortion, make sure that: (ywmax – ywmin)/(xwmax – xwmin) = vpWidth/vpHeight H&B 8-4:265-267 Clipping Clipping window ywmax ywmin xwmin xwmax - Clipping: removing parts outside clipping window - Many algorithms: points, lines, fill-area, curves,… H&B 8-5:274 2D Point Clipping Clipping window ywmax ywmin xwmin xwmax - Trivial: Save point P = (x,y) if it’s in the box: xwmin x xwmax ywmin y ywmax H&B 8-5:274-275 2D Line Clipping 1 ywmax ywmin xwmin xwmax H&B 8-6:275-281 2D Line Clipping 2 ywmax Q u ywmin P xwmin Basic algorithm: Determine interval (u0, u1) in rectangle, by calculating crossings with edges of window. xwmax Line segment: X(u) = P + u (QP), with 0 <= u <= 1 H&B 8-6:275-281 2D Line Clipping 3 ywmax Q u ywmin u1 P xwmin xwmax Line segment: X(u) = P + u (Q-P), with 0 <= u <= 1 u0 := 0; u1 = 1; (* Right side: *) If Px=Qx then If Px > xwmax then return empty else u := (xwmax – Px)/(Qx-Px) if Px < Qx then begin if u < u1 then u1 := u end else if u > u0 then u0 := u; If u0 > u1 then return empty H&B 8-6:275-281 2D Line Clipping 4 ywmax Q u ywmin u1 P xwmin xwmax Line segment: X(u) = P + u (Q-P), with 0 <= u <= 1 (* Left side: *) If Px=Qx then If Px < xwmin then return empty else u := (xwmin – Px)/(Qx-Px) if Px < Qx then begin if u > u0 then u0 := u end else if u < u1 then u1 := u; If u0 > u1 then return empty (* Lower and upper side idem *) H&B 8-6:275-281 2D Line Clipping 5 ywmax ywmin xwmin xwmax - Expensive: calculation crossings - Avoid this by using position end points - For instance: If end points in window, then line in window H&B 8-6:275-281 2D Line Clipping 6 ywmax ywmin 1001 1000 1010 0001 0000 0010 0101 0100 0110 xwmin xwmax bit 4 bit 3 bit 2 bit 1 Top Right Bottom Left Cohen-Sutherland algorithm: - Assign to each point a four-bit region code C; - 1 is outside, 0 is inside H&B 8-6:275-281 2D Line Clipping 7 ywmax ywmin 1001 1000 1010 0001 0000 0010 0101 0100 0110 xwmin xwmax bit 4 bit 3 bit 2 bit 1 Top Right Bottom Left Fast test on status end points line with codes C0 en C1 : C0 bitwise-or C1 = 0000: then completely in; C0 bitwise-and C1 <> 0000: then completely out; Else: fast intersection-calculation using codes. H&B 8-6:275-281 3D Viewing • • • • • Viewing: virtual camera Projection Depth Visible lines and surfaces Surface rendering H&B 10-1:332-334 3D Viewing pipeline 1 • Similar to making a photo – Position and point virtuele camera, press button; Projection plane aka Viewing plane • Pipeline has +/ same structure as in 2D H&B 10-1:332-334 3D Viewing pipeline 2 MC: Modeling Coordinates Apply model transformations WC: World Coordinates To camera coordinates VC: Viewing Coordinates Project PC: Projection Coordinates To standard coordinates NC: Normalized Coordinates Clip and convert to pixels H&B 10-2:334-335 DC: Device Coordinates 3D viewing coordinates 1 Specification of projection: yw P0 : View or eye point P0 N Pref : Center or look-at point V Pref V: View-up vector xw (projection along vertical zw axis) zvp : positie view plane P0, Pref , V: define viewing coordinate system Several variants possible H&B 10-3:336-338 3D viewing coordinates 2 xview yview yw V zw P0 zview N Pref xw P0, Pref , V: define viewing coordinate system Several variants possible H&B 10-4:338-340 3D view coordinates 3 xview yview v yw V zw Derivation axis frame : N P0 Pref u n P0 N Pref zview N n (nx , n y , n z ) N u perpendicular to V and n : xw Vn u (u x , u y , u z ) V v perpendicular to n and u : v n u (v x , v y , v z ) H&B 10-4:338-340 3D viewing coördinaten 4 xview yview v yw V zw u n P0 N Pref xw zview T ransformation world view : First, translate with T(P0 ) Next, rotate with R : ux vx R n x 0 uy vy ny 0 uz vz nz 0 0 0 0 1 Or : MWC,VC RT H&B 10-4:338-340 Projection transformations P2 View plane P2 P’2 P1 P’1 Parallel projection View plane P’2 P1 P’1 Perspective projection H&B 10-6:340-345 Orthogonal projections 1 P2 Parallell projection: Projection lines are parallel P’2 P1 P’1 Orthogonal projection: Projection lines are parallel and perpendicular to projection plane P2 P’2 P1 P’1 z y x Isometric projection: Projection lines are parallel, perpendicular to projection plane, and have the same angle with axes. H&B 10-6:340-345 Orthogonal projections 2 Orthogonal projection: P2 P’2 P1 P’1 Projection of ( x, y, z ) (from view coordinates to projection coordinates) : xp x yp y zp z T rivial! H&B 10-6:340-345 Orthogonal projections 3 View volume Clipping window yview xview Far plane zview Near plane H&B 10-6:340-345 Orthogonal projections 4 View volume (xwmax, ywmax, zfar) yview xview (xwmin, ywmin, znear) zview H&B 10-6:340-345 Orthogonal projections 4 (xwmax, ywmax, zfar) (1,1,1) yview xview (xwmin, ywmin, znear) View volume zview znorm ynorm xnorm (-1,-1,-1) Normalized View volume Translation Scaling From right- to left handed H&B 10-6:340-345 Perspective projection 1 P2 View plane: z = zvp P’2 P1 P’1 Projection reference point yview xview zview View plane: orthogonal to zview axis. H&B 10-8:351-364 Perspective projection 2 View plane: z = zvp P = (x, y, z) R: Projection reference point (xp, yp, zp) (x , y , z ) r r r yview xview To simplify, Assume R in origin zview Question: What is the projection of P on the view plane? H&B 10-8:351-364 Perspective projection 3 Line from R (origin) to P : View plane: z = zvp P = (x, y, z) (xp, yp, zp) X' uP, with 0 u 1, yview or x' ux; y ' uy; and z ' uz. xview R= (0,0, 0) zview At crossing with plane : z ' zvp hence u zvp z . Substitution gives xp z vp z x and y p z vp z y H&B 10-8:351-364 Perspective projection 4 P = (x, y, z) View plane We can see that : y yp z zvp yview y yp z R zvp zview hence yp zvp z y Viewed from the side H&B 10-8:351-364 Perspective projection 5 Clipping window P = (x, y, z) in View plane wmax yview Ratio between W=wmaxwmin and zvp R W zvp determines strenght perspective zview wmin Viewed from the side H&B 10-8:351-364 Perspective projection 6 Clipping window in View plane Ratio between yview W=wmaxwmin and zvp R determines strenght perspective. zview This ratio is 2tan(/2), with the view angle. Viewed from the side H&B 10-8:351-364 Perspective projection 7 yview R zview H&B 10-8:351-364 Perspective projection 7 yview R zview H&B 10-8:351-364 Perspective projection 7 R H&B 10-8:351-364 Perspective projection 8 Option 1: Specify view angle . yview W R zview zvp How to specify the ratio between W en zvp? How to specify ‘how much’ perspective I want to see? H&B 10-8:351-364 Perspective projection 9 yview R 24 mm zview W zvp f (mm) Use camera as metaphor. User specifies focal length f . (20 mm – wide angle, 50 mm – normal, 100 mm – tele). Application adjusts W and/or zvp, such that W/zvp = 24/f. For x: W/zvp = 36/f. H&B 10-8:351-364 View volume orthogonal… View volume Clipping window yview xview Far clipping plane Near clipping plane zview H&B 10-8:351-364 View volume perspective View volume Clipping window yview xview Far clipping plane Near clipping plane R zview H&B 10-8:351-364 To Normalized Coordinates… (1,1,1) yview xview R Rectangular frustum View Volume zview znorm ynorm xnorm (1, 1, 1) Normalized View volume H&B 10-8:351-364 Perspective projection 10 ynorm yview R zview znorm Side view Front view Perspective transformation: Distort space, such that perpendicular projection gives an image in perspective. H&B 10-8:351-364 Perspective projection 10 ynorm yview 2r R zview znorm zn zf Simplest case: Square window, clipping plane coincides with view plane: zn=zvp H&B 10-8:351-364 Perspective projection 11 ((zf /zn)r, (zf /zn)r, zf ) (1,1,1) ynorm yview 2r R zview (-r, z-r, zn) znorm (-1,-1,-1) n zf H&B 10-8:351-364 Perspective projection 11 (rzf /zn, rzf /zn, zf ) (1,1,1) ynorm yview R zview znorm (1, 1, 1) (r, r, zn) Earlier: x p zvp z x, y p zvp z y How to put this transformation in the pipeline? How to process division by z? H&B 10-8:351-364 Homogeneous coordinates (reprise) • Add extra coordinate: P = (px , py , pz , ph) or x = (x, y, z, h) • Cartesian coordinates: divide by h x = (x/h, y/h, z/h) • Points: h = 1 (temporary…) perspective: h = z ! H&B 10-8:351-364 Homogeneous coordinates (reprise) Perspective transform ation can be described by : xh s xx s xy s xz t x xv yh s yx s yy s yz t y yv z s z s s t zy zz z v h zx h 0 1 0 1 0 such that projected coordinates are given by : x p xh / h y p yh / h . z z / h p h H&B 10-8:351-364 Perspective projection 12 (rzf /zn, rzf /zn, zf ) (r, r, zn) xview R zview (1,1,1) ynorm znorm (r, r, zn) (1, 1, 1) First x. Generic form is x p ( s xx x s xy y s xz z t x ) / z. If x r and z z n , then x p 1. If x rz f / z n and z z f , then also x p 1. Elaboration gives : s xx z n / r , s xy s xz t x 0. H&B 10-8:351-364 Perspective projection 13 (rzf /zn, rzf /zn, zf ) (1,1,1) ynorm yview R zview znorm (r, r, zn) (1, 1, 1) Next the y. Same as x, gives : y p ( zn / r ) y / z. Or : s yy ( zn / r ), s yx s yz t y 0. H&B 10-8:351-364 Perspective projection 14 (rzf /zn, rzf /zn, zf ) (1,1,1) ynorm yview R zview znorm (1, 1, 1) (r, r, zn) Finally: z. Generic form is : z p ( s zx x s zy y s zz z t z ) / z. If z z n , then z p 1. If z z f , then z f 1. Elaboration gives s zz zn z f zn z f ,tz 2 zn z f zn z f , s zx s zy 0. H&B 10-8:351-364 Perspective projection 15 Perspective transform ation can hence be described by : 0 0 0 z / r xh n 0 zn / r 0 0 y h zn z f 2 zn z f z 0 0 h zn z f zn z f h 0 0 1 0 where the projected coordinates follow from : x v y v zv 1 x p xv / h y p yv / h . z z / h p v H&B 10-8:351-364 3D Viewport coordinates 1 (xvmax, yvvmax, 1) (1,1,1) znorm ynorm xnorm (1, 1, 1) yv zv xv scaling translation Normalized View volume (xvmin, yvmin, 0) 3D screen H&B 10-9:365-365 3D Viewport coordinaten 2 T ransformation from normalized view coordinates to 3D screen coordinates : xv max xs ys z s h xv min 2 0 0 0 yv max yv min 2 0 0 0 0 0 1 2 0 0 x p 0 yp 1 zp 1 2 1 H&B 10-9:365-365 OpenGL 3D Viewing 1 3D Viewing in OpenGL: - Position camera; - Specify projection. H&B 10-10:365-371 OpenGL 3D Viewing 2 View volume y x z Camera always in origin, in direction of negative z-axis. Convenient for 2D, but not for 3D. H&B 10-10:365-371 OpenGL 3D Viewing 3 Solution for view transform: Transform your model such that you look at it in a convenient way. Approach 1: Do it yourself. Apply rotations, translations, scaling, etc., before rendering the model. Surprisingly difficult and error-prone. Approach 2: Use gluLookAt(); H&B 10-10:365-371 OpenGL 3D Viewing 4 MatrixMode(GL_MODELVIEW); gluLookAt(x0,y0,z0, xref,yref,zref, Vx,Vy,Vz); x0,y0,z0: xref,yref,zref: Vx,Vy,Vz: P0, viewpoint, location of camera; Pref, centerpoint; V, view-up vector. Default: P0 = (0, 0, 0); Pref = (0, 0, 1); V=(0, 1, 0). H&B 10-10:365-371 OpenGL 3D Viewing 5 Orthogonal projection: MatrixMode(GL_PROJECTION); glOrtho(xwmin, xwmax, ywmin, ywmax, dnear, dfar); xwmin, xwmax, ywmin,ywmax: specification window dnear: distance to near clipping plane dfar : distance to far clipping plane y ywmax x xwmax ywmin xwmin z Select dnear and dfar right: dnear < dfar, model fits between clipping planes. H&B 10-10:365-371 OpenGL 3D Viewing 6 Perspective projection: MatrixMode(GL_PROJECTION); glFrustrum(xwmin, xwmax, ywmin, ywmax, dnear, dfar); xwmin, xwmax, ywmin,ywmax: specification window dnear: distance to near clipping plane dfar : distance to far clipping plane y ywmax ywmin x xwmax xwmin Standard projection: z xwmin = -xwmax, ywmin = -ywmax Select dnear and dfar right: 0 < dnear < dfar, model fits between clipping planes. H&B 10-10:365-371 OpenGL 3D Viewing 7 Finally, specify the viewport (just like in 2D): glViewport(xvmin, yvmin, vpWidth, vpHeight); xvmin, yvmin: coordinates lower left corner (in pixel coordinates); vpWidth, vpHeight: width and height (in pixel coordinates); vpHeight vpWidth (xvmin, yvmin) H&B 8-4:265-267 OpenGL 2D Viewing 8 In short: glMatrixMode(GL_PROJECTION); glFrustrum(xwmin, xwmax, ywmin, ywmax, dnear, dfar); glViewport(xvmin, yvmin, vpWidth, vpHeight); glMatrixMode(GL_MODELVIEW); gluLookAt(x0,y0,z0, xref,yref,zref, Vx,Vy,Vz); To prevent distortion, make sure that: (ywmax – ywmin)/(xwmax – xwmin) = vpWidth/vpHeight Make sure that you can deal with resize/reshape of the (OS) window. H&B 8-4:265-267 Next… • We now know how to project objects. • But how to model objects?