COMP Computer Graphics and Image Processing. 5: Viewing 1: The camera. In part 1 of our study of Viewing, we ll look at ˆʹ U ˆ ʹ F ˆʹ S

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1 COMP27112 Û ˆF Ŝ Computer Graphics and Image Processing ˆʹ U ˆ ʹ F C E 5: iewing 1: The camera ˆʹ S Toby.Howard@manchester.ac.uk 1 Introduction In part 1 of our study of iewing, we ll look at iewing in 2D Clipping iewing in 3D iewing in OpenGL Teapot at (0,0,0), but then rotated by 20 o about, and then translated by (0,0,0.2); Camera at (10,10,10), looking at (0,0,0) 2 1

2 iewing in 2D We ll start with looking at viewing in 2D? World coordinates Screen coordinates U We need to specify what we want to see and where we want to see it 3 iewing in 2D How do we specify the mapping from our scene to the display screen?? World coordinates Screen coordinates U We use the analogy of photographing the scene with a camera In this case the camera is confined to the 2D plane 4 2

3 iewing in 2D We specify a window in world coordinates, and a viewport in screen coordinates iewport Window M view World coordinates Screen coordinates We find the matrix M view which transforms the window to the viewport. M view is called the viewing transformation 5 U Window to viewport mapping We can find M view easily, in 3 steps M1 : Translate by (-x0, -y0) to place the window at the origin M2 : Scale the window to be the same shape as the viewport M3 : Shift to the viewport position Window (x1,y1) iewport (u1,v1) M view (x0,y0) World coordinates (u0,v0) Screen coordinates 6 U 3

4 Window to viewport mapping M1 : Translate (-x0, -y0) M2 : Scale (u1-u0/x1-x0, v1-v0/y1-y0) M3 : Translate (u0, v0) M view = M3 * M2 * M1 P screen = M view * P world (x1,y1) Window iewport (u1,v1) M view (x0,y0) World coordinates (u0,v0) Screen coordinates 7 U Window to viewport mapping M view = M3 * M2 * M1 u x u x u0 1 0 x0 0 1 v v1 v y1 y 0 1 y * 0 * M 3 M 2 M 1 So we can now map a world space point P world into screen coordinates P screen as follows. This is our view. P screen = u1 u0 u1 u0 0 x 0* + u0 x1 x0 x1 x 0 v1 v0 v1 v0 0 y 0* + v * P 0 world y1 y0 y1 y M view Note: We rarely multiply matrices by hand like this. The graphics system will multiply all matrices together for us. 8 4

5 Clipping Normally we will want to CLIP against the viewport to remove those parts of primitives whose coordinates are outside the window There are standard algorithms for clipping lines and polygons Window iewport World coordinates Screen coordinates 9 U Multiple windows and viewports Sometimes it s useful to use multiple windows and viewports, to help arrange items on the screen W1 W2 M view2 1 2 M view1 World coordinates Screen coordinates U Real example: the 4 different views in AC3D 10 5

6 11 iewing in 2D: summary We use the analogy of photographing our scene with a 2D camera which can slide in the plane We compute a viewing transformation M view P screen = M view * P world Window iewport M view World coordinates Screen coordinates 12 U 6

7 iewing in 3D In 2D graphics, we view our world by mapping from 2D world coordinates to 2D screen coordinates: easy and obvious In 3D graphics, in order to view our 3D world, we have to somehow reduce our 3D information to 2D information, so that it can be displayed on the 2D display: not so easy and not obvious Here we see a 2D view of an object defined in 3D. It s been projected from 3D to 2D. To specify how this view is created, we again use the analogy of taking a picture using a camera, but this time our camera is like a real-world camera: It has a position and orientation in 3D space, and a particular type of lens. 13 The camera analogy The process of transforming a synthetic 3D model into a 2D view is analogous to using a camera in the real world to take 2D pictures of a 3D scene 14 7

8 The camera analogy Step 1: Arrange the scene into the desired composition 15 The camera analogy Step 2: Position and point the camera at the scene 16 8

9 The camera analogy Step 3: Choose a camera lens (wide-angle? zoom?) 17 The camera analogy Step 4: Decide the size of the final photograph 18 9

10 3D iewing: the camera analogy Real World Computer Graphics 1 Arrange the scene into the desired composition Set Modelling Transformation 2 Point the camera at the scene Set iewing Transformation 3 Choose the camera lens, or adjust the zoom Set Projection Transformation 4 Determine the size and shape of the final photograph Set iewport Transformation 19 3D iewing: the camera analogy Real World Computer Graphics 1 Arrange the scene into the desired composition Set Modelling Transformation 2 Point the camera at the scene Set iewing Transformation 3 Choose the camera lens, or adjust the zoom Set Projection Transformation 4 Determine the size and shape of the final photograph Set iewport Transformation 20 10

11 The 3D iewing Pipeline 3D vertex x y z 1 Modelling transformation M iewing transformation Note: In OpenGL, M and are combined into a single modelview matrix Projection transformation P 4 Clip to view volume Perspective division 5 6 iewport transformation 2D pixel px,py In this lecture we ll cover steps 1 and 2, and look at steps 3 to 6 in the next lecture. 21 The duality of Modelling and iewing Question: to obtain these two images of the model, was the camera moved or was the model moved? Answer: it is impossible to tell, just by looking at the images. It could have been either method. Example: moving the model by (x,y,z) is equivalent to moving the camera by (-x,-y,-z). This also applies to rotations

12 The duality of Modelling and iewing Here, in 2D, we have an object O and a camera C, both sitting on the axis If we keep the object fixed and move the camera by +2, O and C are now 4 units apart. If we keep the camera O C fixed and move the object by -2, O and C are now 4 units apart. O C O Object fixed, camera moved O +2 Camera fixed, object moved -2 C C Whether we move O or C, their relative positions will be the same, so the view from the camera will be the same 23 The duality of Modelling and iewing Now for the KE IDEA, which we ll present in 2D Imagine we have an object O sitting at the origin O We want to view O with a camera C located at position =3 But we don t actually have a camera! O C But we can SIMULATE the effect, by instead moving the object by -3 in O -3 O 24 12

13 Achieving viewing by modelling We ve just seen that we can create the same view from a camera at a certain location and orientation, by instead transforming the object This is exactly what we do in computer graphics However, the idea of having a camera is very natural to us, so we pretend we really have a camera and we express the view we want in terms of camera location and orientation, but to implement this we actually compute a suitable viewing transformation which we apply to the object 25 Specifying the camera We specify where the camera is located in 3D space. This is the eye point, E We specify a centre of interest C, a 3D point at which the camera is looking We specify the up direction of the camera, using a view up vector C E We can then use E,C, and to derive a transformation which, when applied to the model, would give the same view as if we really had this camera

14 The viewing transformation First we use E, C and to derive a coordinate system for the camera : 1. F = C - E, then normalise F 2. Normalise (up vector) 3. Ŝ = ˆF x ˆ (cross product) 4. Û = x ˆF (cross product) We now have a coordinate system for the camera : with axes Ŝ Û ˆF C Û E ˆF Ŝ Û ˆF Ŝ 27 The viewing transformation Next we derive a transformation M that maps the camera coordinate system into world coordinates, in two steps: 1. We translate the origin of the camera system to the origin of the world system 2. We rotate the camera axes to be coincident with the world axes, with aligned with -Z Sˆ ˆ ˆ x Sy Sz 0 Uˆ Uˆ Uˆ 0 x y z M = Fˆ ˆ ˆ x Fy Fz 0 * rotation (we omit the derivation) Ex E y Ez translation ˆF ˆʹ U ˆ ʹ F C ˆʹ S Û E ˆF Ŝ M is the viewing transformation 28 14

15 The viewing transformation: summary The viewing transformation M maps the Ŝ Û ˆF camera axes system to the world coordinates axes system: Û world coordinate system ˆʹ U ˆ ʹ F ˆʹ S Translation and rotation by M E ˆF Ŝ camera coordinate system Z KE IDEA: If we apply M to objects, we will get the same view as if we had a real camera 29 The viewing transformation in OpenGL This OpenGL function takes (E,C,) and computes the viewing transformation we have just seen void glulookat (GLdouble eyex, GLdouble eyey, GLdouble eyez, GLdouble centrex, GLdouble centrey, GLdouble centrez, GLdouble upx, Gldouble upy, Gldouble upz); OpenGL As we have seen with other OpenGL transformation functions, glulookat() creates a temporary matrix T, and then multiplies the modelview matrix by T: M modelview = M modelview * T 30 15

16 The viewing transformation in OpenGL The viewing transformation specifies the location and orientation of the camera (by in fact transforming the model) We incorporate this transformation into the modelview matrix as follows: glmatrixmode(gl_modeliew); glloadidentity(); // M= identity matrix (I) glulookat( stuff ) // M is now I * IEW Because we want the viewing transformation to take place AFTER any true modelling transformations, we need to pre-load the modelview matrix with the viewing transformation And then all subsequent modelling transformations will get multiplied into the modelview matrix 31 Modelling and viewing together // First set the viewing transformation glmatrixmode(gl_modeliew); glloadidentity(); // M= identity matrix (I) glulookat( stuff ) // M is now I * IEW // Now draw a transformed teapot gltranslatef(tx, ty, tz); // OpenGL computes temp translation matrix T, // then sets M= M x T, so now M is (IEW x T) glrotatef(theta, 0.0, 1.0, 0.0); // OpenGL computes temp rotation matrix R, // then sets M= M x R, so M is now (IEW x T x R) glutwireteapot(1.0); So all points P will be transformed first by R, then T, then IEW 32 16

17 Example: Setting a view in OpenGL Here s a real fragment showing the use of a view transformation and a modelling transformation together Note that we also need to set the projection, but we ll cover that in the next lecture, so ignore it for now. glmatrixmode(gl_modeliew); // select modelview matrix glloadidentity(); // initialise it // set the projection (see next lecture) gluperspective( stuff ); // set the view transformation glulookat(10,10,10, 0,0,0, 0,1,0); // move/rotate the model however we want gltranslatef(0.0, 0.0, 0.2); glrotatef(20.0, 0.0, 1.0, 0.0); glutwireteapot(3.0); // draw it See the next slide for a visualisation of this. 33 Teapot at (0,0,0), but then rotated by 20 o about, and then translated by (0,0,0.2); Camera at (10,10,10), looking at (0,0,0) iew seen by camera 34 17

18 Demonstrations Nate Robbins demonstrations will help you to visualise viewing. /opt/info/courses/opengl/tutor (Linux) 35 18