Fundamental Types of Viewing

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1 Viewings

2 Fundamental Types of Viewing Perspective views finite COP (center of projection) Parallel views COP at infinity DOP (direction of projection) perspective view parallel view

3 Classical Viewing Specific relationship between the objects and the viewers

4 Orthographic Projections Projectors are perpendicular to the projection plane preserve both distances and angles orthographic projections temple and three multiview orthographic projections

5 Axonometric Projections (/2) Projection plane can have any orientation with respect to the object projectors are still orthogonal to the projection planes construction top view side view

6 Axonometric Projections Preserve parallel lines but not angles isometric projection plane is placed symmetrically with respect to the three principal faces dimetric two of principal faces trimetric general case

7 Oblique Projections Projectors can make an arbitrary angle with the projection plane preserve angels in planes parallel to the projection plane construction top view side view

8 Perspective Projections (/2) Diminution of size when objects are moved father from the viewer, their images become smaller

9 Perspective Projections One-, two-, and three-point perspectives how many of the three principal directions in the object are parallel to the projection plane vanishing points three-point perspective two-point perspective one-point perspective

10 Positioning of the Camera (/3) OpenGL places a camera at the origin of the world frame pointing in the negative z direction move the camera away from the objects gltranslatef(.,., -d); initial configuration after change in the model-view matrix

11 Positioning of the Camera (2/3) Look at the same object from the positive x axis translation after rotation by 9 degrees about the y axis glmatrixmode(gl_modelview); glloadidentity( ); gltranslatef(.,., -d); glrotatef(-9.,.,.,.); y c x c zc

12 Positioning of the Camera (3/3) Create an isometric view of the cube glmatrixmode(gl_modelview); glloadidentity( ); gltranslatef(.,., -d); glrotatef(35.26,.,.,.); glrotatef(45.,.,.,.);

13 Look-At Function OpenGL utility function glulookat(eyex, eyey, eyez, atx, aty, atz, upx, upy, upz);

14 Others Roll, pitch, and yaw ex. flight simulation Elevation and azimuth ex. star in the sky

15 Simple Perspective Projections (/2) Simple camera projection plane is orthogonal to z axis projection plane in front of COP x x p z p d, z d x y x p, y p z / d z / d three-dimensional view top view side view

16 Simple Perspective Projections (2/2) Homogeneous coordinates Perspective projection matrix w wz wy wx z y x p d z z y x d z z d z y d z x d d z y d z x z y x p p p / / / / / / p / d M projection pipeline Model-view Projection Perspective division

17 Simple Orthogonal Projections Projectors are perpendicular to the view plane Orthographic projection matrix p p p z y y x x z y x z y x p p p

18 Projections in OpenGL Angle of view only objects that fit within the angle of view of the camera appear in the image View volume be clipped out of scene frustum truncated pyramid

19 Perspective in OpenGL (/2) Specification of a frustum glmatrixmode(gl_projection); glloadidentity( ); glfrustum(xmin, xmax, ymin, ymax, near, far); near, far: positive number z max = far z min = near

20 Perspective in OpenGL (2/2) Specification using the field of view glmatrixmode(gl_projection); glloadidentity( ); gluperspective(fovy, aspect, near, far); fov: angle between top and bottom planes fovy: the angle of view in the up (y) direction aspect ratio: width divided by height

21 Parallel in OpenGL Orthographic viewing function glmatrixmode(gl_projection); glloadidentity( ); glortho(xmin, xmax, ymin, ymax, near, far); OpenGL provides only this parallel-viewing function near < far!! no restriction on the sign z max = far z min = near

22 Normalization Rather than derive a different projection matrix for each type of projection, we can convert all projections to orthogonal projections with the default view volume This strategy allows us to use standard transformations in the pipeline and makes for efficient clipping

23 Pipeline View modelview transformation nonsingular projection transformation perspective division 4D 3D clipping against default cube projection 3D 2D

24 Notes We stay in four-dimensional homogeneous coordinates through both the modelview and projection transformations Both these transformations are nonsingular Default to identity matrices (orthogonal view) Normalization lets us clip against simple cube regardless of type of projection Delay final projection until end Important for hidden-surface removal to retain depth information as long as possible

25 Orthogonal Normalization glortho(left,right,bottom,top,near,far) normalization find transformation to convert specified clipping volume to default

26 Orthogonal Matrix Two steps Move center to origin T(-(left+right)/2, -(bottom+top)/2,(near+far)/2)) Scale to have sides of length 2 S(2/(left-right),2/(top-bottom),2/(near-far)) P = ST = 2 right left 2 top bottom 2 near far right left right left top bottom top bottom far near far near

27 Final Projection Set z = Equivalent to the homogeneous coordinate transformation Hence, general orthogonal projection in 4D is M orth = P = M orth ST

28 Oblique Projections The OpenGL projection functions cannot produce general parallel projections such as However if we look at the example of the cube it appears that the cube has been sheared Oblique Projection = Shear + Orthogonal Projection

29 General Shear top view side view

30 Shear Matrix xy shear (z values unchanged) H(q,f) = Projection matrix cotθ cotφ General case: P = M orth H(q,f) P = M orth STH(q,f)

31 Equivalency

32 Effect on Clipping The projection matrix P = STH transforms the original clipping volume to the default clipping volume object top view z = DOP clipping volume near plane far plane x = - z = - DOP x = distorted object (projects correctly)

33 Simple Perspective Consider a simple perspective with the COP at the origin, the near clipping plane at z = -, and a 9 degree field of view determined by the planes x = z, y = z

34 Perspective Matrices Simple projection matrix in homogeneous coordinates M = Note that this matrix is independent of the far clipping plane

35 Generalization N = α β after perspective division, the point (x, y, z, ) goes to x = x/z y = y/z Z = -(a+b/z) M orth N = P = M orth Np We obtain the same result as we would have for a perspective projection. We can normalize the perspective transformation by using the transformation matrix N.

36 Picking a and b If we pick a = b = near far far near 2near far near far the near plane is mapped to z = - the far plane is mapped to z = and the sides are mapped to x =, y = Hence the new clipping volume is the default clipping volume

37 Normalization Transformation distorted object projects correctly original clipping volume original object new clipping volume

38 Normalization and Hidden-Surface Removal Although our selection of the form of the perspective matrices may appear somewhat arbitrary, it was chosen so that if z > z 2 in the original clipping volume then the for the transformed points z > z 2 Thus hidden surface removal works if we first apply the normalization transformation However, the formula z = -(a+b/z) implies that the distances are distorted by the normalization which can cause numerical problems especially if the near distance is small

39 OpenGL Perspective glfrustum allows for an unsymmetric viewing frustum (although gluperspective does not)

40 OpenGL Perspective Matrix The normalization in glfrustum requires an initial shear to form a right viewing pyramid, followed by a scaling to get the normalized perspective volume. Finally, the perspective matrix results in needing only a final orthogonal transformation P = NSH our previously defined perspective matrix shear and scale

41 Why do we do it this way? Normalization allows for a single pipeline for both perspective and orthogonal viewing We stay in four dimensional homogeneous coordinates as long as possible to retain three-dimensional information needed for hidden-surface removal and shading We simplify clipping

42 Walking Though a Scene (/2) 실습 void keys(unsigned char key, int x, int y) { if(key == x ) viewer[] -=.; if(key == X ) viewer[] +=.; if(key == y ) viewer[] -=.; if(key == Y ) viewer[] +=.; if(key == z ) viewer[2] -=.; if(key == Z ) viewer[2] +=.; } void display(void) { glclear(gl_color_buffer_bit GL_DEPTH_BUFFER_BIT); glloadidentity(); glulookat(viewer[], viewer[], viewer[2],,,,,,); glrotatef(theta[],.,.,.); glrotatef(theta[],.,.,.); glrotatef(theta[2],.,.,.); colorcube( ); } glflush( ); glutswapbuffers( );

43 Walking Though a Scene (2/2) 실습 void myreshape(int w, int h) { glviewport(,, w, h); glmatrixmode(gl_projection); glloadidentity( ); if( w <= h ) glfrustum(-2., 2., -2.*(GLfloat)h/(GLfloat)w, 2.*(GLfloat)h/(GLfloat)w, 2., 2.); else glfrustum(-2. *(GLfloat)w/(GLfloat)h, 2. *(GLfloat)w/(GLfloat)h, -2., 2., 2., 2.); } glmatrixmode(gl_modelview);

44 Projections & Shadows (/2) Shadow polygon Steps light source at (x l, y l, z l ) translation (-x l, -y l, -z l ) perspective projection through the origin translation (x l, y l, z l ) / l l l l l l l z y x y z y x PT T M 실습

45 Projections & Shadows (2/2) 실습 GLfloat m[6]; /* shadow projection matrix */ for(i=; i<6; i++) m[i] =.; m[] = m[5] = m[] =.; m[7] = -./yl; glcolor3fv(polygon_color); glbegin(gl_polygon);.. /* draw the polygon normally */. glend( ); glmatrixmode(gl_modelview); glpushmatrix( ); /* save state */ gltranslatef(xl, yl, zl); /* translate back */ glmultmatrixf(m); /* project */ gltranslatef(-xl, -yl, -zl); /* move light to origin */ glcolorfv(shadow_color); glbegin(gl_polygon);.. /* draw the polygon again */. glend( ); glpopmatrix( ); /* restore state */

46 Shadows from a Cube onto Ground 실습

47 Zoom in 2D Zoom? Resizing? Magnifying?

48 Zoom in 2D Zoom? Resizing? Magnifying?

49 Zoom in 3D Zoom? Resizing? Magnifying?

50 Zoom in stereo

51 Zoom in stereo

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