1 OpenGL - column vectors (column-major ordering)
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1 OpenGL - column vectors (column-major ordering) OpenGL uses column vectors and matrices are written in a column-major order. As a result, matrices are concatenated in right-to-left order, with the first operation occuring on the right, the second operation to the left, the next operation to the left, and so on [ Pro ject View World ] x y z = x y z (). World (Model) Transformation Transformation matrices T GL = Translation Scale Rotation X (φ) t x t y t z S GL = s x s y s z Rx GL = cos(φ) sin(φ) sin(φ) cos(φ) Rz GL = Rotation Z (φ) cos(φ) sin(φ) sin(φ) cos(φ) Ry GL = Rotation Y (φ) cos(φ) sin(φ) sin(φ) cos(φ) Complex transformations, ordering of matrices The complex transformations can be decomposed into several basic ones: scalings, rotations, and translations. The basic transformation matrices are then assembled together by multiplicaton, and the order of multiplication matters (Figure ). a) b) Figure : The meaning of matrices order for transformations: a) first rotation, then translation; b) first translation, then rotation. the matrix ordering strongly depends on a way, in which matrices are stored. In OpenGL matrices are stored in columns (column-major order), in DirectX - in rows (row-major order).
2 OPENGL - COLUMN VECTORS (COLUMN-MAJOR ORDERING) The column ordered matrices are assembled from the right to the left (Equation ). Example. For Figure, transformation matrices are as follows: (a) M = T R, (b) M = R T, Example. For example from the lecture, where you were supposed to do at first scaling, then rotation about the X-axis, then about the Y-axis, then about the Z-axis, and at last do translation, transformation matrix is as follows M = T R Z R Y R X S. View Transform The purpose of the view transform is to place the camera at the origin and aim it, to make it look in the direction of the negative Z-axis, with the Y-axis pointing upwards and the X-axis pointing to the right (XYZ is a right-handed coordinate system). The actual position and direction after the view transform has been applied is dependent on the underlying (Application Program Interface) API. An example of the way in which the view transfrom affects the camera and the models are shown in Figure. Y Camera direction Camera direction Y View transform X View frustum X Camera position Z Z Camera position Figure : In the left illustration, the camera is located and oriented as the user wants it to be. The view transform relocates the camera at the origin looking along the negative Z-axis, as shown on the right. From the picture it should be more clear, that in order to relocate the camera we need to undo its previous transformations: first rotation around the origin then translation to desired position (viewer position). In OpenGL it means to apply the matrix M = (T R). These operations result in the following matrix: Mview GL = right x up x viewdir x viewpos x right y up y viewdir y viewpos y right z up z viewdir z viewpos z, () where viewpos is the viewer position, up is up camera vector, viewdir is the camera view direction, right is the right vector. Initial up camera vector can be non-perpendicular to the view direction, in this case we need recompute it. The enumarated vectors are calculated as follows: viewdir = viewpos lookatpos, right = Normalize(up viewdir), up = Normalize(viewDir right) where lookatpos, position at which the camera is looking. Due to the fact, that rotation matrix is orthonormal matrix, the inverse of this matrix is equal to transpose matrix. Translation matrix simply gets inversed signs in the forth column. Therefore view matrix is M view = (R T T ):
3 .3 Projection 3.3 Projection Mview GL = right x right y right z dot(viewpos,right) up x up y up z dot(viewpos,up ) viewdir x viewdir y viewdir z dot(viewpos,viewdir), (3) Before one can actually render a scene, all relevant objects in the scene must be projected onto some kind of plane or into some kind of simple volume. After that clipping and rendering are preformed. In the clipping step objects, which are further than far or closer than near planes are clipped off. Those objects are not rendered. Note, that changing of near parameter does not affect the position of objects in the scene, only objects appearance changes (the parts of the object which are further than far or closer than near planes are simply cut off). Orthographic projection With an orthographic projection, the viewing volume is a rectangular parallelepiped, or more informally, a box (see Figure 3a). The size of the viewing volume doesn t change from one end to the other, so distance from the camera doesn t affect how large an object appears. top toward the vewport left view direction left top w frustum a) bottom right near far b) fov near far near(=view) plane bottom h top Figure 3: Viewing volumes: a) orthographic viewing volume; b) perspective viewing volume. With no other transformations, the direction of projection is parallel to the z-axis, and the viewpoint faces toward the negative z-axis. Note that this means that the values passed in for far and near are used as negative z values if these planes are in front of the viewpoint, and positive if they re behind the viewpoint. The orthographic transform is shown in Equation 4: Port GL = right+le ft top+bottom f ar+near Perspective projection The most unmistakable characteristic of perspective projection (see Figure 3b) is foreshortening: the farther an object is from the camera, the smaller it appears in the final image. This occurs because the viewing volume for a perspective projection is a frustum of a pyramid (a truncated pyramid whose top has been cut off by a plane parallel to its base). Objects that fall within the viewing volume are projected toward the apex of the pyramid, where the camera or viewpoint is. Objects that are closer to the viewpoint appear larger because they occupy a proportionally larger amount of the viewing volume than those that are farther away, in the larger part of the frustum. The perspective transform is shown in Equation 5: Pperspec GL = near near right+le ft top+bottom ( f ar+near) where parameters right, left, top, and bottom are computed as follows: f arnear (4), (5) aspect = width viewport /height viewport top = tan( f ov/) near le ft = bottom aspect bottom = top right = top aspect
4 4 DIRECTX - ROW VECTORS (ROW-MAJOR ORDERING) Note, that usually you preffer to keep the fov-angle unchanged, whenever viewport is resized/reshaped or near/far is modified. In the case of viewport resizing/reshaping, aspect ratio is recomputed, left and right parameters and correspondingly persective transform matrix are recomputed. If near/far are modified, all the parameters excluding aspect ratio are recomputed. After the projection transformation coordinates are clipped: all the objects or parts of the objects with coordinates which do not fit into the viewing volume are cut off and do not proceed further into the pipeline..4 Last Transformations After model-view and projection transformations were applied, object coordinates are still 3D, must be translated into D and rasterized to get the image on the screen. Perspective division Just after the transformation projection we get all the objects projected into the viewing volume. Note, that received coordinates x, y, z, and w are not normalized. In order to provide normalization, we simply divide all coordinates by w: x d y d z d = x c/w c y c /w c z c /w c where x c, y c, z c and w c define the clipped coordinates. Viewport transformation At this step we need to map the normalized viewing volume to the viewport (see Figure 4). Note, that x and y are now normalized and we must translate them to viewport coordinates in pixels. viewing volume, h viewport screen Ox,Oy py, px w pixels Figure 4: Viewport transformation. Moreover, for some API, viewport coordinate system is different: origin is placed in the left upper corner and Y -axes is pointing down. The viewport transformation is determined by the viewport s width P x, height P y, and its center (O x, O y ) in pixels. The vertex s window coordinates are given by: x w y w z w = (p x/) x d + O x (p y /) y d + O y [( f ard neard)/] z d + (neard + f ard)/ (6) where fard, neard lie within [,] and specify the range of valid depth values in the depth buffer. After the viewport transformaion z w coordinate can be written into the depth buffer. Rasterization After all the transformations are done we have D-image. The viewport-coordinates (they are still floats) of the objects in this image pass as input to a rasterization algorithm. This algorithm produces discrete(pixel) version of all the objects in the image. DirectX - row vectors (row-major ordering) DirectX uses row vectors and matrices are written in a row-major order. As a result, matrices are concatenated in left-to-right order, with the first operation occuring on the left, the second operation to the right, the next operation to the right, and so on.
5 . World (Model) Transformation 5 [ x,y,z, World. World (Model) Transformation View Pro ject Model transformation matrices for DirectX are the same as for OpenGL, but transposed: Transformation matrices ] = [ x,y,z, ] (7) T DX = Translation Scale Rotation X (φ) t x t y t z Rz DX = Rotation Z (φ) S DX = cos(φ) sin(φ) sin(φ) cos(φ) s x s y s z Ry DX = Rx DX = Rotation Y (φ) cos(φ) sin(φ) sin(φ) cos(φ) cos(φ) sin(φ) sin(φ) cos(φ) Complex transformations, ordering of matrices The complex transformations for DirectX also can be split into several basic ones: scalings, rotations, and translations. The basic transformation matrices are assembled together by multiplicaton, and the order of multiplication matters (Figure ). But multiplication is now from the left to the right (Equation 7). Example. For Figure, transformation matrices are as follows: (a) M = R T, (b) M = T R, Example. For example from the lecture, where you were supposed to produce at first scaling, then rotation about the X-axis, then about the Y-axis, then about the Z-axis, and at last do translation, transformation matrix is as follows M = S R X R Y R Z T. View Transform The view transformation for DirectX is similar to OpenGL and there are several way to represent it. One way is to translate the viewer position and rotate the corresponding basis (right, up, view direction) around the world origin. Another way is to use DirectX function: D3DXMatrixLookAtLH, which computes the following view matrix: Mview DX = right x up x viewdir x right y up y viewdir y right z up z viewdir z dot(right,viewpos) dot(up,viewpos) dot(viewdir,viewpos), (8) where viewpos, up, viewdir, and right have the same meaning as for Equation and are computed as follows: viewdir = lookatpos viewpos, right = Normalize(up viewdir), up = Normalize(viewDir right)
6 6 REFERENCES.3 Projection DirectX projection matrices are also similar to OpenGL projection matrices. Orthographic projection The orthographic projection is presented as follows: Port DX = near near f ar Perspective projection The perspective transform is shown in Equation : right le ft top bottom Pperspec DX = f ar, () f arnear where parameters right, left, top, and bottom have the same meaning as for OpenGL perspective projection matrix (Equation 5)..4 Last Transformations The last transformations for DirectX are the same as for OpenGL, but viewport coordinate system is different: origin is in the left upper corner and Y -axis is pointing down. This only swaps the y -component in the viewport transformation: x w y w z w = (p x/) x d + O x (p y /) (. y d ) + O y [( f ard neard)/] z d + (neard + f ard)/ (9) () The meaning of parameters P x, P y, O x, O y, fard, and neard is the same as for the OpenGL viewport transformation matrix (Equation 6). References [] Moeller, T., Haines, E.: Real-Time Rendering, 999. AK Peters Ltd [] Segal, M., Akeley, K.: The OpenGL Graphics System: A Specification (Version.5), 3. Silicon Graphics [3] Neider, J., Davis, T.: OpenGL Programming Guide, nd edition, 997. Addison-Wesley Publishing Company [4] Gray, K.: Microsoft DirectX 9 Programmable Graphics Pipeline, 3. Microsoft Corporation [5] Microsoft Visual Studio Help 3
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