Đồ họa thiết kế - Bài 4 Các phép biến đổi Đồ hoạ Transformations

ùng nhìn thấy của Camera 60 Viewing Coordinates z Có thể sử dụng mô hình hệ thống tọa độ quan sát trên các thông số gồm: – Điểm nhìn (VRP View Reference Point ): điểm gốc quan sát. Vị trí của Camera) – Vector mặt phẳng quan sát (VPN View Plane Normal vector ): viewing direction) – Vector hướng quan sát (VUP View UP vector ): establishes orientation of “camera” ),,( 321 nnn== N Nn ),,( 321 uuu=× ×= NV NVu ),,( 321 vvv=×= unv Three mutually orthogonal basis vectors: Khoa CNTT - DDHBK Hà nội hunglt@it-hut.edu.vn 8682595 11 61 61 Ví dụ: Viewing Transformation (i) simple cases ( always view along negative z-axis ) (0, 0, z) Điểm nhìn trên trục Z zW -axis zW xW yW n MWC,VC = T (0, 0, z,) (a, b, c) Nhìn ở vị trí (a, b, c) theo trục Z và ngược hướng zW-axis MWC,VC = T (-a, -b, -c ) zW xW yW 62 62 .and , , 3 2 1 3 2 1 3 2 1 ⎟⎟ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎝ ⎛ = ⎟⎟ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎝ ⎛ = ⎟⎟ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎝ ⎛ = n n n v v v u u u nvu Ma trận biến đổi tọa độ thế giới sang tọa độ quan sát : MWC,VC = R T (xw, yw, zw) and (u, v, n) 2 hệ tọa độ với cùng 1 gốc. Vector đơn vị (u, v, n) được định nghĩa theo WCS 63 3D Viewing Transformation z Với điểm quan sát VRPoint (x0, y0, z0) the Viewing Transformation can be defined by composition of the following two transformations: ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ − − − = 1000 100 010 001 0 0 0 z y x T ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ = 1000 0 0 0 321 321 321 nnn vvv uuu R TRM •=VCWC , This is used to determine the location of each vertex in the camera co-ordinate system xx vcwc rr ,M=′ 64 Viewing Transformation z We might also achieve this in terms of the following transformation sequence: – Translate the view reference point to the origin of the world co- ordinate system – Apply Rotations to align xv, yv and zv, axes, respectively (x0, y0, z0) ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ − − − 1000 100 010 001 0 0 0 z y x 65 65 where , 1000 0 0 0 321 321 321 ⎟⎟ ⎟⎟ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜⎜ ⎝ ⎛ = nnn vvv uuu R Verification: etc , ? = ⎥⎥ ⎥⎥ ⎦ ⎤ ⎢⎢ ⎢⎢ ⎣ ⎡ = 1 3 2 1 u u u R uR pV = MWC,VC pW xW zW yW pM pW pV yV xV -zV 66 • Recall we’ve re-expressed our world geometry in eye’s frame reference. • To do projection, we use a synthetic camera by introducing a viewing frustum after the viewing transformation. 66 xW zW yW yM xMzM pM yV xV -zV pV pW xW zW yW pM pW Khoa CNTT - DDHBK Hà nội hunglt@it-hut.edu.vn 8682595 12 67 Camera trong OpenGL z Các hàm cơ bản – glOrtho() and gluLookAt() – these created parallel projections z We now want to modify a scene to give us perspective projections. – lines converge as they get further away 68 68 Demo : Viewing from Nate Robins (Utah) 69 The Camera z Khởi tạo Camera với phép chiếu phối cảnh – Thay cho glOrtho() bằng: – gluPerspective() z gluPerspective – viewAngle – aspectRatio (W/H) – near plane – far plane 70 The Camera z The Perspective Camera glMatrixMode(GL_PROJECTION); glLoadIdentity(); gluPerspective(viewAngle, aspectRatio, N, F); 71 The Camera z Positioning the Camera – The camera is positioned using a combination of translations and rotations. – Think if the camera being in the same location as the viewers eye. 72 The Camera eye(x,y,z) lookat(x,y,z) up(x,y,z) Khoa CNTT - DDHBK Hà nội hunglt@it-hut.edu.vn 8682595 13 73 The Camera glMatrixMode(GL_MODELVIEW); glLoadIdentity(); gluLookAt(eye.x, eye.y, eye.z, look.x, look.y, look.z, up.x, up.y, up.z); 74 The Camera z The camera can have an arbitrary orientation and position. z Therefore we can think of the camera having its own axes. 75 Camera Movement z A camera has six degrees of freedom: 1. slid in 3 directions; and 2. rotated in 3 directions z Camera Movement – The camera can move along its axes. – This is called sliding the camera. 76 The Camera z Camera Movement – To move the camera along the u axis you would simply perform z eye = eye + Du – To move slide the camera in any direction: eye.x = dU*u.x + dV*v.x + dN*n.x eye.y = dU*u.y + dV*v.y + dN*n.y eye.z = dU*u.z + dV*v.z + dN*n.z z Camera Movement – Besides physically moving the camera to another location – the camera can be tilted in different directions to look at different parts of the scene. u v n 77 The Camera z Camera Movement – Besides physically moving the camera to another location – the camera can be tilted in different directions to look at different parts of the scene. 78 The Camera - Camera Movement We use a plane analogy to describe the cameras movement. z a rotation from the horizontal along the length is called PITCH z a rotation from the horizontal along the width is called ROLL z a rotation around the vertical is called YAW Khoa CNTT - DDHBK Hà nội hunglt@it-hut.edu.vn 8682595 14 79 The Camera z Pitch v’ = cos(θ)v – sin(θ)n n’ = sin(θ)n + cos(θ)v n v 80 The Camera z Roll u’ = cos(θ)u + sin(θ)v v’ = cos(θ)v - sin(θ)u u v 81 The Camera z Yaw n’ = cos(θ)n - sin(θ)u u’ = sin(θ)n + cos(θ)u u n 82 A 3D Scene z Notice the presence of the camera, the projection plane, and the world coordinate axes z Viewing transformations define how to acquire the image on the projection plane 83 Viewing Transformations z Create a camera-centered view z Camera is at origin z Camera is looking along negative z-axis z Camera’s ‘up’ is aligned with y-axis 84 2 Basic Steps z Align the two coordinate frames by rotation Khoa CNTT - DDHBK Hà nội hunglt@it-hut.edu.vn 8682595 15 85 2 Basic Steps z Translate to align origins 86 Creating Camera Coordinate Space z Specify a point where the camera is located in world space, the eye point z Specify a point in world space that we wish to become the center of view, the lookat point z Specify a vector in world space that we wish to point up in camera image, the up vector z Intuitive camera movement 87 Constructing Viewing Transformation, V z Create a vector from eye-point to lookat-point z Normalize the vector z Desired rotation matrix should map this vector to [0, 0, -1]T Why? 88 Constructing Viewing Transformation, V z Construct another important vector from the cross product of the lookat-vector and the vup- vector z This vector, when normalized, should align with [1, 0, 0]T Why? 89 Constructing Viewing Transformation, V z One more vector to define z This vector, when normalized, should align with [0, 1, 0]T z Now let’s compose the results 90 Compositing Vectors to Form V z We know the three world axis vectors (x, y, z) z We know the three camera axis vectors (r, u, l) z Viewing transformation, V, must convert from world to camera coordinate systems Khoa CNTT - DDHBK Hà nội hunglt@it-hut.edu.vn 8682595 16 91 Compositing Vectors to Form V z Remember – Each camera axis vector is unit length. – Each camera axis vector is perpendicular to others z Camera matrix is orthogonal and normalized – Orthonormal z Therefore, M-1 = MT 92 Compositing Vectors to Form V z Therefore, rotation component of viewing transformation is just transpose of computed vectors 93 Compositing Vectors to Form V z Translation component too z Multiply it through 94 Final Viewing Transformation, V z To transform vertices, use this matrix: z And you get this: