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Exploring Viewing Transformations in Computer Graphics

Understand how to transform 3D world to 2D surface for graphics display devices, Camera Analogy, OpenGL’s 'look at' point, Viewing and Projection transformations explained.

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Exploring Viewing Transformations in Computer Graphics

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  1. Viewing • Viewing and viewing space (camera space) • World space to viewing space transformation

  2. world coordinates VIEWINGTRANSFORMATION viewing coordinates World to Viewing Coordinates • Graphics display devices are 2D rectangular screens. Hence we need to understand how to transform our 3D world to a 2D surface • Viewing the desired scene is analogous to taking pictures using a camera • We now need to

  3. Camera Analogy • View is described in terms of: • camera position in world coordinate system • camera direction (viewing direction) • camera orientation: usually defined by the up vector • aperture size: field of view

  4. yW P0 N xW V zW yW P0 yW P0 N xW xW zW zW View Coordinate System • specify a view reference point in the world coordinate system. This can be any point along the camera direction, or the camera position itself • specify the view plane normal N - this gives the camera, or Z direction • specify the view-up direction, V - this gives the camera up direction, or Y direction v’ v n v=v’-(v.n) n

  5. V U P0 N Viewing Coordinate System • We can construct a vector U perpendicular to both V and N, and this will correspond to the Xv axis. How? • We can define U as right, V as up, and N as towards the viewer: a right handed system UV=N • We can also define U as right, V as up and N as into the scene: a left handed system VU=N, in which bigger N values mean points are further away • OpenGL is right handed yW xW zW

  6. yW P0 Q xW zW View Coordinate System • Some systems (e.g., OpenGL) allow you to specify a ‘look at’ point, Q, from which N is calculated as the direction to the ‘look at’ point from the view reference point

  7. World to Viewing • Objects must be viewed in the viewing space • This can be done by aligning the view coordinate system with world coordinate system, e.g., view reference point is transformed to world origin, and U, V, N are aligned with X, Y, Z directions through rotations y (x0, y0, z0) x z

  8. V Ux Uy Uz 1 U (Ux, Uy, Uz,1)T = P0 N y x z World to Viewing • Translate view origin to world origin, then align U, V and N axes with X, Y and Z directions by rotation R = Rz. Ry. Rx • rotate around X to bring N into the X-Z plane • rotate around Y to align N with Z • rotate around Z to align V with Y • An easier way to work out the rotation matrix R: • U in world space should be (1,0,0) in view space • V should be (0,1,0) • N should be (0,0,1) • So we have the following equations 1. R*(Ux, Uy, Uz,1)T = (1,0,0,1) 2. R*(Vx, Vy, Vz,1)T = (0,1,0,1) 3. R*(Nx, Ny, Nz,1)T = (0,0,1,1)

  9. World to Viewing Transformation • Remember U.U=1, U.V=0, U.N=0, V.N=0, so if we choose the rotation matrix as • The equations 1,2,3 will be satisfied • The rotation matrix is a “change of basis” matrix • So viewing transformation from world space to viewing space is: M = RT R =

  10. y Pw=(1,1,0) v x u n (0,0,1) z What’s Pw in viewing coordinates? Intuitively, P in viewing coordinate is (-1,1,1), but how do we derive it? 1. Translate view origin to world origin with translation vector (0, 0, -1) 2. Multiply Pw by matrix M below to align viewing axes with world axes -1 0 0 0 M = 0 1 0 0 0 0 -1 1 So Pw in viewing space is: Pv = M T Pw u, v, n in world space: u=(-1,0,0) v=(0,1,0) n=(0,0,-1)

  11. yv xv zv OpenGL Viewing Coordinate System • The default camera is placed at the coordinate origin of world space (U aligned with the X axis, V aligned with Y, and N aligned with Z), looking along the negative z-direction, and the view plane is perpendicular to the viewing direction

  12. Projection • We need to transform from a special viewing coordinate system (camera on z-axis pointing along the axis) into a projection coordinate system viewing coordinates projection coordinates PROJECTIONTRANSFORMATION

  13. P1 P2 P2 view plane Parallel Projection - Two types • In parallel projection, the observer position is at an infinite distance, so the projection lines are parallel • Orthographic parallel projection has view plane perpendicular to direction of projection • Oblique parallel projection has view plane at an oblique angle to direction of projection

  14. yv xv zv Parallel Projection Calculation looking along x-axis P (x,y,z) (xp,yp,d) yV viewing space d o - zV view plane yP = y Similarly, xP = x

  15. Parallel Projection Calculation • So • x = xp • y = yp • z = d • The projection transformation matrix is simply If view plane is xoy plane, Then d=0 x y z 1 xp yp d 1 • 1 0 0 0 • 0 1 0 0 • 0 0 d/z 0 • 0 0 0 1 = projection space projection matrix viewing space

  16. P1 P1’ camera P2 P2’ view plane Perspective Projection • In a perspective projection, object positions are projected onto the view plane along lines which converge at the observer, or Centre of Projection (COP) • Perspective projection gives realistic views, but does not preserve proportions - projections of distant objects are smaller than projections of objects of the same size which are closer to the view plane (fore-shortening)

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