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Interactive Computer Graphics Viewing

Interactive Computer Graphics Viewing. James Gain and Edwin Blake Department of Computer Science University of Cape Town July 2002 jgain@cs.uct.ac.za. Map of the Lecture. Classical Viewing: Orthographic, Axonometric, Oblique, Perspective Camera Positioning and Viewing APIs

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Interactive Computer Graphics Viewing

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  1. Interactive Computer GraphicsViewing James Gain and Edwin Blake Department of Computer ScienceUniversity of Cape Town July 2002jgain@cs.uct.ac.za Collaborative Visual Computing Laboratory

  2. Map of the Lecture • Classical Viewing: • Orthographic, Axonometric, Oblique, Perspective • Camera Positioning and Viewing APIs • Simple Projection Matrices • Implementing Projections in OpenGL • Projections and Shadows Transformations Viewing Shading Interactive Computer Graphics Contents

  3. Viewing Principles • Image formed by intersection of projection plane and lines of projection from object to viewer • Perspective Viewing: • Projectors converge to a finite Centre of Projection (COP) • Objects in the distance are shrunk • Parallel Viewing: • As COP is moved to infinity, projectors become parallel • Projectors oriented along a Direction of Projection (DOP) • Preserves relative lengths and angles Perspective Projection Parallel Projection Interactive Computer Graphics Contents

  4. Classical Viewing: Parallel I • Unlike CG, Classical viewing depends on a specific relationship between object and viewer • Many objects (e.g. buildings) have a box-like shape with 6 principal faces oriented relative to the projection plane • Orthographic: • projection plane is parallel to a single principal face • Preserves both distance and angles • But usually need multiple projections because only on face visible at a time Interactive Computer Graphics Contents

  5. Classical Viewing: Parallel II • Axonometric: • Projectors are orthogonal to projection plane • Angles are no longer preserved • Isometric: three principal faces symmetric with respect to the plane • Dimetric: two prinicipal faces symmetric with respect to plane • Trimetric: general case • Oblique: • Projectors are parallel to each other but make an arbitrary angle with the projection plane • Not physically realistic Interactive Computer Graphics Contents

  6. Classical Viewing: Perspective • Characterized by diminution • Distortion is non-linear: • Unable to take measurements • But more realistic • Categorised according to number of visible faces and consequent number of vanishing points for principal directions • All classical views are simply a subset of either general parallel or perspective CG viewing Three-point Two-Point One-Point Interactive Computer Graphics Contents

  7. Transformation Pipeline Modelling Transform Object in object co-ordinates Object in world co-ordinates Viewing Transform Object in 2D screen co-ordinates Perspective Projection Object in viewing co-ordinates Interactive Computer Graphics Contents

  8. Positioning of the Camera • The viewpoint (camera) is located at the origin and the projection plane is parallel to the plane at a distance along the negative axis • Camera Transformation: • Must be able to position and orient the camera arbitrarily • Transform objects from modelling coordinates to camera coordinates • Need an appropriate way of specifying the camera position and orientation • OpenGL: • Camera has a left-hand coordinate system • Camera transformation is part of the GL_MODELVIEW state Interactive Computer Graphics Contents

  9. u-v-n Viewing • Create a camera frame (point and 3 orthonormal vectors): • Positioned at a view-reference point (VRP) • Viewing pointed along the view- plane normal ( ). Determines the orientation of the viewplane • Viewing oriented by the view-up vector (VUP). Sets the roll of the viewplane • Construct a camera frame • = Projection of VUP onto plane • = Normalizaion of Interactive Computer Graphics Contents

  10. Look-At Viewing • Camera located at an eyepoint (eye) specified in world coordinates, pointed towards an atpoint (at) with a view-up vector (up) • gluLookAt(eyex, eyey, eyez, atx, aty, atz, upi, upj, upk); • Alters the modelview matrix to match this camera transform Interactive Computer Graphics Contents

  11. Euler and Polar Viewing • Other applications require a non-rectilinear coordinate system • Flight Simulation: • Three Euler angles (roll, pitch and yaw) relative to a vehicle’s centre of mass • Stellar Cartography: • Polar coordinates (azimuth, elevation and distance) relative to a ground plane and position Interactive Computer Graphics Contents

  12. Perspective Projection • From 3D world co-ordinates to 2D screen co-ordinates • Perspective projection provides an illusion of depth in an image by giving distant objects a smaller projected screen size • The screen co-ordinates of a point are calculated using similar triangles Interactive Computer Graphics Contents

  13. Geometry of Perspective Projection • Triangle is similar to . • Similarly: • The Projection Transform is: Interactive Computer Graphics Contents

  14. Exercise: Binocular Projection • Calculate the separate screen projections and for the left and right eyes of a point . The eyes are located at and Interactive Computer Graphics Contents

  15. Solution: Binocular Projection • Right eye: • Triangle is similar to . • Left eye: • By a similar derivation • The co-ordinate projection is unchanged because our eyes are only separated horizontally. Interactive Computer Graphics Contents

  16. Orthogonal Projection • Projectors are perpendicular to the viewplane • Points retain their and values: Interactive Computer Graphics Contents

  17. Projections in OpenGL • Need to consider focal length and size of the film plane • View Volume: • Determines the field of view for clipping objects • A truncated pyramid (frustum) with apex at the COP • Specified by front and back clipping planes and a horizontal and vertical angle of view Interactive Computer Graphics Contents

  18. Perspective Viewing in OpenGL • glFrustum(xmin, xmax, ymin, ymax, near, far); • near and far must both be positive. Note: they are specified in camera coordinates, so • gluPerspective(fovy, aspect, near, far); • fovy is the angle between the top and bottom frustum planes, aspect = width / height • Must make sure to first select the GL_PROJECTION matrix mode: • glMatrixMode(GL_PROJECTION); Interactive Computer Graphics Contents

  19. Parallel Viewing in OpenGL • glOrtho(xmin, xmax, ymin, ymax, near, far); • Again • But near and far are not restricted to being positive Interactive Computer Graphics Contents

  20. Hidden Surface Removal • Opaque surfaces cause occlusion • Use Hidden-Surface Removal to delete invisible surfaces • Z-buffer algorithm: • As polygons are rasterized store depth value of projected points. Only render a point with current minimum depth • Worst-case complexity equals number of polygons • Need specialized memory - depth or z buffer • Amenable to hardware implementation z-buffer Algorithm • OpenGL z-buffer: • glutInitDisplayMode(GLUT_DEPTH); initialize z-buffer in display • glEnable(GL_DEPTH_TEST); enable hidden surface removal • glClear(GL_DEPTH_BUFFER_BIT); clear z-buffer before rendering Interactive Computer Graphics Contents

  21. Projection Normalization • Projection Requirements: • Handle general view volumes • Retain depth information (required by HSR) as far down the pipeline as possible • Projection Solution: • Decompose projection into a distortion and canonical orthographic projection • Canonical View Volume glOrtho(-1.0, 1.0, -1.0, 1.0, -1.0, 1.0); Perspective Projection = Distortion and Orthographic projection Interactive Computer Graphics Contents

  22. Orthogonal Projection • glOrtho(xmin, xmax, ymin, ymax, near, far) •  glOrtho(-1.0, 1.0, -1.0, 1.0, -1.0, 1.0) • Translate: • Centre of view volume to origin • Scale: • View volume to cube with sides of length 2 Interactive Computer Graphics Contents

  23. Perspective Normalization • Convert perspective view frustrum to orthographic cube • gluPerspective(90.0, 1.0, zmin, zmax) •  glOrtho(-1.0, 1.0, -1.0, 1.0, -1.0, 1.0) • Near plane at zmin, far plane at zmax, fovy = 90 deg, aspect = square • Mapping is non-linear but preserves the ordering of depth Interactive Computer Graphics Contents

  24. Projection and Shadows • Shadows are useful shape cues • Fully general shadowing algorithms are expensive • But simple shadows (from a point light source onto a flat plane) can be simulated with projection • Set COP to light source, apply perspective transform to polygon in front of projection plane to create shadow polygon • Example: • Light source • Shadow projection onto • Exercise: Write out Interactive Computer Graphics Contents

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