CG-lecture03

CG-lecture03 - Computer Graphics Lecture 5 Angel:...

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Unformatted text preview: Computer Graphics Lecture 5 Angel: Interactive Computer Graphics Camera Setting up 3D Geometric Primitives Pipeline (for direct illumination) Rendering 3D Modeling Transformation Lighting Transform into 3D world coordinate system Illuminate according to lighting and reflectance Viewing Transformation Transform into 3D camera coordinate system Projection Transformation Transform into 2D screen coordinate system Clipping Scan Conversion Image Clip primitives outside camera’s view Draw pixels (includes texturing, hidden surface, ...) A 3D Scene Notice the presence of the camera, the projection plane, and projection the world the coordinate axes Viewing transformations define how to acquire the image Viewing on the projection plane on Viewing Transformations Create a camera-centered view Camera is at origin Camera is looking along negative z-axis Camera’s ‘up’ is aligned with y-axis 2 Basic Steps Align the two coordinate frames by rotation 2 Basic Steps Translate to align origins Creating Camera Coordinate Space Specify a point where the camera is located in world Specify space, the eye point eye Specify a point in world space that we wish to become Specify the center of view, the lookat point lookat Specify a vector in world space that we wish to space point up in camera image, the up vector up Intuitive camera Intuitive movement movement Constructing Viewing Transformation, V Create a vector from eye-point to lookat-point. centerx − eye x look = centery − eye y center − eye z z Normalize and reverse the vector to obtain z axis of the camera Normalize system (Described in world system). system − look n= look Constructing Viewing Transformation, V Construct another important vector from the cross Construct product of the up-vector and the z-axis. product up × n = u This is the x-axis of the camera system (Described in This world system). world Constructing Viewing Transformation, V One more vector to define, and it is the y axis of the One camera system (Described in world system) camera n×u = v Now let’s compose the results Constructing Viewing Transformation, V Therefore, we obtain the rotation component of Therefore, viewing transformation. viewing V Rotate u = v n Note these are row vectors Final Viewing Transformation, V To transform vertices, use this matrix: x ' u x ' y vx z ' = n x 1 0 uy uz vy ny 0 vz nz 0 And you get this: − u ⋅ eye x y − v ⋅ eye − n ⋅ eye z 1 1 3D Geometric Primitives Pipeline (for direct illumination) Rendering 3D Modeling Transformation Lighting Transform into 3D world coordinate system Illuminate according to lighting and reflectance Viewing Transformation Transform into 3D camera coordinate system Projection Transformation Transform into 2D screen coordinate system Clipping Scan Conversion Image Clip primitives outside camera’s view Draw pixels (includes texturing, hidden surface, ...) Taxonomy of Projections FVFHP Figure 6.10 Taxonomy of Projections Parallel Projection Center of projection is at infinity • Direction of projection (DOP) same for all points View Plane DOP Angel Figure 5.4 Orthographic Projections DOP perpendicular to view plane Front Front Angel Figure 5.5 Top Side Orthographic Projection Simple Orthographic Transformation Perspective Transformation First discovered by Donatello, Brunelleschi, and DaVinci during First Renaissance Renaissance Objects closer to viewer look larger Parallel lines appear to converge to single point (Vanish point) What is perspective? Perspective works by representing the light that passes Perspective from a scene, through an imaginary rectangle (the painting), to the viewer's eye. 1) Objects are drawn smaller as their distance from the observer increases. observer 2) Spatial foreshortening, which is the distortion of 2) items when viewed at an angle. items One-point perspective One vanishing point is One typically used for roads, railroad tracks, or buildings viewed so that the front is directly facing the viewer. Two-point perspective Two-point perspective can be used to draw the same objects as onesame point perspective, point rotated: looking at the corner of a house, or looking at two forked roads shrink into the distance, for example. Three-point perspective Three-point perspective is usually used for buildings seen from above. Perspective Projection How many vanishing points? 3-Point Perspective 2-Point Perspective 1-Point Perspective Angel Figure 5.10 Perspective vs. Parallel Perspective projection + Size varies inversely with distance - looks realistic – Distance and angles are not (in general) preserved – Parallel lines do not (in general) remain parallel Parallel projection + Good for exact measurements + Parallel lines remain parallel – Angles are not (in general) preserved – Less realistic looking Less Projection Matrix We talked about geometric transforms, focusing on We modeling transforms modeling • Ex: translation, rotation, scale, gluLookAt() Ex: gluLookAt() • These are encapsulated in the OpenGL modelview matrix These modelview Projection is also represented as a matrix Next few slides: representing orthographic and Next perspective projection with the projection matrix projection Perspective Projection In the real world, objects exhibit perspective In foreshortening: distant objects appear foreshortening distant smaller smaller The basic situation: Perspective Projection When we do 3-D graphics, we think of the When screen as a 2-D window onto the 3-D world: screen How tall should this bunny be? Perspective Projection The geometry of the situation is that of similar triangles. The similar View from above: View View plane X x’ = ? (0,0,0) What is x’ ? P (x, y, z) d Z Perspective Projection Desired result for a point [x, y, z, 1]T projected onto the Desired view plane: view x' x =, d z d ⋅x x x' = = , z zd y' y = d z d⋅y y y' = = , z zd What could a matrix look like to do this? z' = d Reminder: Homogeneous Coords What effect does the following matrix have? x ' 1 y ' 0 = z ' 0 w' 0 0 1 0 0 0 x y 0 0 z 0 10w 0 0 1 Conceptually, the fourth coordinate w is a bit like a scale Conceptually, factor factor A Perspective Projection Matrix Answer: 1 0 Mperspective = 0 0 00 1 0 01 0 1d 0 0 0 0 A Perspective Projection Matrix Example: x 1 y 0 = z 0 z d 0 00 1 0 01 0 1d Or, in 3-D coordinates: 0 x y 0 0 z 01 x z d , y , zd d A Perspective Projection Matrix Consider the aspect ratio difference between retinal Consider plane and screen (or window region): plane 1 x ' aspect y 0 z' = 0 w' 0 ' 0 0 1 0 0 1 ' θH 0 − 2tg 2 0 x y 0 0 z 0 w Projection Matrices Now that we can express perspective projection Now as a matrix, we can compose it onto our other matrices with the usual matrix multiplication matrices End result: a single matrix encapsulating End modeling, viewing, and projection transforms modeling, ...
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This note was uploaded on 06/12/2011 for the course ECON 101 taught by Professor Professor during the Spring '10 term at Cisco Junior College.

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