Assignment 4

# Assignment 4 - Assignment 4(Non-Programming Programming...

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1 Assignment 4 (Non-Programming & Programming) Worth: 10 points Due Date and Time: Before class begins at 12:20 pm on 06/28/2007. Copy zipped programming submission to CS250 web page. Place your stapled non-programming submission on instructor’s desk in Michelangelo. Topics Introduction to Graphics Pipe Transforms: Scaling, Rotation, and Translation View Transform Perspective Transform Viewport Transform First Person Camera Hidden Surface Removal using Back-Face Removal & Depth Buffer Triangle Rasterization with Linear (Depth and Color) Interpolation Non-Programming Statement 1. Let r q p C F ˆ , ˆ , ˆ , = be the notation used to describe an orthonormal right-handed reference frame where r q p C ˆ , ˆ , ˆ , are the origin, X axis (oriented right), Y axis (oriented into paper), and Z axis (oriented up), respectively. The implicit world reference frame is defined as k j i O F W ˆ , ˆ , ˆ , = where ( ) 0 , 0 , 0 = O , ( ) 0 , 0 , 1 ˆ = i , ( ) 0 , 1 , 0 ˆ = j , and ( ) 1 , 0 , 0 ˆ = k . In world frame, a camera is located at position W C with view and up vectors v r and up , respectively. Let w v u C F W V ˆ , ˆ , ˆ , = be the view frame within static world frame W F with origin at W C , X axis u ˆ oriented right, Y axis v ˆ oriented into paper, and Z axis w ˆ oriented up. 1.1. To completely define V F , compute normalized basis vectors u ˆ and w ˆ . Draw a neat picture to illustrate your answer. (2.5 points) 1.2. Next, compute the affine matrix required to transform a point from world frame W F into view frame V F . Draw a neat picture to illustrate your answer. (2.5 points) 1.3. In view frame V F , horizontal field-of-view is o θ , aspect ratio is a , near and far clip planes are n and units from viewpoint along Y axis v ˆ , respectively. Assume near clip plane is also the projection plane. Right-handed NDC frame is defined by the following inequalities: 1 0 x (width), 1 0 z (height), 1 0 y (depth). Starting from first principles, derive the perspective matrix required to transform a view frame point into clip frame. Draw a neat picture to illustrate your answer. (5 points) 1.4. Now, suppose in view frame V F , the bottom-left and top-right coordinates of view window are ( ) b n l , , and ( ) t n r , , , respectively. Let far clip plane be f units from viewpoint along Y axis v ˆ . Right-handed NDC frame is defined by the following inequalities: 1 1 x (width), 1 1 z (height), f y n

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2 (depth). Starting from first principles, derive the homogeneous perspective
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Assignment 4 - Assignment 4(Non-Programming Programming...

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