--- title: "2D viewing pipeline with matrices" subject: "Computer Graphics" chapter: 3 key: "semester_5/computer_graphics:viewing-pipeline-2d-matrices" source: https://www.bctengineer.com/q/semester_5/computer_graphics/viewing-pipeline-2d-matrices site: BCT Engineering --- # 2D viewing pipeline with matrices Computer Graphics · Chapter 3: Two –Dimensional Transformations - Asked in: 2073 Shrawan, 5 marks; 2076 Chaitra, 8 marks; 2068 Bhadra (old course), 10 marks; 2066 Magh (old course); 2069 Chaitra, 4 marks (asked 5 times) - Video: [Graphics | 9 | Two Dimensional Viewing Pipeline](https://www.youtube.com/watch?v=kasPoa9x0ko) — Dr. Mohammed El-Said - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/computer_graphics/viewing-pipeline-2d-matrices - Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-3 ## Question Explain the two-dimensional viewing pipeline (world to screen / window-to-viewport transformation): describe the different steps with matrix representation at each step. ## Short answer The 2D viewing pipeline maps world coordinates to device coordinates: modelling transformation places objects in world coords; viewing transformation aligns the window using R(-theta)T(-xw,-yw); clipping removes outside; window-to-viewport maps via M=T(xvmin,yvmin) S(sx,sy) T(-xwmin,-ywmin), with sx=(xvmax-xvmin)/(xwmax-xwmin), sy=(yvmax-yvmin)/(ywmax-ywmin); then device coords. ## Answer The **two-dimensional viewing pipeline** is the sequence of transformations that converts a 2D scene described in world coordinates into the pixels of a viewport on the display device. #### Steps of the 2D viewing pipeline ```text Modelling Modelling World Viewing coordinates -> transform -> coordinates -> transform | Device <- Window-to-viewport <- Clipping <- Viewing coordinates mapping (window) coordinates ``` 1. **Modelling transformation:** each object is defined in its own modelling coordinates and is placed into the common world coordinate system (WC) by translation, rotation and scaling. 2. **World to viewing coordinates (viewing transformation):** a window (the part of the world to be displayed) is defined in WC. If the window is rotated by an angle θ and has reference point (xw, yw), the window is aligned with the axes: M = R(−θ) · T(−xw, −yw). 3. **Clipping:** everything outside the window is removed (point, line, polygon clipping). 4. **Window-to-viewport mapping:** the window is mapped to the viewport (the area of the screen on which the image appears), normally via normalised coordinates 0 to 1. 5. **Device coordinates:** the viewport coordinates are given to the display device. #### Matrices ```text Translate T(−xwmin, −ywmin): | 1 0 −xwmin | | 0 1 −ywmin | | 0 0 1 | Scale S(sx, sy): | sx 0 0 | sx = (xvmax−xvmin)/(xwmax−xwmin) | 0 sy 0 | sy = (yvmax−yvmin)/(ywmax−ywmin) | 0 0 1 | Translate T(xvmin, yvmin): | 1 0 xvmin | | 0 1 yvmin | | 0 0 1 | Composite window-to-viewport: M = T(xvmin,yvmin) · S(sx,sy) · T(−xwmin,−ywmin) | sx 0 xvmin − sx·xwmin | M = | 0 sy yvmin − sy·ywmin | | 0 0 1 | Window-to-viewport equations: xv = xvmin + (xw − xwmin)·sx yv = yvmin + (yw − ywmin)·sy ``` If sx = sy, the shape is preserved; otherwise the picture is stretched or compressed. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The correct order of the 2D viewing pipeline is: A. Modelling → World → Viewing → Clipping → Window-to-viewport → Device B. World → Modelling → Clipping → Viewing → Device C. Device → Viewport → Window → World D. Modelling → Viewing → Device → Clipping Answer: A. Objects are placed in world coordinates, converted to viewing coordinates, clipped, then mapped to the viewport and device. (Key point: Order) 2. The window-to-viewport matrix is composed as: A. M = T(−xwmin,−ywmin) · S(sx,sy) · T(xvmin,yvmin) B. M = T(xvmin,yvmin) · S(sx,sy) · T(−xwmin,−ywmin) C. M = S(sx,sy) · T(xvmin,yvmin) D. M = R(θ) · S(sx,sy) Answer: B. Translate the window corner to the origin, scale, then translate to the viewport corner. (Key point: Composite matrix) 3. The scale factor sx is: A. (xwmax − xwmin)/(xvmax − xvmin) B. xvmax/xwmax C. (xvmax − xvmin)/(xwmax − xwmin) D. (xvmin − xwmin) Answer: C. Viewport width divided by window width. (Key point: Scale factor) 4. What happens when sx ≠ sy? A. The picture is rotated B. The picture is clipped C. The shape is preserved D. The picture is stretched or compressed Answer: D. Shape is preserved only when sx = sy. (Key point: Aspect ratio) ## Related in Computer Graphics - [3D world to viewing coordinate matrix](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aworld-to-viewing-3d-matrix) — Computer Graphics, chapter 4, asked 2× - [Short note: 2D viewing pipeline](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Ashort-note-2d-viewing-pipeline) — Computer Graphics, chapter 3, asked 2× - [3D window to viewport transformation](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Awindow-viewport-3d-matrix) — Computer Graphics, chapter 4 - [Rectangle window to viewport mapping](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aviewing-transform-rectangle-window-viewport) — Computer Graphics, chapter 3 - [Triangle window to viewport mapping](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aviewing-transform-triangle-window-viewport) — Computer Graphics, chapter 3 - [2D reflection matrices and line y+x=0](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Areflection-matrices-axes-y-plus-x) — Computer Graphics, chapter 3 ## Related in other subjects - [Steps for free body diagram; equilibrium 2D/3D](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Afbd-steps-equilibrium-2d-3d) — Applied Mechanics, chapter 2, asked 2× - [Stiffness matrix: element matrices, member forces, BMD](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Astiffness-element-matrices-600kn-bmd) — Theory of Structures II, chapter 3 - [Bar stiffness matrix and 2D truss element](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Abar-stiffness-truss-rotation) — Computational Techniques in Civil Engineering, chapter 4 - [Add two matrices](https://www.bctengineer.com/question.md?key=semester_1%2Fcomputer_programming%3Amatrix-add-two) — Computer Programming, chapter 7