--- title: "Triangle window to viewport mapping" subject: "Computer Graphics" chapter: 3 key: "semester_5/computer_graphics:viewing-transform-triangle-window-viewport" source: https://www.bctengineer.com/q/semester_5/computer_graphics/viewing-transform-triangle-window-viewport site: BCT Engineering --- # Triangle window to viewport mapping Computer Graphics · Chapter 3: Two –Dimensional Transformations - Asked in: 2073 Chaitra, 8 marks - Video: [#32 Lec-32 Solved Numerical on window to view-port mapping](https://www.youtube.com/watch?v=cKYzhL-gQMk) — sks knowledge point - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/computer_graphics/viewing-transform-triangle-window-viewport - Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-3 ## Question A triangle A (15, 20), B (20, 30) and C (30, 20) lies inside a window (10, 10), (40, 50). Find the final image of this triangle after transforming into the viewport (0, 0), (20, 20). Show all transformation steps. ## Short answer After scaling and translating the triangle from window (10,10)-(40,50) to viewport (0,0)-(20,20), the transformed vertices are A'(3.33,5), B'(6.67,10), C'(13.33,5). ## Answer **Given:** window (xwmin, ywmin) = (10,10), (xwmax, ywmax) = (40,50); viewport (xvmin, yvmin) = (0,0), (xvmax, yvmax) = (20,20). Triangle A(15,20), B(20,30), C(30,20) is inside the window. #### Step 1: scale factors ```text sx = (xvmax − xvmin)/(xwmax − xwmin) = 20/30 = 0.6667 sy = (yvmax − yvmin)/(ywmax − ywmin) = 20/40 = 0.5 ``` #### Step 2: transformation steps 1. Translate the window's lower left corner to the origin: T(−10, −10) 2. Scale by (sx, sy) = (2/3, 1/2) about the origin. 3. Translate to the viewport's lower left corner: T(0, 0) (no change here). ```text M = T(xvmin,yvmin) · S(sx,sy) · T(−xwmin,−ywmin) | 1 0 0 | | 2/3 0 0 | | 1 0 −10 | | 2/3 0 −20/3 | = | 0 1 0 | · | 0 1/2 0 | · | 0 1 −10 | = | 0 1/2 −5 | | 0 0 1 | | 0 0 1 | | 0 0 1 | | 0 0 1 | ``` #### Step 3: apply to the vertices xv = (x − 10)·(2/3), yv = (y − 10)·(1/2) | Point | After translation | After scaling | Viewport (final) | |---|---|---|---| | A (15,20) | (5, 10) | (3.33, 5) | (3.33, 5) | | B (20,30) | (10, 20) | (6.67, 10) | (6.67, 10) | | C (30,20) | (20, 10) | (13.33, 5) | (13.33, 5) | **Answer: the image in the viewport is the triangle A'(3.33, 5), B'(6.67, 10), C'(13.33, 5).** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What are sx and sy for window (10,10)-(40,50) and viewport (0,0)-(20,20)? A. sx = 0.5, sy = 0.667 B. sx = 0.667, sy = 0.5 C. sx = 1.5, sy = 2 D. sx = 0.667, sy = 0.667 Answer: B. sx = 20/30 and sy = 20/40. (Key point: Scale factors) 2. What is the correct order of the viewing transformation? A. Scale about the origin, then translate window to origin B. Translate window corner to origin, scale, translate to viewport corner C. Translate viewport to origin, scale by window size D. Rotate, scale, translate Answer: B. The window's lower left corner goes to the origin, then scaling, then translation to the viewport corner. (Key point: Transformation steps) 3. What is the viewport image of B(20,30)? A. (6.67, 10) B. (13.33, 5) C. (10, 20) D. (6.67, 20) Answer: A. xv = (20-10)(2/3) = 6.67 and yv = (30-10)(1/2) = 10. 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