--- title: "Rotation about a pivot point" subject: "Computer Graphics" chapter: 3 key: "semester_5/computer_graphics:rotation-about-pivot-point-derive" source: https://www.bctengineer.com/q/semester_5/computer_graphics/rotation-about-pivot-point-derive site: BCT Engineering --- # Rotation about a pivot point Computer Graphics · Chapter 3: Two –Dimensional Transformations - Asked in: 2070 Asar, 8 marks; 2078 Bhadra, 6 marks (asked 2 times) - Video: [Computer Graphics:-Composite Transformations:- General 2D Pivot Point Rotation](https://www.youtube.com/watch?v=TkFjM4u2FeM) — Priyanka Pujari - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/computer_graphics/rotation-about-pivot-point-derive - Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-3 ## Question Derive the 2D transformation matrix for rotating an object by angle θ about a pivot point (xr, yr) (composite matrix for rotation about an arbitrary point, clockwise). ## Short answer Rotation about pivot (xr,yr) is M=T(xr,yr)*R(-theta)*T(-xr,-yr) with R(-theta)=[[cos theta, sin theta,0],[-sin theta, cos theta,0],[0,0,1]]. Thus x'=xr+(x-xr)cos theta+(y-yr)sin theta, y'=yr-(x-xr)sin theta+(y-yr)cos theta; for counter-clockwise replace theta by -theta. ## Answer To rotate an object about a pivot point (xr, yr), the pivot is first moved to the origin, the object is rotated about the origin, and then the pivot is moved back. This is a composite transformation of three basic transformations. #### Basic matrices (homogeneous, column vectors) For **clockwise** rotation by θ, the angle is −θ in the standard (counter-clockwise positive) convention: ```text Rotation about origin, clockwise θ: R(−θ) = | cosθ sinθ 0 | | −sinθ cosθ 0 | | 0 0 1 | Translation T(tx, ty) = | 1 0 tx | | 0 1 ty | | 0 0 1 | ``` #### Steps 1. Translate the pivot to the origin: T(−xr, −yr) 2. Rotate about the origin: R(−θ) 3. Translate back: T(xr, yr) ```text M = T(xr, yr) · R(−θ) · T(−xr, −yr) | 1 0 xr | | cosθ sinθ 0 | | 1 0 −xr | = | 0 1 yr | · | −sinθ cosθ 0 | · | 0 1 −yr | | 0 0 1 | | 0 0 1 | | 0 0 1 | ``` Multiplying the last two matrices first: ```text | cosθ sinθ −xr·cosθ − yr·sinθ | | −sinθ cosθ xr·sinθ − yr·cosθ | | 0 0 1 | ``` Then adding the translation by (xr, yr): ```text | cosθ sinθ xr(1 − cosθ) − yr·sinθ | M = | −sinθ cosθ yr(1 − cosθ) + xr·sinθ | | 0 0 1 | ``` So a point P(x, y) goes to x' = xr + (x − xr)cosθ + (y − yr)sinθ y' = yr − (x − xr)sinθ + (y − yr)cosθ **Check:** for (x, y) = (xr, yr): x' = xr, y' = yr, so the pivot does not move. For counter-clockwise rotation, replace θ by −θ (the sign of the sinθ terms changes). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the correct order of operations for rotation about a pivot (xr, yr)? A. Rotate, translate pivot to origin, translate back B. Translate pivot to origin, rotate about origin, translate back C. Scale, rotate, translate D. Translate by (xr, yr), rotate, translate by (xr, yr) again Answer: B. The pivot is moved to the origin, the rotation is done, then the pivot is moved back. (Key point: Composite steps) 2. For clockwise rotation by theta, which matrix is used for the rotation step? A. Matrix with cosθ, -sinθ in the first row (counter-clockwise) B. Matrix with cosθ, sinθ in the first row and -sinθ, cosθ in the second C. Matrix with -cosθ on the diagonal D. Matrix with sinθ on the diagonal Answer: B. Clockwise by theta is R(-theta), which has cosθ, sinθ in the first row and -sinθ, cosθ in the second. (Key point: Clockwise rotation) 3. How is the composite matrix written with column vectors? A. M = T(-xr,-yr) · R(-θ) · T(xr,yr) B. M = R(-θ) · T(xr,yr) · T(-xr,-yr) C. M = T(xr,yr) · R(-θ) · T(-xr,-yr) D. M = T(xr,yr) · T(-xr,-yr) · R(-θ) Answer: C. With column vectors the first operation is the rightmost matrix, so T(-xr,-yr) is on the right. (Key point: Matrix product order) 4. How can you check the final matrix? A. The origin maps to itself B. The matrix has determinant 0 C. All points move by (xr, yr) D. The pivot point maps to itself Answer: D. Substituting (xr, yr) gives x' = xr and y' = yr, so the pivot stays fixed. 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