Computer Graphics · Unit 4
2D Geometric Transformations
Exam-focused notes for 2D Geometric Transformations (Computer Graphics, BIT304): what the TU syllabus asks and how it has actually been tested, with 6 solved past questions from this unit.
What this unit covers
- Translation transformation
- Rotation transformation and matrices
- Scaling transformation and matrices
- Shearing transformation
- Homogeneous coordinates
- Composite transformations
- Successive rotations and angle addition
Scaling transformation and matrices
What is scaling in 3D graphics? Derive the scaling matrix for scaling along the X, Y, and Z axes. Apply the scaling transformation to a point (2,3,4) with scaling factors $S_x = 2$, $S_y = 3$, $S_z = 4$. [10]
- Point: $P = (2, 3, 4)$ - Scaling factors: $Sx = 2$, $Sy = 3$, $Sz = 4$ Scaling is a geometric transformation that alters the size of an object by multiplying each coordinate of its points by scaling factors along the coordinate axes. If all factors are eq...
Full solved answer →Translation transformation
Explain the concept of translation transformation in 2D graphics. Discuss the effect of translation on objects. [5]
Translation is a geometric transformation that moves every point of an object by a fixed distance in a specified direction. It shifts an object from one position to another without changing its shape, size, or orientation. For a point P(x, y), translation b...
Full solved answer →Rotation transformation and matrices
List the rotation matrix in clockwise direction with respect to x-axis, y-axis and z-axis. Rotate the object (0, 0, 0), (2, 3, 0), (5, 4, 0) about the rotation axis y = 4.[10]
Object vertices (2D points in xy-plane, z = 0): - $P1 = (0, 0, 0)$ - $P2 = (2, 3, 0)$ - $P3 = (5, 4, 0)$ Rotation axis: the line $y = 4$ Rotation angle: NOT specified in the question. Note on missing data: The angle of rotation is not given. Since the point...
Full solved answer →Shearing transformation
Reflect a line segment having end points (9,3) and (12,10) about a line X = 7. Draw initial and final result graph as well. [5]
- Endpoint A = $(9, 3)$ - Endpoint B = $(12, 10)$ - Line of reflection: $x = 7$ (vertical line, $k = 7$) For any point $(x, y)$ reflected about the vertical line $x = k$: $$(x, y) \rightarrow (2k - x,\ y)$$ The $y$-coordinate is unchanged; the $x$-coordinat...
Full solved answer →Write short notes on: a. Shearing and Scaling b. Line Clipping [5]
Scaling is a transformation that changes the size of an object by multiplying coordinates by scale factors. Transformation equations: - x' = sx · x - y' = sy · y Where sx and sy are scale factors for x and y directions respectively. Matrix form: Properties:...
Full solved answer →Successive rotations and angle addition
Prove that successive rotation is equal to addition of angles. [5]
Theorem: When two rotations are applied successively about the same point, the result is equivalent to a single rotation by an angle equal to the sum of the individual rotation angles. --- Let us consider a point P in a 2D plane and apply two successive rot...
Full solved answer →Make Unit 4 stick
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