Q1Computer Graphics and Multimedia
Question
(a) Explain 3D Transformations in detail (Translation, Rotation about X, Y, Z axes, Scaling).
(b) Discuss the concept of Projection (Parallel and Perspective) with examples.
Answer
3D Transformations use 4×4 homogeneous matrices for Translation, Rotation, and Scaling; Projection transforms 3D scenes to 2D using parallel or perspective methods.
In 3D graphics, transformations are applied to objects using 4×4 homogeneous transformation matrices. A 3D point P(x,y,z) is represented in homogeneous form as [x, y, z, 1]ᵀ. The transformed point P' = M × P.
Moves an object by vector (tx, ty, tz). Matrix T(tx,ty,tz): [[1,0,0,tx],[0,1,0,ty],[0,0,1,tz],[0,0,0,1]]. Result: x'=x+tx, y'=y+ty, z'=z+tz.
Resizes object by factors (sx, sy, sz). Matrix S(sx,sy,sz): [[sx,0,0,0],[0,sy,0,0],[0,0,sz,0],[0,0,0,1]]. Uniform scaling: sx=sy=sz. For scaling about a fixed point, translate to origin, scale, translate back.
Rx(θ): [[1,0,0,0],[0,cosθ,-sinθ,0],[0,sinθ,cosθ,0],[0,0,0,1]]. Result: x'=x, y'=y·cosθ-z·sinθ, z'=y·sinθ+z·cosθ.
Ry(θ): [[cosθ,0,sinθ,0],[0,1,0,0],[-sinθ,0,cosθ,0],[0,0,0,1]]. Result: x'=x·cosθ+z·sinθ, y'=y, z'=-x·sinθ+z·cosθ.
Rz(θ): [[cosθ,-sinθ,0,0],[sinθ,cosθ,0,0],[0,0,1,0],[0,0,0,1]]. Result: x'=x·cosθ-y·sinθ, y'=x·sinθ+y·cosθ, z'=z.
Projection is the transformation that maps 3D objects onto a 2D view plane (projection plane or picture plane). Two fundamental types exist: Parallel and Perspective.
In parallel projection, all projectors (lines from object points to the projection plane) are parallel to each other. The size of an object in the projection is independent of its distance from the plane — no foreshortening. Types:
- Orthographic Projection: Projectors perpendicular to the projection plane. Views: front, top, side. Used in engineering drawings. Formula: x'=x, y'=y (dropping z).
- Oblique Projection: Projectors at an angle to the projection plane. Subtypes: Cavalier (scale factor 1 for receding lines) and Cabinet (scale factor 0.5 for receding lines).
- Isometric Projection: Special orthographic projection where all three axes are foreshortened equally — common in technical illustrations and isometric games.
In perspective projection, all projectors converge at a single point called the center of projection (COP) or eye point. Objects farther away appear smaller (foreshortening), mimicking human vision and creating a sense of depth. Formula for 1-point perspective (COP at origin, projection plane at z=d): x'=x·d/z, y'=y·d/z.
- 1-Point Perspective: One vanishing point — projectors parallel to two axes meet at one point.
- 2-Point Perspective: Two vanishing points — for objects rotated about one axis.
- 3-Point Perspective: Three vanishing points — most realistic, for objects at arbitrary angles.
- Produces realistic images; used in rendering, photography simulation, and 3D games.
