BSc CSIT (TU) Science Computer Graphics (BSc CSIT, CSC209) Question Paper 2075 Nepal

2D Geometric Transformation

A 2D geometric transformation is an operation that changes the position, orientation, or size of an object in the 2D plane by mapping each point $(x, y)$ to a new point $(x', y')$. The basic transformations are translation, rotation and scaling.

Using homogeneous coordinates, a point $(x, y)$ is written as $(x, y, 1)$ so that all three transformations can be expressed as a single $3 \times 3$ matrix multiplication and composed easily.

1. Translation

Moves an object by displacement $(t_x, t_y)$:

2. Rotation

Rotation by angle $\theta$ (anticlockwise) about the origin:

3. Scaling

Scales an object by factors $s_x, s_y$ about the origin:

If $s_x = s_y$ the scaling is uniform; otherwise it is differential (non-uniform). Composite transformations are obtained by multiplying these matrices in the required order.

Cohen-Sutherland Line Clipping Algorithm

The Cohen-Sutherland algorithm clips a line against a rectangular window with boundaries $x_{min}, x_{max}, y_{min}, y_{max}$. It divides the plane into 9 regions, assigning each endpoint a 4-bit region (outcode).

Region Codes (bits = TBRL)

The central (inside) region has code 0000.

Algorithm

1. Compute outcodes O1, O2 for endpoints P1, P2.
2. If O1 OR O2 == 0000  -> line is fully inside: ACCEPT (trivial).
3. Else if O1 AND O2 != 0000 -> both endpoints share an outside region:
      line is fully outside: REJECT (trivial).
4. Else (line straddles a boundary):
      - Pick the endpoint that is outside (outcode != 0).
      - Find intersection with the corresponding window edge using:
          x = x1 + (x2-x1)*(y_edge - y1)/(y2-y1)   for top/bottom
          y = y1 + (y2-y1)*(x_edge - x1)/(x2-x1)   for left/right
      - Replace the outside endpoint with the intersection point,
        recompute its outcode, and repeat from step 2.

Example

Window: $x_{min}=10,\ x_{max}=100,\ y_{min}=10,\ y_{max}=100$. Clip line $P_1(5,5)$ to $P_2(60,60)$.

• Outcode of $P_1(5,5)$: left ($5<10$) and bottom ($5<10$) $\Rightarrow 0101$.
• Outcode of $P_2(60,60)$: inside $\Rightarrow 0000$.
• Logical AND $=0000 \Rightarrow$ not trivially rejected; $P_1$ is outside.
• Clip $P_1$ against left edge $x=10$: $y = 5 + (60-5)\cdot(10-5)/(60-5) = 5 + 55\cdot 5/55 = 10$. New point $(10,10)$, outcode $0000$.
• Now both endpoints are inside $\Rightarrow$ ACCEPT clipped segment from $(10,10)$ to $(60,60)$.

Projections in 3D Graphics

A projection maps 3D points onto a 2D view (projection) plane along projectors. There are two main classes.

Parallel Projection

• Projectors are parallel to each other (centre of projection at infinity).
• Preserves relative proportions; parallel lines remain parallel.
• Does not give a realistic depth sense; used in engineering/CAD drawings.
• Types: Orthographic (projectors perpendicular to plane) and Oblique (projectors at an angle).

Perspective Projection

• Projectors converge at a single centre of projection (COP / eye) at a finite distance.
• Objects farther away appear smaller (foreshortening); creates realistic depth.
• Parallel lines (not parallel to the plane) converge to vanishing points (one, two or three point perspective).

Derivation of Perspective Projection Matrix

Let the centre of projection be at the origin and the projection plane be at $z = d$ (in front of the eye). A point $P(x, y, z)$ projects to $P'(x_p, y_p, d)$.

By similar triangles along the $z$-axis:

In homogeneous coordinates this is written as:

Dividing by the homogeneous factor $h = z/d$:

which matches the similar-triangle result, giving the required perspective projection matrix.

Sutherland-Hodgman Polygon Clipping

The Sutherland-Hodgman algorithm clips a polygon against a convex clipping window by processing the polygon against one edge of the window at a time (left, right, bottom, top). The output vertex list of one stage becomes the input of the next.

Rules for each polygon edge (from vertex $S$ to vertex $P$)

For the current clip boundary:

1. In $\to$ In: output $P$.
2. In $\to$ Out: output intersection point $I$.
3. Out $\to$ In: output intersection $I$ and then $P$.
4. Out $\to$ Out: output nothing.

After all four window edges are processed, the remaining vertex list is the clipped polygon.

Example

A triangle with vertices $A(5,5), B(15,15), C(25,5)$ is clipped against the window $x_{max}=20$.

• Edge $A\to B$: both inside ($x<20$) $\Rightarrow$ output $B$.
• Edge $B\to C$: $B$ inside, $C$ outside ($25>20$) $\Rightarrow$ output intersection at $x=20$.
• Edge $C\to A$: $C$ outside, $A$ inside $\Rightarrow$ output intersection at $x=20$ and then $A$.

The triangle is thus clipped into a quadrilateral whose right side lies on $x=20$. Repeating the same procedure for the other three edges yields the final clipped polygon.

Limitation: works correctly only for convex clipping windows.

RGB and CMY Color Models

RGB (Additive) Model

• Based on the primary colors of light: Red, Green, Blue.
• Colors are formed by adding light: $\text{R}+\text{G}+\text{B} = \text{White}$, and absence of all = Black.
• Represented as a unit cube with axes R, G, B (each $0$ to $1$); the diagonal from black to white is the gray scale.
• Used by emissive devices: monitors, TVs, cameras, scanners.
• Examples: R+G = Yellow, G+B = Cyan, R+B = Magenta.

CMY (Subtractive) Model

• Based on the secondary colors / pigment primaries: Cyan, Magenta, Yellow.
• Colors are formed by subtracting (absorbing) light from white: $\text{C}+\text{M}+\text{Y} = \text{Black}$, absence of all = White (paper).
• Used by reflective devices: printers and hardcopy output.

Relationship

CMY is the complement of RGB. (In printing, CMYK adds a black ink K for true blacks and cost saving.)

3D Transformation Matrices (Homogeneous Coordinates, $4\times4$)

Translation by $(t_x, t_y, t_z)$

Scaling by $(s_x, s_y, s_z)$ about the origin

Rotation about the x-axis by angle $\theta$

The $x$-coordinate is unchanged while $y$ and $z$ rotate:

A transformed point is obtained as $[x'\ y'\ z'\ 1]^T = M\,[x\ y\ z\ 1]^T$.

Window to Viewport Transformation

• A window is a rectangular region selected in the world coordinate system that defines what is to be displayed.
• A viewport is a rectangular region on the device/screen that defines where it is displayed.
• The window-to-viewport transformation (viewing transformation) maps a point inside the window onto the corresponding point inside the viewport, preserving relative position.

Mapping

Let the window be $(x_{wmin}, y_{wmin})$ to $(x_{wmax}, y_{wmax})$ and the viewport $(x_{vmin}, y_{vmin})$ to $(x_{vmax}, y_{vmax})$. To keep relative position equal:

Solving:

Here $s_x, s_y$ are the scaling factors. If $s_x \ne s_y$ the image is distorted; equal scale factors preserve the aspect ratio. The mapping is a translate-scale-translate sequence.

Parallel vs Perspective Projection

Summary: parallel projection keeps true size and parallelism but looks flat, while perspective projection mimics how the human eye sees, shrinking distant objects toward vanishing points.

Cathode Ray Tube (CRT)

A CRT is a vacuum tube display device that produces images by directing a controlled electron beam onto a phosphor-coated screen.

Main Components

1. Electron gun – contains a heated cathode (filament) that emits electrons by thermionic emission, and a control grid that regulates beam intensity (brightness).
2. Focusing system – electrostatic/magnetic lenses that converge the electrons into a fine beam.
3. Deflection system – horizontal and vertical deflection plates/coils that steer the beam to any point on the screen.
4. Phosphor-coated screen – glows when struck by the electron beam, producing a spot of light.

Working

• The heated cathode emits electrons, which are accelerated by a high positive voltage toward the screen.
• The control grid varies the number of electrons, controlling the brightness of the spot.
• The focusing system narrows the beam; the deflection system positions it on the screen.
• When electrons strike the phosphor, light is emitted. Because the glow fades quickly (persistence), the image must be refreshed (typically 60+ times/sec) to avoid flicker.
• In color CRTs, three guns (R, G, B) and a shadow mask with phosphor triads are used.

(Diagram: cathode and control grid on the left feeding into focusing and deflection sections, with the electron beam striking the phosphor screen on the right.)

Object-Space vs Image-Space Methods (Hidden Surface Removal)

Summary: Object-space methods compare 3D surfaces directly and are resolution-independent but costlier for many objects, while image-space methods determine visibility at each pixel and are simpler to implement (e.g., Z-buffer) though tied to screen resolution.

Spline; Interpolation vs Approximation Splines

Spline

A spline is a smooth curve constructed from a set of control points using piecewise polynomial functions (commonly cubic) joined with continuity at the joints. The name comes from the flexible draftsman's strip used to draw smooth curves. Splines are defined so as to maintain parametric/geometric continuity ($C^0, C^1, C^2$) between curve segments.

Interpolation vs Approximation Splines

Note: Approximation splines (Bézier, B-spline) lie inside the convex hull of their control points, giving designers smooth, predictable control.

Key Frame System in Computer Animation

A key frame is a frame that defines the important (key) positions/states of an object at specific instants of an animation sequence. A key frame system is an animation technique in which the animator specifies only these key frames, and the system automatically generates the in-between frames.

Working

1. The animator defines key frames at chosen times $t_1, t_2, \dots$ giving the object's shape, position, color, etc.
2. The intermediate frames between two key frames are generated automatically by in-betweening (tweening), usually by interpolation of the key parameters.
3. Linear interpolation for a parameter $P$ between key frames at times $t_a$ and $t_b$:

4. Non-linear (spline) interpolation is used for smooth acceleration/deceleration (ease-in/ease-out).

Advantages

• Reduces the animator's effort (only key frames drawn).
• Produces smooth motion through automatic interpolation.
• Widely used in 2D and 3D character animation (e.g., morphing is a related key-frame technique).