Unit 6: Formula Sheet
2025, December 14
1. Hardware & Display
Dot Clock Frequency
$$Dot~Clock \approx Total~Pixels \times Refresh~Rate$$
Frame Buffer Memory (Bits)
$$Memory = Width \times Height \times Bits\_per\_pixel$$
Frame Time
$$Frame~Time = \frac{1}{Refresh~Rate}$$
Aspect Ratio
$$AR = \frac{Width}{Height}$$
mindmap
root((Hardware Math))
Pixels
Resolution (WxH)
Aspect Ratio (W/H)
Speed
Refresh Rate (Hz)
Frame Time (1/Hz)
Dot Clock (Px * Hz)
Storage
Frame Buffer
W * H * Depth
2. Line & Circle Algorithms
DDA Line
Steps Calculation
If $|dx| > |dy| \implies Steps = |dx|$
Else $\implies Steps = |dy|$
Else $\implies Steps = |dy|$
Increments
$$X_{inc} = \frac{dx}{Steps}, \quad Y_{inc} = \frac{dy}{Steps}$$
Bresenham Line
Initial Decision ($P_0$)
$$P_0 = 2dy - dx$$
Next Decision ($P_{k+1}$)
If $P_k < 0: P_{k+1}=P_k + 2dy$
If $P_k \ge 0: P_{k+1} = P_k + 2dy - 2dx$
If $P_k \ge 0: P_{k+1} = P_k + 2dy - 2dx$
Mid-Point Circle
Initial Decision ($P_0$)
$$P_0 = 1 - r$$
Next Decision ($P_{k+1}$)
If $P_k < 0: P_{k+1}=P_k + 2x + 1$
If $P_k \ge 0: P_{k+1} = P_k + 2x - 2y + 1$
If $P_k \ge 0: P_{k+1} = P_k + 2x - 2y + 1$
mindmap
root((Algo Math))
Lines
DDA (Decimals)
Bresenham (2dy - dx)
Circles
Mid-Point (1-r)
Next if <0 (2x+1)
Next if >0 (2x-2y+1)
3. Transformations (2D & 3D)
2D Basic
| Type | Formula |
|---|---|
| Translation | $x' = x + t_x$ $y' = y + t_y$ |
| Scaling | $x' = x \cdot s_x$ $y' = y \cdot s_y$ |
| Rotation | $x' = x\cos\theta - y\sin\theta$ $y' = x\sin\theta + y\cos\theta$ |
Review of Reflection
Reflection about $y=x$
$$x' = y, \quad y' = x$$
Reflection about $y=k$ (Line)
$$x' = x, \quad y' = -y + 2k$$
Shearing
X-Shear
$$x' = x + sh_x \cdot y, \quad y' = y$$
Y-Shear
$$x' = x, \quad y' = y + sh_y \cdot x$$
3D Rotation
About Z-Axis
$x' = x\cos\theta - y\sin\theta$
$y' = x\sin\theta + y\cos\theta$
$z' = z$
$y' = x\sin\theta + y\cos\theta$
$z' = z$
About X-Axis
$x' = x$
$y' = y\cos\theta - z\sin\theta$
$z' = y\sin\theta + z\cos\theta$
$y' = y\cos\theta - z\sin\theta$
$z' = y\sin\theta + z\cos\theta$
About Y-Axis
$x' = z\sin\theta + x\cos\theta$
$y' = y$
$z' = z\cos\theta - x\sin\theta$
$y' = y$
$z' = z\cos\theta - x\sin\theta$
mindmap
root((Matrix Math))
Move
Translate (+)
Resize
Scale (*)
Spin
Rotate (Trig)
Distort
Shear (Add factor)
4. Projections & VSD
Perspective (Simple)
$$x' = x \cdot \frac{d}{z}, \quad y' = y \cdot \frac{d}{z}$$
Oblique Projection
$$x' = x + L \cdot z \cos\phi$$
$$y' = y + L \cdot z \sin\phi$$
Back-Face Test
$$V \cdot N > 0 \implies Hidden$$
Plane Equation
$$Ax + By + Cz + D = 0$$
$$z = \frac{-(Ax + By + D)}{C}$$
mindmap
root((3D Formulas))
Depth
Plane Eq (Ax+By..)
Z-Buffer (Compare Z)
View
Perspective (div Z)
Oblique (+ Offset)
Hidden
Dot Product
V dot N