CSCE 441 Lecture 20

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End Exam 1 content

Exam Review

6 questions, some with multiple parts (max sub-parts = 18)

• Explanations, short-answer
• Computation
• No programming

50 minutes to complete

Know Algorithms

• Scan conversion
• Clipping

Clipping

Cohen-Sutherland

Region Codes where 1 bit means "outside the window"

• If bitwise AND between region codes of endpoints is non-zero, then trivially reject
• If bitwise OR is identical to zero, then trivially accept
• Otherwise, have to split line into two segments via intersection

Liang-Barsky

1. Set up parametric equation of line for $t\in [0,1]$
2. Find intersection parameter with boundary.
3. Reject if $x_{min}>x_{max}$

Transformations

Taking vector equations and turning them into matrix Form

Cross Product Matrix

$(v\times ◻)=\left[\begin{array}{ccc}0& -v_z& v_y\\ v_z& 0& -v_x\\ -v_y& v_x& 0\end{array}\right]$

Non-Uniform Scaling

$\hat{p}=\overrightarrow{p}+(\alpha -1)(\overrightarrow{v}\cdot (\overrightarrow{p}-\overrightarrow{o}))\overrightarrow{v}$

What's multiplied?

• $I$
• $(\alpha -1)\overrightarrow{v}{\overrightarrow{v}}^T$

What's added?

• $(1-\alpha )\overrightarrow{v}(\overrightarrow{v}\cdot \overrightarrow{o})$

$\left[\begin{array}{c}\hat{p}\\ 1\end{array}\right]=\left[\begin{array}{cc}I+(\alpha -1)\overrightarrow{v}{\overrightarrow{v}}^T& (1-\alpha )\overrightarrow{v}(\overrightarrow{v}\cdot \overrightarrow{o})\\ 0& 1\end{array}\right]\left[\begin{array}{c}\overrightarrow{p}\\ 1\end{array}\right]$

3D Rotations

$\hat{p}=\overrightarrow{o}+(1-\cos \theta )(\overrightarrow{v}\cdot \overrightarrow{q})\overrightarrow{v}+(\overrightarrow{v}\times \overrightarrow{q})\sin \theta +\overrightarrow{q}\cos \theta$

$\hat{q}=(1-\cos \theta )(\overrightarrow{v}\cdot \overrightarrow{q})+(\overrightarrow{v}\times \overrightarrow{q})\sin \theta +\overrightarrow{q}\cos \theta$

What is multiplied by $\overrightarrow{q}$?

• $\cos \theta$, but it's not a scalar, so we need an identity factor: $\cos \theta I$
• $\sin \theta (\overrightarrow{v}\times ◻)$
• $(1-\cos \theta )\overrightarrow{v}{\overrightarrow{v}}^T$

We add all of these things together (esentially factoring out $\overrightarrow{q}$) and fill in the upper-left block of our matrix

$\left[\begin{array}{cc}\cos \theta I+\sin \theta (\overrightarrow{v}\times ◻)+(1-\cos \theta )\overrightarrow{v}{\overrightarrow{v}}^T& 0\\ 0& 1\end{array}\right]$

Color

• Half-toning
• Dithering
• Error diffusion
• models of color
• Equations (total of 3; one per color)
• Properties of light

Lighting

(last topic covered chronologically)

Know definitions and ways to compute

• Ambient
• Diffuse
• Specular