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 $t\in \left[0,1\right]$
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

$\left(v\times \Box \right)={\begin{bmatrix}0&-v_{z}&v_{y}\\v_{z}&0&-v_{x}\\-v_{y}&v_{x}&0\end{bmatrix}}$

Non-Uniform Scaling

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

What's multiplied?

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

What's added?

• $(1-\alpha )\,{\vec {v}}\,\left({\vec {v}}\cdot {\vec {o}}\right)$

${\begin{bmatrix}{\hat {p}}\\1\end{bmatrix}}={\begin{bmatrix}I+\left(\alpha -1\right)\,{\vec {v}}\,{\vec {v}}^{T}&(1-\alpha )\,{\vec {v}}\,\left({\vec {v}}\cdot {\vec {o}}\right)\\0&1\end{bmatrix}}\,{\begin{bmatrix}{\vec {p}}\\1\end{bmatrix}}$

3D Rotations

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

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

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

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

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

${\begin{bmatrix}\cos {\theta }\,I+\sin {\theta }\,({\vec {v}}\times \Box )+(1-\cos {\theta })\,{\vec {v}}\,{\vec {v}}^{T}&0\\0&1\end{bmatrix}}$

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