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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{p} = \vec{p} + (\alpha - 1) \, (\vec{v} \cdot (\vec{p} - \vec{o})) \, \vec{v}}

What's multiplied?

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\alpha - 1) \, \vec{v} \, \vec{v}^T}

What's added?

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-\alpha) \, \vec{v} \, \left( \vec{v} \cdot \vec{o} \right)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{q}} ?

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos{\theta}} , but it's not a scalar, so we need an identity factor: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos{\theta} \, I}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin{\theta} \, \left( \vec{v} \times \Box \right)}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-\cos{\theta}) \, \vec{v} \, \vec{v}^T}

We add all of these things together (esentially factoring out Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{q}} ) and fill in the upper-left block of our matrix

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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