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Commit 823ed8e2 authored by Anderson Tavares's avatar Anderson Tavares
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<div class="content">
<div>
<div id="reg01"></div><h2>Background and Overview</h2>
<h3 class="hidden">Euclidean Geometry</h3>
<div class="blocks2 hidden">
<h3>Euclidean Geometry</h3>
<div class="blocks2">
<div class="block2 reg01euclid row2">
<div class="figure">
<img src="figs/euclid.jpg">
......@@ -91,8 +91,8 @@
<div class="figcaption">Solid angles and Volumes</div>
</div>
</div>
<h3 class="hidden">Perspective</h3>
<div class="blocks2 hidden">
<h3>Perspective</h3>
<div class="blocks2">
<div class="block2 col2">
<div class="figure"><img src="figs/optics.svg"></div>
<div class="figcaption">Euclid's&nbsp;<span class="emph">Optics</span> &nbsp;book</div>
......@@ -117,8 +117,8 @@
<div class="figcaption">Leon Battista Alberti's&nbsp; <span class="emph">De Pictura</span> (1435)</div>
</div>
</div>
<h3 class="hidden">Projective Geometry</h3>
<div class="blocks2 hidden">
<h3>Projective Geometry</h3>
<div class="blocks2">
<div class="block2 row2">
<div class="figure"><img src="figs/kepler.jpg"></div>
<div class="figcaption">Johannes Kepler (1571-1630)</div>
......@@ -183,8 +183,8 @@
</ul>
</div>
</div>
<h3 class="hidden">Photogrammetry</h3>
<div class="blocks2 hidden">
<h3>Photogrammetry</h3>
<div class="blocks2">
<div class="block2 col2 row2">
<div class="figure"><img src="figs/cameraobscura.jpg"></div>
<div class="figcaption">Photography: Camera Obscura</div>
......@@ -241,8 +241,8 @@
</div>
</div>
<h3 class="hidden">Computer vision</h3>
<div class="blocks2 hidden">
<h3>Computer vision</h3>
<div class="blocks2">
<div class="block2 row2">
<div class="figure"><img src="figs/first-scan.jpg"></div>
<div class="figcaption">First digitized image (1957)</div>
......@@ -288,7 +288,7 @@
<p class="part">Representations</p>
<div id="reg02"></div><h2>Cartesian Representations</h2>
<div class="blocks2 hidden">
<div class="blocks2">
<div class="block2 row2 col2">
<div class="figure"><img src="figs/reg02-cartesian.svg"></div>
<div class="figcaption"><span class="kw">Cartesian coordinate system</span></div>
......@@ -305,9 +305,13 @@
<div class="figure"><img src="figs/lefthand.svg"></div>
<div class="figcaption">Left hand system</div>
</div>
<div class="block2">
<div class="figure"><img src="figs/lefthand.svg"></div>
<div class="figcaption">Left hand system</div>
</div>
</div>
<h3 class="hidden">Points in <span class="keq">\mathbb{E}^2</span></h3>
<div class="blocks2 hidden">
<h3>Points in <span class="keq">\mathbb{E}^2</span></h3>
<div class="blocks2">
<div class="block2 row3 col2">
<p><span class="kw">Point</span> in <span class="keq">\mathbb{E}^2\Rightarrow</span> <span class="kw">Vector</span> <span class="keq">\vec{x}\in \mathbb{R}^2</span>, <span class="keq">\vec{x}=\begin{bmatrix}x_1\\x_2\end{bmatrix}</span></p>
<p><span class="kw">Vector sum</span>: <span class="keq">\vec{z}=\vec{x}+\vec{y}=\begin{bmatrix}x_1+y_1\\x_2+y_2\end{bmatrix}</span></p>
......@@ -357,8 +361,8 @@
<div class="figcaption">Left hand system</div>
</div>
</div>
<h3 class="hidden">Lines in <span class="keq">\mathbb{E}^2</span></h3>
<div class="blocks2 hidden">
<h3>Lines in <span class="keq">\mathbb{E}^2</span></h3>
<div class="blocks2">
<div class="block2 row2 col2">
<p><span class="kw">Line</span> in <span class="mr-math"><span class="mr-bb"><span class="mr-mord">E</span></span><span class="mr-msupsub"><span class="mr-msup"><span class="mr-mtext">2</span></span></span></span>: <span class="keq">\vec{x}</span> satisfying the <span class="kw">equation of the line</span>:</p>
<div class="eq keq">l_1x_1+l_2x_2=\vec{l}\cdot\vec{x}=\Delta</div>
......@@ -483,6 +487,50 @@
<p>Line as intersection of planes</p>
</div>
</div>
<h3>Basic geometric operations</h3>
<div class="blocks2">
<div class="block2 col2 row2">
<h4 class="center">Line from two points in <span class="keq">\mathbb{E}^2</span></h4>
<div class="eq keq">\overbrace{\vec{x}-\vec{y}=\begin{bmatrix}x_1-y_1\\x_2-y_2\end{bmatrix}}^{\text{Tangent vector}}\hspace{10px}\overbrace{\vec{l}=\pm\begin{bmatrix}y_2-x_2\\x_1-y_1\end{bmatrix}}^{\text{Normal vector}}</div>
<div class="eq keq">\hat{l}=\displaystyle\frac{\vec{l}}{\|\vec{l}\|}, \Delta=\hat{l}\vec{x}</div>
<p class="center"><span class="eq keq">\vec{x}=\vec{y}\Rightarrow \vec{l}=\begin{bmatrix}0&0\end{bmatrix}^T</span> (not a line normal)</p>
</div>
<div class="block2 row2">
<div class="figure"><img src="figs/placeholder.svg"></div>
<div class="figcaption">Vector form</div>
</div>
<div class="block2 col2 row2">
<h4 class="center">Intersection between two lines in <span class="keq">\mathbb{E}^2</span></h4>
<p>Two normalized lines: <span class="keq">(\hat{l},\Delta_l)</span> and <span class="keq">(\hat{m},\Delta_m)</span></p>
<p><span class="keq">x_l=\begin{bmatrix}\hspace{-3px}l_2&\hspace{-5px}l_1\Delta_l\\\hspace{-3px}-l_1&\hspace{-5px}l_2\Delta_l\end{bmatrix}\hspace{-5px}\begin{bmatrix}s\\1\end{bmatrix}\hspace{-3px}=\hspace{8px}\begin{bmatrix}m_2&m_1\Delta_m\\-m_1&m_2\Delta_m\end{bmatrix}\hspace{-5px}\begin{bmatrix}t\\1\end{bmatrix}=x_m</span></p>
<p>Use <span class="keq">l_1^2+l_2^2=m_1^2+m_2^2=1</span> and solve for <span class="keq">s</span> and <span class="keq">t</span></p>
<div class="eq keq">\displaystyle\begin{bmatrix}s\\t\end{bmatrix}\hspace{-2px} =\hspace{-2px} \frac{1}{l_2m_1-l_1m_2}\hspace{-1px}\begin{bmatrix}-\Delta_ll_1m_1-\Delta_ll_2m_2+\Delta_m\\-\Delta_l+\Delta_2l_1m_1+\Delta_2l_2m_2\end{bmatrix}</div>
<div class="eq keq">x = x_l = x_m = \displaystyle \frac{1}{l_2m_1-l_1m_2}\begin{bmatrix}\Delta_ml_2-\Delta_lm_2\\\Delta_lm_1-\Delta_ml_1\end{bmatrix}</div>
</div>
<div class="block2 row2">
<div class="figure"><img src="figs/placeholder.svg"></div>
<div class="figcaption">Intersection between two lines</div>
</div>
<div class="block2 col2 row2">
<h4 class="center">Distance between point and line</h4>
<p>Point <span class="keq">\vec{x}</span> and line <span class="keq">(\hat{l},\Delta)</span></p>
<p>Tangent vector <span class="keq">\hat{t}=\left[-l_2, l_1\right]\Rightarrow \hat{t}\perp\hat{l}</span></p>
<p><span class="keq">\{\hat{l},\hat{t}\}</span> is a basis for <span class="keq">\mathbb{R}^2</span></p>
<p><span class="keq">\vec{x}=(\hat{l}\cdot\vec{x})\hat{l}+(\hat{t}\cdot\vec{x})\hat{t}=\vec{x}_l+\vec{x}_t</span></p>
<p>Point on line closest to origin: <span class="keq">\vec{x}_0=\Delta\hat{l}</span></p>
<p><span class="keq">\{\vec{x}_0, \vec{x}_l, [0,0]\}</span> are co-linear (<span class="keq">\vec{x}_0\perp \hat{l}</span> and <span class="keq">\vec{x}_l\perp \hat{l}</span>)</p>
<p><span class="keq">
\begin{aligned}
d &= \|\vec{x}_l-\vec{x}_0\|=\|(\hat{l}\cdot\vec{x})\hat{l}-\Delta\hat{l}\|\\&=\|\hat{l}(\hat{l}\cdot\vec{x}-\Delta)\|=|\hat{l}\cdot\vec{x}-\Delta|
\end{aligned}
</span></p>
</div>
<div class="block2 row2">
<div class="figure"><img src="figs/placeholder.svg"></div>
<div class="figcaption">Distance between point and line</div>
</div>
</div>
<div id="reg03"></div><h2>Homogeneous Representations in 2D</h2>
......
tsbb06.mr
+ 63
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......@@ -62,8 +62,8 @@
\div.content{
\div{
\div#reg01\h2{Background and Overview}
\h3.hidden{Euclidean Geometry}
\div.blocks2.hidden{
\h3{Euclidean Geometry}
\div.blocks2{
\div.block2.reg01euclid.row2{
\div.figure{
\img(src="figs/euclid.jpg")
......@@ -91,8 +91,8 @@
\div.figcaption{Solid angles and Volumes}
}
}
\h3.hidden{Perspective}
\div.blocks2.hidden{
\h3{Perspective}
\div.blocks2{
\div.block2.col2{
\div.figure{\img(src="figs/optics.svg")}
\div.figcaption{Euclid's&nbsp;\span.emph{Optics} &nbsp;book}
......@@ -117,8 +117,8 @@
\div.figcaption{Leon Battista Alberti's&nbsp; \span.emph{De Pictura} (1435)}
}
}
\h3.hidden{Projective Geometry}
\div.blocks2.hidden{
\h3{Projective Geometry}
\div.blocks2{
\div.block2.row2{
\div.figure{\img(src="figs/kepler.jpg")}
\div.figcaption{Johannes Kepler (1571-1630)}
......@@ -183,8 +183,8 @@
}
}
}
\h3.hidden{Photogrammetry}
\div.blocks2.hidden{
\h3{Photogrammetry}
\div.blocks2{
\div.block2.col2.row2{
\div.figure{\img(src="figs/cameraobscura.jpg")}
\div.figcaption{Photography: Camera Obscura}
......@@ -241,8 +241,8 @@
}
}
\h3.hidden{Computer vision}
\div.blocks2.hidden{
\h3{Computer vision}
\div.blocks2{
\div.block2.row2{
\div.figure{\img(src="figs/first-scan.jpg")}
\div.figcaption{First digitized image (1957)}
......@@ -288,7 +288,7 @@
\p.part{Representations}
\div#reg02\h2{Cartesian Representations}
\div.blocks2.hidden{
\div.blocks2{
\div.block2.row2.col2{
\div.figure{\img(src="figs/reg02-cartesian.svg")}
\div.figcaption{\span.kw{Cartesian coordinate system}}
......@@ -305,9 +305,13 @@
\div.figure{\img(src="figs/lefthand.svg")}
\div.figcaption{Left hand system}
}
\div.block2{
\div.figure{\img(src="figs/lefthand.svg")}
\div.figcaption{Left hand system}
}
}
\h3.hidden{Points in \span.keq{\\mathbb\{E\}^2}}
\div.blocks2.hidden{
\h3{Points in \span.keq{\\mathbb\{E\}^2}}
\div.blocks2{
\div.block2.row3.col2{
\p{\span.kw{Point} in \span.keq{\\mathbb\{E\}^2\\Rightarrow} \span.kw{Vector} \span.keq{\\vec\{x\}\\in \\mathbb\{R\}^2}, \span.keq{\\vec\{x\}=\\begin\{bmatrix\}x_1\\\\x_2\\end\{bmatrix\}}}
\p{\span.kw{Vector sum}: \span.keq{\\vec\{z\}=\\vec\{x\}+\\vec\{y\}=\\begin\{bmatrix\}x_1+y_1\\\\x_2+y_2\\end\{bmatrix\}}}
......@@ -357,8 +361,8 @@
\div.figcaption{Left hand system}
}
}
\h3.hidden{Lines in \span.keq{\\mathbb\{E\}^2}}
\div.blocks2.hidden{
\h3{Lines in \span.keq{\\mathbb\{E\}^2}}
\div.blocks2{
\div.block2.row2.col2{
\p{\span.kw{Line} in $\bb{E}^2$: \span.keq{\\vec\{x\}} satisfying the \span.kw{equation of the line}:}
\div.eq.keq{l_1x_1+l_2x_2=\\vec\{l\}\\cdot\\vec\{x\}=\\Delta}
......@@ -483,6 +487,50 @@
\p{Line as intersection of planes}
}
}
\h3{Basic geometric operations}
\div.blocks2{
\div.block2.col2.row2{
\h4.center{Line from two points in \span.keq{\\mathbb\{E\}^2}}
\div.eq.keq{\\overbrace\{\\vec\{x\}-\\vec\{y\}=\\begin\{bmatrix\}x_1-y_1\\\\x_2-y_2\\end\{bmatrix\}\}^\{\\text\{Tangent vector\}\}\\hspace\{10px\}\\overbrace\{\\vec\{l\}=\\pm\\begin\{bmatrix\}y_2-x_2\\\\x_1-y_1\\end\{bmatrix\}\}^\{\\text\{Normal vector\}\}}
\div.eq.keq{\\hat\{l\}=\\displaystyle\\frac\{\\vec\{l\}\}\{\\|\\vec\{l\}\\|\}, \\Delta=\\hat\{l\}\\vec\{x\}}
\p.center{\span.eq.keq{\\vec\{x\}=\\vec\{y\}\\Rightarrow \\vec\{l\}=\\begin\{bmatrix\}0&0\\end\{bmatrix\}^T} (not a line normal)}
}
\div.block2.row2{
\div.figure{\img(src="figs/placeholder.svg")}
\div.figcaption{Vector form}
}
\div.block2.col2.row2{
\h4.center{Intersection between two lines in \span.keq{\\mathbb\{E\}^2}}
\p{Two normalized lines: \span.keq{(\\hat\{l\},\\Delta_l)} and \span.keq{(\\hat\{m\},\\Delta_m)}}
\p{\span.keq{x_l=\\begin\{bmatrix\}\\hspace\{-3px\}l_2&\\hspace\{-5px\}l_1\\Delta_l\\\\\\hspace\{-3px\}-l_1&\\hspace\{-5px\}l_2\\Delta_l\\end\{bmatrix\}\\hspace\{-5px\}\\begin\{bmatrix\}s\\\\1\\end\{bmatrix\}\\hspace\{-3px\}=\\hspace\{8px\}\\begin\{bmatrix\}m_2&m_1\\Delta_m\\\\-m_1&m_2\\Delta_m\\end\{bmatrix\}\\hspace\{-5px\}\\begin\{bmatrix\}t\\\\1\\end\{bmatrix\}=x_m}}
\p{Use \span.keq{l_1^2+l_2^2=m_1^2+m_2^2=1} and solve for \span.keq{s} and \span.keq{t}}
\div.eq.keq{\\displaystyle\\begin\{bmatrix\}s\\\\t\\end\{bmatrix\}\\hspace\{-2px\} =\\hspace\{-2px\} \\frac\{1\}\{l_2m_1-l_1m_2\}\\hspace\{-1px\}\\begin\{bmatrix\}-\\Delta_ll_1m_1-\\Delta_ll_2m_2+\\Delta_m\\\\-\\Delta_l+\\Delta_2l_1m_1+\\Delta_2l_2m_2\\end\{bmatrix\}}
\div.eq.keq{x = x_l = x_m = \\displaystyle \\frac\{1\}\{l_2m_1-l_1m_2\}\\begin\{bmatrix\}\\Delta_ml_2-\\Delta_lm_2\\\\\\Delta_lm_1-\\Delta_ml_1\\end\{bmatrix\}}
}
\div.block2.row2{
\div.figure{\img(src="figs/placeholder.svg")}
\div.figcaption{Intersection between two lines}
}
\div.block2.col2.row2{
\h4.center{Distance between point and line}
\p{Point \span.keq{\\vec\{x\}} and line \span.keq{(\\hat\{l\},\\Delta)}}
\p{Tangent vector \span.keq{\\hat\{t\}=\\left[-l_2, l_1\\right]\\Rightarrow \\hat\{t\}\\perp\\hat\{l\}}}
\p{\span.keq{\\\{\\hat\{l\},\\hat\{t\}\\\}} is a basis for \span.keq{\\mathbb\{R\}^2}}
\p{\span.keq{\\vec\{x\}=(\\hat\{l\}\\cdot\\vec\{x\})\\hat\{l\}+(\\hat\{t\}\\cdot\\vec\{x\})\\hat\{t\}=\\vec\{x\}_l+\\vec\{x\}_t}}
\p{Point on line closest to origin: \span.keq{\\vec\{x\}_0=\\Delta\\hat\{l\}}}
\p{\span.keq{\\\{\\vec\{x\}_0, \\vec\{x\}_l, [0,0]\\\}} are co-linear (\span.keq{\\vec\{x\}_0\\perp \\hat\{l\}} and \span.keq{\\vec\{x\}_l\\perp \\hat\{l\}})}
\p{\span.keq{
\\begin\{aligned\}
d &= \\|\\vec\{x\}_l-\\vec\{x\}_0\\|=\\|(\\hat\{l\}\\cdot\\vec\{x\})\\hat\{l\}-\\Delta\\hat\{l\}\\|\\\\&=\\|\\hat\{l\}(\\hat\{l\}\\cdot\\vec\{x\}-\\Delta)\\|=|\\hat\{l\}\\cdot\\vec\{x\}-\\Delta|
\\end\{aligned\}
}}
}
\div.block2.row2{
\div.figure{\img(src="figs/placeholder.svg")}
\div.figcaption{Distance between point and line}
}
}
\div#reg03\h2{Homogeneous Representations in 2D}
......
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