Title: CS4277_3CV / The Principle of Point/Line Duality (ECE 661)
On every note of this deck, unless a note says otherwise:
Title: CS4277_3CV::The Principle of Point/Line Duality (ECE 661)

Front: How is a conic defined with points?
Back: [$$]x^TCx = 0[/$$]
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Front: How is a conic defined with lines?
Back: [$$]l^tC^*l = 0[/$$]
Where [$]C^* = C^{-1}[/$]
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Front: What is [$]C^*[/$]?
Back: The dual conic.
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Front: What are Circular Points?
Back: Points of intersection of any circle in a plane with [$]l_\infty[/$].
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Front: How many are Circular Points are there?
Back: Since there can be at most two intersection points between a line and a circle, we have two circular points.
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Front: What are the two circular points?
Back: [$]\vec{I} = \begin{pmatrix} 1 \\ i \\ 0\end{pmatrix}, \ \vec{J} = \begin{pmatrix} 1 \\ -i \\ 0\end{pmatrix}[/$]
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Front: What is the dual degenerate conic?
Back: It is the degenerate conic where each line is replaced by a point, the circular points.
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Front: What is the HC representation of a dual degenerate conic?
Back: [$$]C^*_\infty = \vec{I}\vec{J}^T + \vec{J} \vec{I}^T[/$$]
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Front: What is the HC representation of a dual degenerate conic in matrix form?
Back: [$$]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}[/$$]
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