Title: CS4277_3CV / Points, Lines, and Conics (ECE 661)
On every note of this deck, unless a note says otherwise:
Title: CS4277_3CV::Points, Lines, and Conics (ECE 661)

Front: Given any two lines [$]l_1 = \begin{pmatrix} a_1 \\ b_1 \\ c_1 \end{pmatrix}[/$] and [$]l_2 = \begin{pmatrix} a_2 \\ b_2 \\ c_2 \end{pmatrix}[/$], what is the point at the intersection?
Back: [$]x = l_1 \times l_2[/$]
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Front: Given any two points [$]x_1 = \begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix}[/$] and [$]x_2 = \begin{pmatrix} x_2 \\ y_2 \\ z_2 \end{pmatrix}[/$], what is the line passing through both points?
Back: [$]l = x_1 \times x_2[/$]
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Front: What is the scalar triple dot product identity?
Back: [$]\vec{A} \cdot (\vec{A} \times \vec{B}) = \vec{B} \cdot (\vec{A} \times \vec{B}) = 0[/$]
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Front: What are ideal points?
Back: [$]\begin{pmatrix} u \\ v \\ 0 \end{pmatrix}[/$] are known as Ideal Points.
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Q: All ideal points form a [...].
A: straight line in R^2
Title: CS4277_3CV::Points, Lines, and Conics (ECE 661)

Front: What is the line at infinity?
Back: [$]\begin{pmatrix} 0 & 0 & 1 \end{pmatrix}[/$]
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Front: How is the camera image of the line at infinity called?
Back: Vanishing Line.
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Front: What is a conic?
Back: A curve that is described by a second-degree implicit algebraic form in a plane.
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Front: What is the implicit form for a conic in the physical plane [$]\mathbb{R}^2[/$]?
Back: [$$]ax^2 + bxy + cy^2 + dx + ey + f = 0[/$$]
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Front: What is the HC representation of a conic?
Back: [$$]\begin{bmatrix} a & b/2 & d/2 \\ b/2 & c & e/2 \\ d/2 & e/2 & f\end{bmatrix}[/$$]
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Front: What is the line tangent to a conic C at point x?
Back: [$$]l = Cx[/$$]
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Front: What is a degenerate conic?
Back: When you slice the double cones with a plane that passe through the axis of the double cones, you will now see two intersecting straight lines.
These two lines constitute a degenerate conic.
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Front: What is the HC representation of a degenerate conic?
Back: [$$]C = lm^T + ml^T[/$$]
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Front: How to constrain a conic C given five points?
Back: 1. Stack constraints from five points to matrix A
2. Obtain the right null-space of this 5x6 matrix
3. Reconstruct C from the solution
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