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# 1. Conic

• Conic is a curve on a plane which is of form ax^2+bxy+cy^2+dx+ey+f=0.
• Let (x_1,x_2,x_3)^t ~ (frac{x_1}{x_3},frac{x_2}{x_3},1)^t  be the homogeneous coordinates of a point, then  ax_1^2+bx_1x_2+cx_2^2+dx_1x_3+ex_2x_3+fx_3^2=0.
• For matrix representation, it is of form x^tCx=0.
$$\begin{pmatrix} x_1 & x_2 & x_3 \end{pmatrix} \begin{pmatrix} a & \frac{b}{2} & \frac{d}{2} \\ \frac{b}{2} & c & \frac{e}{2} \\ \frac{d}{2} & \frac{e}{2} & f \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} =0 \Rightarrow x^tCx=0$$
• C is the conic as matrix representation which is symmetric. In addition, all kC (kne0) are the same conic.

# 2. The tangent to the conic

• Let the point x be on the conic C and on the line l. It means x^tCx=0 and l^tx=0. If there exists another point y such as x,  y^tCy=0 and l^ty=0.
• For any alpha>0, (x+alphay)^tC(x+alphay)=x^tCx+alpha^2y^tCy+alphay^tCx+alphax^tCy=0+0+0+0=0. It implies that all the points on the line passing through points x and y is also on C, so it is a contradiction. Therefore, l is the tangent to the conic and l=Cx.

# 3. Degenerate conic

• When a double conic is divided by a plane passing through its center, its cross section is two lines, not parabola, hyperbola, circle, or ellipse. These two lines l and m are a degenerate conic C_infty such that C_infty=lm^t+ml^t.
• Since x^tC_inftyx=0, l^tx=0, and m^tx=0,  C_infty=lm^t+ml^t and rank(C_infty)=2.

# 3. Dual conic

• A conic on a plane is a set of points, but it can be viewed as a set of tangents to each point which is called dual conic.
• The conic C satisfies x^tCx=0 for a point x on C, and the dual conic C^{ast} satisfies l^tC^{ast}l=0 for the tangent to the x.
• If a point x is on the conic C, then the tangent l to x is l=Cx, so x=C^{-1}l.
• x^t=l^tC^{-t}=l^tC^{-1} since C is symmetric. It implies that x^tCx=(l^tC^{-1})l=0. Therefore, C^{ast}=C^{-1}.

# 4. Homography of conic

• Assume that a point x is on the conic C and a line l is on the dual conic C^{ast} of C. These are trasformed to xprime, Cprime, lprime, and C^{ast prime} by a homography H.
• From x^tCx and xprime=Hx, quad x^{primet}H^{-t}CH^{-1}xprime=0, so Cprime=H^{-t}CH^{-1}.
• The tangent l to x satisfies l^tx=0, so l^tx=l^t(H^{-1}xprime)=(l^tH^{-1})xprime=0=l^{primet}xprime. It means lprime=H^{-t}l.
• From l^tC^{ast}l=0 and lprime=H^{-t}l, quad l^{primet}HC^{ast}H^tlprime=0, so C^{ast prime}=HC^{ast}H^t.

5. Circular points: all the circles intersects with l_infty at two points

• All the circles have b=0 and a=c from the conic equation  ax_1^2+bx_1x_2+cx_2^2+dx_1x_3+ex_2x_3+fx_3^2=0.
• If we set a=1, x_1^2+x_2^2+dx_1x_3+ex_2x_3+fx_3^2=0.
• The intersection point with l_infty is at infinity, so x_3=0 and it reduces to x_1^2+x_2^2=0.
• In homogeneous coordinate, this has two solutions, (x_1, x_2)={(1, i), (1, -i)}.
• Therefore, the circular points are (1,i,0)^t and (1,-i,0)^t.

# 6. Dual degenerate conic

• It is the dual of degenerate conic C_infty=lm^t+ml^t.
• Degenerate conic is defined by lines, so its dual is defined by points.
• Let the circular points be U and V, the dual degenerate conic C_{infty}^{ast} satisfies C_{infty}^{ast}=UV^t+VU^t.
• As the dual of x^tCx=0 form is l^tC^{ast}l=0, C_{infty}^{ast} consists of the lines such that l^tC_{infty}^{ast}l=0.
• C_{infty}^{ast} as matrix representation is the following:
\begin{align} C_{\infty}^{\ast} &= UV^t+VU^t= \begin{pmatrix} 1 \\ i \\ 0 \end{pmatrix} \begin{pmatrix} 1 & -i & 0 \end{pmatrix} + \begin{pmatrix} 1 \\ -i \\ 0 \end{pmatrix} \begin{pmatrix} 1 & i & 0 \end{pmatrix} \\ \\ &= \begin{pmatrix} 1 & -i & 0 \\ i & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} + \begin{pmatrix} 1 & i & 0 \\-i & 1 & 0 \\0 & 0 & 0 \end{pmatrix} \equiv \begin{pmatrix} 1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} I & \vec{0} \\ \vec{0} & 0 \end{pmatrix} \end{align}
• C_{infty}^{ast} is invariant to similarity homography. This homography H is
$$H= \begin{pmatrix} A & \vec{v} \\ \vec{0} & 1 \end{pmatrix} \quad \text{where} \quad A^tA=\lambda^{2}I^{2}.$$     Since C_{infty}^{ast prime}=HC_{infty}^{ast}H^t,  \begin{align} C_{\infty}^{\ast \prime} &= \begin{pmatrix} A & \vec{v} \\ \vec{0} & 1 \end{pmatrix} \begin{pmatrix} I & \vec{0} \\ \vec{0} & 0 \end{pmatrix} \begin{pmatrix} A^t & \vec{0} \\ \vec{v}^t & 1 \end{pmatrix} = \begin{pmatrix} A & \vec{v} \\ \vec{0} & 1 \end{pmatrix} \begin{pmatrix} A^t & \vec{0} \\ \vec{0} & 0 \end{pmatrix}= \begin{pmatrix} \lambda^2I & \vec{0} \\ \vec{0} & 0 \end{pmatrix} \\ &\equiv \begin{pmatrix} \vec{I} & \vec{0} \\ \vec{0} & 0 \end{pmatrix} = C_{\infty}^{\ast} \end{align}

• Quadric is similar to conic and is defined in one more higher dimension than that of conic.
• Quadric Q is a 4times 4 symmetric matrix and x^tQx=0 for the point x on Q.
• As rank(C)=3 for a conic C in general except for the degenerate conic whose rank(C_infty)=2, quadric Q is, in general, rank(Q)=4 except for the degenerate quadric whose rank(Q_infty)=3.

# 8. The intersection of quadric and plane is a conic

• For the non-colinear points A, B, and C are on a plane Pi, a point x on Pi is represented as x=uA+vB+wC for some u, v, c in mathbb{R}. In other words, x=(A,B,C)(u,v,w)^t=Mvec{p} where M is a 4times3 matrix and vec{p} is a 3times1 matrix.
• If x is on the intersection area of a quadric Q and Pi, x^tQx=0 and vec{p}^tM^tQMvec{p}=0.
• In fact, vec{p} can be represented in homogeneous coornidate which is composed of A, B, and C. It means that x=mvec{e_1}+nvec{e_2}=m(A-C)+n(B-C)=mA+nB-(m+n)C where m=u, n=v, and w=-m-n.
• Therefore, vec{p}^t(M^tQM)vec{p} can be considered as the conic C where C=M^tQM for the point vec{p} on C.

# 9. The tangent to the quadric

• For the point x on the tangent plane Pi to the quadric Q,  x^tQx=0 and Pi^tx=0. It implies that Pi=Q^tx=Qx. This property is correspoding to that of conic.

• A quadric is a set of points, but it can be viewed as a set of tangent planes to each point which is called dual quadric.
• The quadric Q satisfies x^tQx=0 for a point x on Q, and the dual quadric Q^{ast} satisfies Pi^tQ^{ast}Pi=0 for the tangent plane to the x.
• The tangent plane Pi on the quadric Q satisfies Pi=Qx, so x^tQx=Pi^tQ^{-t}Q Q^{-1}Pi=0. It means Q^{ast}=Q^{-t}=Q^{-1}. This property is correspoding to that of conic.

• Assume that a point x is on the quadric Q and a plane Pi is on the dual quadric Q^{ast} of Q. These are trasformed to xprime, Qprime, Piprime, and Pi^{ast prime} by a homography H.
• From x^tQx and xprime=Hx, quad x^{primet}H^{-t}QH^{-1}xprime=0, so Qprime=H^{-t}QH^{-1}.
• FromPi^tQ^{ast}Pi=0 and Piprime=H^{-t}Pi, quad Pi^{primet}HQ^{ast}H^tPiprime=0, so Q^{ast prime}=HQ^{ast}H^t.

# 12. What Qx stands for where the point x is NOT on the quadric Q

• Qx means a polar plane when the point x is not on the quadric Q.
• In other words, the polar plane consists of the planes which passes through x and tangent to Q. This property is corresponding to that of conic which is called a polar line.
•  Assume that the point y is on Q and its tangent plane Qy passes through x. So, (Qy)^tx=0=y^tQ^tx=y^tQx. Since y^tQx is a scalar, y^tQx=(y^tQx)^t=x^tQ^ty=(Qx)^ty.  Therefore, Qx is the polar plane.
• Assume that the point y is on a conic C and its tangent line Cy passes through x. So, (Cy)^tx=0=y^tC^tx=y^tCx. Since y^tCx is a scalar, y^tCx=(y^tCx)^t=x^tC^ty=(Cx)^ty.  Therefore, Cx is the polar line.

# 13. Absolute conic:  all the spheres intersect with Pi_infty

• A sphere is a kind of a quadric, so the intersection of a sphere and Pi_infty=(0,0,0,1)^t is a conic.
• All the spheres satisfy x_1^2+x_2^2+x_3^2+dx_1x_4+ex_2x_4+fx_3x_4+gx_4^2=0 for the point (x_1, x_2, x_3, x_4)^t in homogeneous coordinate.
• The intersection with Pi_infty is at infinity, so x_4=0 and it reduces to x_1^2+x_2^2+x_3^2=0.
• This form can be changed as (x_1, x_2, x_3)I(x_1, x_2, x_3)^t=0 which is of conic form. This conic is on Pi_infty and consists of the only imaginary part, which is called the absolute conic.
• The absolute conic is denoted by Omega_infty.
• The dual of  Omega_infty is called absolute dual quadric Omega_infty^{ast},
$$\begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$
• For all the planes Pi=(x_1, x_2, x_3, x_4)^t such that (x_1, x_2, x_3)(x_1, x_2, x_3)^t=0, Pi is tangent to the Omega_infty. Moreover, this Pi satisfies Pi^t Omega_infty^{ast} Pi=0, which is the dual quadric form.
• Geometrically, Omega_infty^{ast} consists of all the tangent planes of Omega_infty.

# 14. Projection of Omega_infty

• When the point x=(x_1, x_2, x_3, 0)^t on Pi_infty is projected by P, the image of x is
$$PX=KR[\ I\ |\ -E\ ] \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ 0 \end{pmatrix}=KR \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} =H\bar{x}$$
where K is the intricsic matrix of the camera, R is the rotation matrix, E is the camera center, H=KR and bar{x}=(x_1,x_2,x_3)^t. It means that bar{x} is the direction to the intersection with Pi_infty.
• Omega_infty is defined by this x which satisfies bar{x}^tOmega_inftybar{x} . Moreover, Omega_infty can be transformed by this H as follows:
\begin{align} H^{-t}\Omega_{\infty}H^{-1} &= H^{-t}IH^{-1}=(KR)^{-t}(KR)^{-1}=(R^tK^t)^{-1}(KR)^{-1} \\ &= K^{-t}R^{-t}R^{-1}K^{-1}=K^{-t}(RR^t)^{-1}K^{-1}=K^{-t}K^{-1}=\omega \end{align}
• bar{x} and Omega_infty are transformed to omega and Hbar{x} by H. Since bar{x} is on the Omega_infty, Hbar{x} should be on omega which means (Hbar{x})^tw(Hbar{x}).
• (Hbar{x})^tw(Hbar{x})=(Hbar{x})^t(K^{-t}K^{-1})(Hbar{x})>0 yields that K^{-t}K^{-1} is positive definite. Therefore, all this kind of Hbar{x} are imaginary.

15. All the planes intersect with Omega_infty at circular points

• Since Omega_infty is on Pi_infty, l_infty is also on Pi_infty. An arbitrary plane Pi includes any circles, so these circles intersect with l_infty at circular points. It means that this Pi includes l_infty.

• For a circle on Pi, there exists the sphere including this circle. This sphere also intersects with Pi_infty because all the spheres intersect with Pi_infty at absolute conic Omega_infty. Therefore, this sphere includes Omega_infty. As a result, the intersection of Omega_infty and l_infty is circular points.

# ︎Reference

[1] https://engineering.purdue.edu/kak/computervision/ECE661Folder/Index.html
[2] Hartley, R. and Zisserman, A. (2003) Multiple View Geometry in Computer Vision. 2nd Edition, Cambridge University Press, Cambridge.

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