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In general I'd write a projective transformation in homogeneous coordinates as $$\begin{pmatrix}x\\y\\z\end{pmatrix}\mapsto \begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}\cdot \begin{pmatrix}x\\y\\z\end{pmatrix}$$ or, in inhomogeneous coordinates, as $$\begin{pmatrix}x\\y\end{pmatrix}\mapsto \begin{pmatrix}(ax+by+c)/(gx+hy+i)\\(dx+... 3 The general linear group GL_n(k) is the group of automorphisms (in a suitable sense) of the n-space k^n. Similarly, the projective linear group PGL_n(k) is the group of automorphisms (in a suitable sense*) of the projective n-space \mathbb{P}^n(k). In projective geometry, two nonzero points are identified if they are on a same line going through ... 1 The projective plane over \mathbb F arises from \mathbb F^3 by excluding the origin and then identifying vectors that are multiples of each other. Similarly, PGL_n(\mathbb F) arises from the set of n\times n matrices by excluding the singular matrices and then identifying matrices that are multiples of each other (because the center of the general ... 0 Another approach to calculate the multiplicities is to use Lemma 4.4 (page 45). We calculate the multiplicities of the points [\alpha:1:0] and show they are d. First we need a local representation of F near [\alpha:1:0]. In the patch where y\neq 0, this corresponds to (\alpha, 0) in \mathbb C^2 (with coordinates x, z), cut out by x^d+1+z^d=... 3 First, note that a given triangle and a given point of perspectivity determine three rays making some set of angles; we'll say that the "outer" rays form respective angles \alpha and \beta with the "middle" ray. (Having taken opposite rays if necessary, we may assume \alpha + \beta \leq 180^\circ.) Also, if a target triangle has a "similar copy" that's ... 1 Yes, it is possible: Problem. Given two triangles ABC and A^{\prime}B^{\prime}C^{\prime} and a point P in the plane such that P does not lie on any of the lines BC,CA,AB. Then, there exist three points A^{\prime\prime},B^{\prime\prime },C^{\prime\prime} on the lines AP,BP,CP, respectively, such that triangle A^{\prime\prime}B^{\prime\prime}C^{... 0 The first time I came across this theorem, it was in Undergraduate Algebraic Geometry by Miles Reid. Printed by Cambridge University Press in 1988, originally in the series ' London Mathematical Society Student Texts'. Section 2.11 in this book is called 'Pascal's theorem', but contains and proves the statement in both directions. It thus proves both Pascal'... 0 A circular helix has the following parametric representation$$ x=r\cos t,\; y=r\sin t,\; z=ct $$Our goal is to figure out r,c. r As the helix lies on the cylinder, they both have the same radius, therefore r={7.5\over2}cm. c Suppose \cdot represents differentiation with respect to t. The tangent vector to the helix is given by$$ (\...

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You can look at Liu, Algebraic Geometry and Arithmetic Curves, prop 7.4.1 p285. The idea is use the Riemann-Roch theorem in order to show that the anticanonical divisor of the curve $C$ induces a closed immersion $C \hookrightarrow \mathbf{P}^2_K$. Thus $C$ is isomorphic to a plane curve $C'$. This curve is defined by a homogeneous polynomial of some degree ...

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Consider the equation $x_0^2 - x_1^2 = rx_2^2$ (visualize this as a cone in any way you choose) and then let $r\to 0$ to see how the cone degenerates into two intersecting planes.

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Let $S=k[x,y,z]$ and let $I$ be a homogeneous ideal such that the projective variety defined by $I$ has dimension $0$. Then $S/I$ has Krull dimension $1$ and so its Hilbert polynomial is a constant, i.e., the Hilbert function is eventually a constant. Suppose that $H_{S/I}(n) = H_{S/I}(n_0), \, \forall n \ge n_0$. Now let $R = k[x,y]$ and let $J$ be the ...

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The question is equivalent to its dual: Let $g_1,g_2,g_3$ be three distinct points on a line $A$, and let $h_1,h_2,h_3$ be three distinct points on a line $B$ and suppose $A\cap B\notin\{g_1,g_2,g_3,h_1,h_2,h_3\}$. Let $C_{ij}$ be the line $(g_i,h_j)$. Are $C_{12}\cap C_{21}$, $C_{13}\cap C_{31}, C_{23}\cap C_{32}$ collinear? But this is just ...

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I found what I was looking for in "Harley & Zisserman" (I must overlooked it). The projected line in the image can be calculated by: $$[l]_x = P\cdot L\cdot P^T$$ and $$[l]_x =\begin{bmatrix} 0&-c&b&\\ c&0&-a\\ -b&a&0 \end{bmatrix}$$ where $(a,b,c)$ are coefficients in the line equation $ax+by+c=0$ and $P$ is the ...

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Okay then. One way to achieve what you want is via the following: First use a combination of $x$ and $y$ rotations to align the $x$ axis with the minor axis of the ellipse. Then rotate about the $x$-axis by angle $\phi$. (Remark: the following paragraph is new) To find out the angle $\phi$, we start by thinking about what happens if we wrote a flat ...

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\begin{cases} x(t) = \frac{1+t^2}{2+t^3}\\ y(t) = \frac{t}{2+t^3} \end{cases} ASYMPTOTE : $x$ and $y$ tend to infinity when $\begin{cases}(2+t^3)\to 0 \\ t\to -2^{1/3} \end{cases}$ Let $t=-2^{1/3}+\epsilon \quad$ the series expansion leads to : \begin{cases} x=\frac{1}{6}(2^{1/3}+2)\frac{1}{\epsilon} -\frac{1}{6}(2^{2/3}-1) +O(\epsilon) \\ y=-\frac{1}{...

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To the third question: Entering into M2 R=QQ[x,y] S=QQ[t] KS=frac S f=map(KS,R,{(1+t^2)/(2+t^3),t/(2+t^3)}) ker f we see $2x^3-6x y^2+5y^3-x^2-2x y+2y^2+y=0$ is the implicit equation. The partial derivatives $6x^2-6y^2-2x-2y,-12xy+15y^2-2x+4y+1$ have a common zero at $(x,y)=(2/3,1/3)$ which is on the curve. This is your singular point ($t=1$). Moving ...

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Is the camera plane the projective space of the real world? It depends. Usually a physical camera has a limited sensor, so the thing where the image appears is too bounded to be an affine or projective plane. You can say “well, simply extend that plane infinitely”. Then it depends on how exactly you define that extension, whether you end up with an ...

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Start by computing the plane spanned by the line and the point of the camera. If you don't know how to compute such a join using Plücker coordinates, I can expand on this. The projection of the line is the intersection of that plane with the plane of your image. If you can choose the embedding of your drawing plane, embed it at $z=0$ in such a way that the $... 0 Let$L$be the set of$35$lines of$PG(3,2)$. For each plane$\pi$contained in$PG(3,2)$, define$P_\pi$to be the set of$7$lines contained in$\pi$, and for each point$p \in \pi$, define$L_p = \{\ell \in P_\pi:p \in \ell\}$. Then the incidence structure$(P_\pi,L_\pi = \{L_p:p \in \pi\})$is a Fano plane (axioms follow from the dual statements for$\...

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It turns out that it doesn’t matter what units you use for the extrinsic matrix as long as they’re consistent with the units that you use for world coordinates. For simplicity, let’s assume that there’s no rotation, as is the case for your matrices. That part of the extrinsic matrix isn’t affected by the units of measurement, anyway. We have some ...

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