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Questions tagged [projective-geometry]

Projective geometry is closely related to perspective geometry. These types of geometry originated with artists around the 14th century.

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Equation of a ray passing through the camera center and an image point

If C is the camera center in homogeneous world coordinates and x is a point on the image plane, the ray passing through C and x is said to be: $$ X(\lambda) = P^{+}x + \lambda C $$ where P is the ...
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The 2-Dimensional Projective Space $\mathbb{P}^2$ Has the Basis with *Three* (?) Vectors $\mathbf{e}_i$, $i = 1, \dots, 3$?

My textbook says the following: Consider a set of basis vector $\mathbf{e}_i$, $i = 1, \dots, 3$ for a 2-dimensional projective space $\mathbb{P}^2$. For reasons to become clear, we will write the ...
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All the possible configurations in $\Bbb P^4$ of two planes whose sum is $\Bbb P^4$ and a line

Consider the real projective space $\Bbb P^4$ and let $\alpha,\beta$ be planes such that $\alpha+\beta=\Bbb P^4$. What are the possible configurations of $\alpha, \beta$ and a generic line $\ell$? ...
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Rational bezier curves and pdfs

My goal is to write out the 2D projection of a 3D bezier curve to a pdf document. I am told that the 2D perspective (conic) projection of a 3D bezier curve is a rational bezier curve. There are ...
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Tangent lines of elliptic curve with a fixed point and Weierstress form

I'm reading Ian Connell's Elliptic Curve Handbook for the details of Nagell's algorithm, which can construct the birational map from an elliptic curve to its Weierstress form. At the bottom of Page ...
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Connection between number theory and projective geometry?

Consider the directed graphs $G_n^p$ with node set $\{0,1,\dots,p-1\}$, $0 \leq n < p$ with an arrow from $a$ to $b$ if $na=b\operatorname{mod}p$. These graphs are graphical multiplication tables ...
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Show the one-point set $\{a\}$ is a projective variety, and compute explicit generators for the ideal $I_p(a)⊴K[x_0,…,x_n]$.

I think I have it, but I'm also new to this so I'm looking for verification. To be a projective variety, $\{a\}$ needs to be the zero locus of a set $S$, meaning $\{a\}=V(S)=\{x∈P:f(x)=0 \text{ for ...
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Help filling in gaps for a lemma in the introduction to Hulek's elementary algebraic geometry

This is a step in Hulek's Elementary Algebraic geometry that I'm trying to fill in the gaps for. Any help is appreciated. This is Lemma 0.11 in the book. Let $p,q\in\mathbb{C}[t]$ be coprime. If ...
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Projection of Uniform Distribution between 2 Simplex

$X$ is a uniform distribution on the standard simplex $L$. For any point $\vec{x}$ on $X$, all entries of $\vec{x}$ are between 0 and 1 and $\vec{x}\cdot \vec{1} = 1$. $S$ is the standard simplex ...
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Coordinate ring of complete intersection Calabi Yau (CICY)

I am quite new in commutative algebra so I would be grateful if you could clarify to me this issue. I would like to construct the coordinate ring, $A$, of a Calabi Yau threefold $X$ which is a ...
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Projection of convex polytope onto plane

A convex polytope is given by a system of linear inequalities in 4D space. How do I find its projection onto plane given by equation $x_1 + x_2 = 0$? If it was $x_1 = 0$, I'd use Fourier-Motzkin ...
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Understanding the definition of the projective sphere and quotient spaces

Munkres Topology: Corollary 60.4 says that $p^{-1}(y)$ is a 2 point set. Why is this? I understand $P^2$ to consist of equivalence classes of $S^2$ which are 2 point sets of antipodal points $\{x,-x\}...
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Under a central collineation, show that a line (not on the center), and its image meet in a fixed point.

This is probably easy, but I have no idea. If $T$ is the center, then any lines passing through $T$ are mapped to themselves. This isn't the case for this problem, though. I suppose that this proof ...
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Intuition about definition of projective line and relation to the sphere

I have a definition of the projective line given in my lecture notes as: $ \mathbb{P}^1=(\mathbb{C}^2 $ \ $ \{0\})/\mathbb{C}^\times $ my notes then state that this is the space of lines through the ...
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Find a locus of points

Given a triangle $ABC$, $A'$ and $B'$ halves $BC$ and $AC$. We have a variable point on a line $AB$. Parallel to $AA'$ and $BB'$ through $P$ cuts $AC$ in $E$ and $BC$ in $F$. Now line $EF$ cuts $AA'$ ...
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Scalar question regarding homogeneous coordinates

Let $x$, $y$, $z$ be three collinear points in homogeneous coordinates. Show there are scalars $j$, $k$, and $l$ such that $jx + ky = lz$, with $j$, $k$, and $l$ not all $0$. I started with the ...
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System of homogeneous equations and irreducibility

Is a system of homogeneous polynomial equations of n eqs and n+1 variables, over complexes, always irreducible? Any reference?
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Show the space of flat connections $(\mathbb T^2)^N / {\mathfrak S}_N$ is a $(N-1)$-complex projective space

Is it true that $$ \mathbb{T}^2/(\mathbb{Z}/2)=\mathbb{CP}^1=S^2? $$ $$ \mathbb{T}^2/(\mathbb{Z}/2)=\mathbb{CP}^1=S^2? $$ I am trying to digest the following statements: $$ M_{\rm flat} =\mathbb E ...
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What are the transformations that preserve cross ratios on a sphere in higher dimensions?

If we have four points $x,y,z,w$ on a sphere, then the cross ratio is $\frac{|x-z|}{|x-w|}\frac{|y-w|}{|y-z|}$. If we consider $S^1 \subseteq \mathbb{C}$, then the transformations of $\mathbb{C}$ ...
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How to prove $proj_{W^\bot}x$ = $perp_{W}x$?

This was a homework question given in today's class, but I'm really stuck on it. I tried to use the formula for projection of a vector onto a subspace to manipulate it, but got stuck. Is this a smart ...
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Projective transformation with parabola and circle

Let $y=x^2$ and $x^2+y^2=2$ be a parabola and circle in the standard embedding plane $\{[x:y:1]\mid x,y\in\mathbf{R} \}\subset \mathbf{RP}^2$. Give a projective transformation $t_A$ which maps the ...
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Drawing Incidence Structure

I'm new in Incidence Geometry and I came across with a question as: Let $T_k $be the set of all subsets of {1,2,....,n} with k elements. Draw the incidence structure $ (T_2, T_3,/subseteq) $ for n=3. ...
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Find the viewing angles of a cube from a projection of its edges

If I project a cube's corner on a plane by photographing it from a distance I get three lines radiating from a point. From two angles $A$, $B$ made by these three lines in the plane, I should be able ...
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Uniqueness of the quadratic form associated with a quadric

Let two quadratic forms $\Phi_1:E\to\mathbb{K}$ and $\Phi_2:E\to\mathbb{K}$ be, such that $\{u\in E:\Phi_1(u)=0\}=\{u\in E:\Phi_2(u)=0\}$. Then, it is known that $\Phi_1=\lambda\Phi_2$ for some $\...
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How to prove that in PG(2,R), if a projectivity takes 3 points to themselves, then the projectivity takes all points to themselves?

If I have $(P,Q,R,X,Y,Z,...)\barwedge(P,Q,R,X',Y',Z',...)$ and all points lie on line $l$, How do I go about proving $(X,Y,Z,...)=(X',Y',Z',...)$? By a theorem, there exists a unique projectivity ...
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Rotating a Circle in $\mathbb R^3$ using $(\theta,\phi,\psi)$ and projecting onto $\mathbb R^2$

Description I am working on creating a wireframe sphere that showcases latitude and longitude lines. A visual representation of what I'm attempting to achieve can be found here. To be clear, I'm ...
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How to visualize the pole/polar of a line/point in projective space PG(2, R) not in contact with the curvature?

So, I understand that given a curvature $\mathbb{C}$. The polar of a point on the conic is tangent to the conic at that point and the pole of a line touching the conic only once and tangent to the ...
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Map a Conic to the Normalized Image Plane

I have a problem with understanding a specific operation related to the mapping of an ellipse, captured by a camera, to a plane: According to this paper, given the calibration matrix $K$ of a camera, ...
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Equations plane in projective space

suppose I have a point $(0:0:y:z)$ and a line described by $x_2=x_3=0$ in $\mathbb{P}^3(\mathbb{C})$. How do I write the equation of the plane that contains both? Then I want the intersection with the ...
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Prove polynomial on $\mathbb{CP}^1$ has finite singular values

Suppose I have a polynoial $p:\mathbb{C} \to \mathbb{C}$. There's a natural way to extend this polynomial to a polynomial $p:\mathbb{CP}^1 \to \mathbb{CP}^1$. If you have a polynomial $p(z) = a_nz^n + ...
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Stabilize positions of a cluster of points in three dimensions

I'm a statistics/machine-learning guy with a geometry problem I'm hoping to get some help on. I am working on an embedding model that takes objects in a 26-dimensionsal space and, by embedding, ...
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Visualization of p-adic numbers

I try to understand and get a feeling which gaps p-adic numbers fill to complete $\mathbb{Q}$. In the course of this I depicted (for $p = 2$) the "base" $\{p^k\}_{k\in\mathbb{Z}}$ with respect to ...
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What are all the functions that preserve the cross ratio?

Suppose a function $f:\mathbb {RP}^1\to \mathbb {RP}^1$ satisfy: $$ \left[f(a),f(b);f(c),f(d)\right]=\left[a,b;c,d\right] $$ for all $a,b,c,d \in \mathbb {RP}^1$. What can the function be in general? ...
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Intersection of ray with plane given in homogeneous coordinates

Given a plane $p\in\mathbb{P}^3$ expressed in homogeneous coordinates ($p_x$, $p_y$, $p_z$, $p_w$) a ray expressed as a source point $s\in\mathbb{P^3}$ expressed in homogeneous coordinates ($s_x$, $...
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Equivalent definitions of Dual of a variety.

Let $X$ be a closed irreducible projective variety of dimension $n$ in $\mathbb{P}_N.$ The dual of a projective variety is defined in following 2 ways. Def 1:If $X$ is nonsingular then the dual is the ...
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Hyperplane tangent to a projective variety.

Let $X$ be a projective variety in projective space $P_n$. Let $H$ be a hyperplane, we say $H$ is tangent to $X$ at $p$ if $T_p(X)+T_p(H)\neq T_p(\mathbb{P}_N).$ My question is to prove a hyperplane $...
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Informal definition of the real projective line $\mathbb{R}\mathbb{P}^1$

In his book Four pillars of geometry, John Stillwell defines informally (p. 107): "[The real projective line] is the set $\mathbb{R} \cup \{\infty\}$ together with all the linear fractional ...
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Projective classification of quadric from affine classification

Currently I'm taking a first course on Projective Geometry and I'm working on the following problem : Given a homogeneous polynomial $F$ of degree $2$ and $n+1$ variables we consider the quadric ...
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Homogeneous coordinate representation of a vertical line

Is there homogenous coordinate representation for a vertical line passing through an arbitrary point on the x axis (say C). Generally this is represented as: $x = C$ in euclidean geometry would ...
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Intersection of projective quadric and affine plane.

I'm stuck in trying to understand the graphical part of the following problem. Let $\mathcal C = \{ [x:y:w:z] \in \mathbb P ^3: x^2 +xy +yw -w^2 = 0 \}$. Graph the intersection $\mathcal C \cap \...
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What is the surface area an ellipse takes up on a sphere?

Take $\Sigma$ a $k\times k$ positive-definite real matrix and $E$ to be an associated ellipse: $$E:=\{(x_1,\dots, x_N): \frac{1}{N}\sum_n x_n^\dagger \Sigma x_n \leq 1\}.$$ Now take $z$ uniform on $\...
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If angles $A$, $B$, $C$ of convex quadrilateral $\square ABCD$ are equal, then $D$ lies on the Euler line of $\triangle ABC$

In a convex quadrilateral $ABCD$ angles at $A,B,C$ are equal. Prove that vertex $D$ lies on the Euler line of triangle $ABC$. My try: We can use complex numbers. Set circumcirle of triangle $ABC$ as ...
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Why is there only one point at infinity in the extended complex plane, but one in each direction in the real projective plane?

It is a question that my friends recently discussed. My opinion is that, by using one single point at infinity to form $\hat{\mathbb{C}}$, the behavior of functions such as $f(z)=z^2$ is just like ...
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Compute real length from perspective

Given the two green lines and the real distance between them (blue lines), is it possible to calculate where in the picture would be placed a point 4.5m away from a certain initial point (red line)? ...
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Conics and a group of projective maps

Let $C$, $D$ be conics in a projective plane, which have exactly four common points. Find four projective transformations which form a group $G$ (under composition) such that for all $\tau \in G$ we ...
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relationship between the point m and line in projective space

a 3-vector [a b c]T can be either a point in projective space of dimension 2(p2) or a line in P2. What is the relationship between the point m ~ [a b c]T and the line l ~ [a b c]T ?(explain) (Hint: ...
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Polynomial that vanishes identically on a line

Let $L \subset \mathbb{P}^n$ be a line. This is from Joe Harris's Algebraic Geometry, the paragraph right before exercise 1.3. ...it's not hard to see that a polynomial $F(Z)$ of degree $d - 1$ ...
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Projecting 8D dataset on onto skew coordinates

I have an 8 dimensional dataset (as an Nx8 matrix), and I am hypothesising that much of the dataset can be described simply by linear addition of two known non-orthogonal vectors (i.e. two 1x8 vectors)...
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What are the ranges of $x$ and $y$ in the Mercator projection?

What are the ranges of $x$ and $y$ in the Mercator projection ? I searched about it but I couldn't find anything. Any ideas? Thanks in advance
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Automorphism group of union of varieties

A projective hypersurface $\mathcal{V}(F)$, given by a homogeneous polynomial $F$, can always be expressed as the union of its affine components $\mathcal{V}(F_{i})$, where $F_{i}=F(x_{1},\ldots,x_{i-...