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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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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 [duplicate]

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-...
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Shadow terminator plane in a bowl

When sunlight casts a shadow on an axi-symmetric inside surface we have a $ plane$ determined by rim boundary $B$ projection $E$ and mid centers $C$ on rim that divides illuminated (white) and dark ...
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Calculate a location on a square, on square's projection on a plane

I have the projection of a square on a plane. I know it's four corners' coordinates on said plane. (This is from a picture of a square taken by a camera at an unknown relation to the square.) I now ...
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Explicit homeomorphism between $\mathbb{S^2}$ and $\mathbb{P^1(C)}$

I know that $\mathbb{P^1(C)} \cong \mathbb{P^1(C)} \cup \{N\} $, where $N$ is the north-pole of the sphere, is homeomorphic to the sphere $S^2$ thanks to the stereographic projection, but I am not ...
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How to find the coordinates of the fourth vertex of a square in a projective perspective

If I suppose that I know the coordinate of three points (eg. A'B'C') in a 2D coordinate system of A'B'C'D', my question is how ...
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Converting $X, Y$ on plane to world $X, Y, Z$ coordinates

I working on program for displaying 3-dimensional objects on the computer screen using Ray Tracing method. And now I came across a problem with throwing rays from the orthogonal camera. So, as you ...
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What is an improper affine space at infinity? Schouten: Tensor Analysis for Physicists

Edit to add: This clearly deals with projective geometry. If I figure it out well enough to post a satisfactory answer, I will do so. If someone else posts a decent answer before then, I will be ...
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Other intersection of common tangents

Core question Suppose I have two conics, $A$ and $B$. In general these have four common tangents. Let $p$ be the point of intersection of two of these tangents, and $q$ the intersection of the other ...
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Making plane parallel to $xy$ axis

I want to rotate plane: $2x + 4y + 2z + 4 = 0$ to make it parallel to $xy$ axis by using rotation matrix. In specific, how to calculate angles of rotation? I can't find any good explanation on the web,...
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Geometric interpretation of the tensor product of two projectif spaces $\mathbb{R} P^n \otimes \mathbb{R} P^n$ [duplicate]

We define a Projective space of a vector space as follow : http://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_vector_spaces Given a vector space $V$ over a field $\mathbb{K}$, its ...
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Projective Transformation between two matched 3D point sets

I have two point sets of about 800 3D-points that are matched. That means, I know the corresponding points in the two point clouds. Now I want to minimize the distance between the corresponding points ...
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Some basic questions regarding varieties in biprojective space (product of two $\mathbb{P}^m$'s)

I am just learning about product of projective spaces and I have some basic questions I would like to figure out. I will be working with $\mathbb{P}^m \times \mathbb{P}^m$. And by bihomogeneous form I ...
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Degrees of Freedom in Affine Transformation and Homogeneous Transformation

I understand that a 2D Affine Transformation has 6 DOF and a 2D Homogeneous Transformation has 8 DOF. However, how can I identify what those independent paramters are? If we consider Euclidean ...
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Projective cubic curve passing through seven points in $\mathbb{C}^2$

Let $x_1,...,x_7$ be distinct points in $\mathbb{C}^2$. Prove that there exists a cubic curve passing through these points which has a singularity at the point $x_1$. My attempt so far... A related ...
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Subset of C2 such that any vanishing cubic polynomial is reducible [closed]

I'm looking for a subset of $ \mathbb C^2$ consisting of four points such that any polynomial of degree 3 vanishing on this set is reducible. Don't really know where to start, any ideas on how to ...
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Smooth quartic surface in $\mathbb{P}^{3}$ that contains a smooth curve of genus 2 and degree 6.

I am reading through an article by Matsumura and Monsky on Automorphisms of Hypersurfaces in which they state that there exist quartic surfaces is $\mathbb{P}^{3}$ which have infinite automorphism ...
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dimension tangent space equals 1+dimension variety

I'm reading some lecture notes (which I won't link because they're not in English), where there's this statement (without proof): Suppose we have $X\subset\mathbb{P}(\mathbb{C}^n)$ variety which is ...
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Asymptotes of an implicit curve

Per the method described at How to find asymptotes of implicit function? , I proceeded to find the asymptotes of $$ x^3 + 3x^2y - 4y^3 - x + y + 3 = 0 $$ Whilst, it correctly generates the asymptote $...
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Transversal intersection of complex projective varieties

Let $X$ and $Y$ be two complex projective varieties, which intersect transversally at $x$. We can also think of X and Y as smooth manifolds. Now when considered as smooth manifolds do they intersect ...
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Probability of chosing an affine point in a projective space

Consider a projective space $\mathbb{P}^n(k)$ of dimension $n > 0$ over an infinite field $k$. Let $\Delta$ be a hyperplane in $\mathbb{P}^n(k)$. Assume that we chose a point $P$ "at random" (we ...
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Projecting boundary constraints to lower dimensions / Interior point of a n-D parallelepiped

I need to solve an estimation problem of the form $Ax = b$ ($A\in \mathbb{R}^{m\times n}, m < n$) where I have a lower dimensional projection $b$ and need to compute the best guess of $x$ within ...
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Internal ellipse formed by vertices of a pentagonal star

$(A,C,D,E)$ are four fixed points on a fixed outer (red) ellipse and $B$ is a variable point lying between $A$ and $C$. Alternate vertices are joined to form a pentagonal star. (drawn on Geogebra). A ...
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Secant variety of 2-Veronese embedding

I want to prove that $\sigma_s(\nu_2(\mathbb{P}^n))$ has not the expected dimension; where $\sigma_s$ is the $s$-secant variety, and $\nu_2$ is the 2-Veronese embedding. I tried to prove it by myself ...
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The relevance of Pappus' and Desargues' theorems

Pappus' and Desargues' theorems are two notable theorems in projective/affine geometry. I am trying to understand their relevance and significance in the context of (projective) geometry. ...
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Fermat Point of a Tetrahedron

Here's a curious set of vertices for a tetrahedron: {{-22, -25, 4}, {-12, 15, -6}, {8, 5, -6}, {18, -15, 24}} The Fermat point of a tetrahedron minimizes the total distances from the point to the ...
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Instrinsic Camera Parameters

I am trying to understand intrinsic camera parameters. Specifically I not able to understand the skew factor and pixel scaling derivation in it. Example: Most of the literature explanation starts ...
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contact geometry good reference

I am working on which seems to be a classical result : the isomorphism between the cotangent bundle of projective space (with zero section removed) and the cotangent bundle of its dual projective ...
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Does a projection matrix mean $P=P^T$?

I know that a matrix $P$ is a projection matrix IFF $P=P^2$, but is it also true that it must be such that $P=P^T$? I thought that was only true for orthogonal projection matrices. But i am ...
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Projecting a surface on another surface

I would like to know if there is a method, and what is called the area of mathematics that studies this kind of things, for projecting surfaces on other surfaces. Example: suppose to have a half-...
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1answer
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Determine lines intersecting four skew lines in $\mathbb{P}^3$

Let $l_1, l_2, l_3, l_4$ be four skew lines in a projective space $\mathbb{P}^3$ (meaning $l_i \cap l_j = \varnothing \;\forall i≠j$). Let $R = \{ r : r \cap l_i ≠ \varnothing,\;i=1,...,4 \}$ be ...
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Projection of a plane onto a surface of sphere

What is the effect on the plane equation of projecting a plane onto the surface of a sphere. Assume we are starting with the standard point-normal plane equation: $$ax + by + cz + d = 0$$ or ...
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1answer
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Projection of circular motion on a perpendicular plane

First of all, I think the following question is more of a Mathematical question than physics question, so I am asking in math stackexchange. Assume that a particle is rotating in $x$-$y$ plane about ...
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Two skew lines and two lines in \mathbb{P}^3

Given two lines $r, r' \in \mathbb{P}^3$ such that $r \cap r' = \varnothing$ and two other lines $s, s'$ such that $A=r \cap s, B=r \cap s', C = r' \cap s, D = r' \cap s', A≠B≠C≠D$. Show that $A, ...
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Does identifying opposite points in Euclidean space result in a smooth manifold?

Taking Euclidean space $\mathbb{R}^n$ and identifying all pairs of points $\{\mathbf{x}, -\mathbf{x}\}$ results in a topological quotient space $\mathbb{R}^n/\mathbb{Z}_2$. Is this quotient space a ...
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Proof that quadrics of signature $(2, 2, 0)$ in $\mathbb{R}\mathbb{P}^3$ is homeomorphic to torus

I know that a quadrics of signature $(2, 2, 0)$ in $\mathbb{R}\mathbb{P}^3$ is homeomorphic to a torus. However, I know of no simple proof of this fact. I have heard that one proof uses the fact that ...
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$G(x_1,\dots, x_n)$ homogeneous, degree $d$ over algebraically closed field $K$. $[c_i,d_i]\in P^1_K$, then $G(c_iu+d_iv)\in K[u,v]$ is degree $d$?

$G(x_1,\dots, x_n)\in K[x_1,\dots, x_n]$ is a homogeneous of degree $d$ polynomial over algebraically closed field $K$. $[c_i,d_i]\in P^1_K$ where $P^1_K$ is indicating $c_i,d_i$ cannot be ...
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Construct the 4th point if we know the cross ratio!

Given 3 collinear points A, B,C. Find D (by constructing) if R(ABCD)=1.7/3 (projective geometry). Can someone help me with this problem.
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Projective space definitions

My questions are as follows: Are all these different definitions of projective space equivalent? For example, Bezout's theorem holds under all 4 definitions (with an appropriate change in ...
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2answers
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Definition of dimension of a projective variety

The title is quite self-explanatory: I'm an undergraduate maths student who hasn't attended (yet) a course in Algebraic Geometry (but I did a course on commutative algebra), however I have to write a ...
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How is this geometrical invariant called in English?

I am studying affine and projective geometry and I have encountered some invariant: the cross ratio, which in Italia is called "birapporto" and another one which I do not know the name of in English. ...