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

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Prerequisite of Projective Geometry for Algebraic Geometry

I studied Euclidean Geometry in high-school, and I have not studied anything relates to geometry since I started studying in university. I am now intending to study Algebraic Geometry, however, I ...
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How to calculate the dimension of the intersection of projection of varieties?

In $\mathbb P_{n+1}$ we consider $d$ varieties $V_1,\ldots V_d$, each of them is defined by $d-1$ equations, as follows: we have $d-1$ polynomials $f_2(X_0,\ldots,X_n),\ldots,f_d(X_0,\ldots,X_n)$ and ...
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Existence of a special set of q+2 points in the finite affine plane over $\mathbb F_q$

I am working in the finite affine plane over $\mathbb F_q$ with $q=2^n$. Such a plane has $q^2$ points, $q^2+q$ lines, each line has $q$ points, and by a point is passing $q+1$ lines. There are $q+1$ ...
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Harris, Exercise 10.28 (weighted projective spaces)

So I recently started teaching myself about weighted projective spaces from Harris' Algebraic geometry. It was going well until I came across this exercise, which has me stumped: "Show that any ...
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89 views

Describing $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as gluing of affine charts

How can we describe $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as a gluing of affine charts? I'm having trouble with this problem, perhaps because I don't understand ...
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Properties of divisors when moving from char 0 to char p.

Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...
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202 views

Hirzebruch surfaces

How can I express the 2nd Hirzebruch surface, $F_{2}$ in terms of $SO(3)$. Is it true that $F_{2}$ is the total space of a bundle with fibre SO(3) over $\mathbb{R}_{+}$?
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A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
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134 views

Irreducible closed subsets of projective varieties

I want to prove the following lemma: Let $X \subset \mathbb{P}^n$ be a projective variety. Let $W \subset X$ be a closed irreducible set. Then $W$ is also a projective variety. My idea is as ...
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Geometric interpretation of a certain property of graded rings

Let $k$ be a field, and $A$ a (commutative associative unital) $k$-algebra. It is known that there is a natural isomorphism between ‘regular’ (in the sense of regular maps of varieties) actions of ...
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129 views

Projecting Projective Curves

I've been stuck for quite a while on what is probably a trivial problem. Let $X\subset\mathbb{P}^n$ be a smooth projective curve, and let $$\mathcal{I}=\{(p,q,r):p,q\in X,p\neq ...
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The geometry of $\operatorname{PSO}(4)$ and the quaternions

Question: Given a twist of the projective space, how do I find unit quaternions that represent it? Backgroud and what do I mean: Following Conway & Smith's On Quaternions and Octonions, every ...
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A question from “Foundations of Projective Geometry” by Hartshorne.

"Foundations of Projective Geometry" by Hartshorne says the following: The completion of the affine plane of four points is a projective plane with 7 points. The affine plane of $4$ points is ...
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20 views

Theorem about two quadrilaterals with parallel edges

I'm looking for a name for the following theorem: If $abAB$ lie on one line and $cdCD$ lie on another line, and furthermore $ac\Vert AC,ad\Vert AD,bc\Vert BC$, then $bd\Vert BD$. One can ...
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128 views

Projection and Pseudocontraction on Hilbert space

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
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28 views

Projection from a point to a plane - confused about terminology.

Edit: It seems rude to delete the question, but I have my answer now thanks to rghthndsd. I'm a bit unsure about the terminology in a homework question I'm doing, and I can't find any clear answers ...
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Projectivizing a group: how to go from AGL(n,K) to PGL(n+1,K)

$ \newcommand{\GL}{\operatorname{GL}} \newcommand{\AGL}{\operatorname{AGL}} \newcommand{\PGL}{\operatorname{PGL}} $Given an irreducible matrix group $G_{\infty,0} \leq \GL(n,K)$, I form the group ...
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Projective Geometry Question.

I want to learn projective gemetry. I have the coxeter book and found some youtube videos. I have just started. I have tried answering this Challege question from the video, and so far. I have ...
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209 views

Good textbook on geometries

I am looking for a good textbook that thoroughly covers euclidean, affine, projective and non-euclidean geometries. I will be starting graduate school in algebraic geometry next year and I would like ...
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52 views

Finding the equations of a variety under a projection map

Suppose I have a projective variety $X$ in $\mathbb{P}^N$ ($N >2$, say) defined as the zero set of some homogeneous polynomials $f_1, \ldots, f_r$. Consider the projection map $[x_0: x_1 : \cdots ...
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Are there in $(\mathbb{C}[x,y,z]/(x^3+y^3+z^3))_{x}$ exactly $12$ lines?

Let $R=\mathbb{C}[x,y,z]/(x^3+y^3+z^3)$ be the coordinate ring of the affine variety defined by the equation $x^3+y^3+z^3=0$. We can consider the localization in the element $x$, denoted by $R_x$. I ...
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Configuration analogues of projective spaces?

In a configuration, each point is incident to the same number of lines and each line is incident to the same number of points. The Fano plane is a configuration, with 3 points on each line, and 3 ...
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How can I compute the multiplicities of the projection map over a smooth projective plane curve?

Suppose that $F$ is a non-singular homogeneous polynomial in $\mathbb{C}^3$ and let $X$ be its zero locus, which is well-defined in $\mathbb{P}^2$. Consider the function $\pi:X\rightarrow\mathbb{P}^1$ ...
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Do we have homogeneous coordinates for probabilities?

As a roboticist, implementing visual odometry on a robot, homogeneous coordinates are convenient for projections of a non-moving object on an image sensor at $t$ and $t+1$ to estimate its position, ...
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51 views

Isomorphism of $P(V)$ and $P(V^*)$

Let $V$ be a finite-dimensional left vector space over a division ring $K$, and let $V^*$ the dual right vector space (consisting of all linear functions from $V$ to $K$). We can (and will) treat ...
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185 views

There is some intuitive idea of Pascal's 's theorem in Projective Geometry?

In projective geometry, Pascal's theorem (formulated by Blaise Pascal when he was 16 years old) determines that a hexagon inscribed in a conic, the lines that contain the opposite sides intersect in ...
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127 views

Intersection of a cone $x^2+y^2-z^2$ and a generic plane in $\mathbb{RP}^3$

If we take the zero locus of $x^2+y^2-z^2$ to be our cone, I'd like to know how to go about finding the intersection of the cone and a generic plane $Ax+By+Cz+Dw=0$. The result will be a conic, but ...
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Solving some geometry problems using involutions

Some geometry problems ( like this and this ) have short solutions if we use involutions. What references are there for solving geometry problems using involutions? I am particularly interested in ...
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On existence of a tangent line passing through a given point

Question Suppose $k$ is an algebraically closed field of characteristic $0$, and $C\subseteq\mathbb P^2(k)$ is an irreducible projective plane curve of degree $n>1$, and $P$ is a point on $\mathbb ...
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line at infinity

I tried solving the following question, could you have a look at my answer and tell me whether it's right or wrong? All input is appreciated. Question: Let $ABCD$ be the vertexs of a parallelogram in ...
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classify antiholomorphic involutions of projective space

On $\mathbb{CP}^1$ with Fubini-Study metric, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\overline w, \overline z] \qquad \eta ...
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Intuition behind using projective geometry for defining the addition on an elliptic curve

We already had the chord-and-tangent construction that can be used to define a way of "adding" points on an elliptic curve. Also this addition satisfies all the group laws. Still why one needs to ...
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Problem about $\mathbb{P}^3(K)$

Show that four skew lines in $\mathbb{P}^3$ have two transversals in common. I know that exist a quadric which contains three of the four lines....but i'm stuck EDIT: If the skew lines are ...
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Is it possible to transform projections?

From linear algebra I know how to transform a point or a direction from one space to another using transformation matrices (and I think it's pretty awesome by the way). But what about projections? Is ...
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Plücker coordinates of the Clifford parallels

Let $$q=\cos\theta+(x_q\textbf{i}+y_q\textbf{j}+z_q\textbf{k})\sin\theta$$ be a unit quaternion parameterised by $\theta\in\mathbb{R}$, where $(x_q,y_q,z_q)$ is fixed and $x_q^2+y_q^2+z_q^2=1$, and ...
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Quotient of a proj variety by an involution

Usually, if you have an affine variety defined by some equations and have an involution on it, it's quite easy to immediately see what the equations of the quotient of the variety by the involution ...
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Projective homologies

In a projective plane (i.e. two-dimensional) $\mathbb P$, we call a general homology a projective transformation $h:\mathbb P\to\mathbb P$ such that $h$ has a line of fixed points $L$ called the axis ...
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Classification of Projective transformation according to Jordan form

Say we have the projective space $\mathbb P^2_{\mathbb R}$ = $\mathbb P(\mathbb R^3) \stackrel {\text{def.}}{=} \{\text{span(u)}\mid u\in\mathbb R^3\smallsetminus\{0\}\}$. Denote $[u]$ for an element ...
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Question 1.1 in Projective Geometry by Coxeter

Thus projective geometry deals with triangles, quadrangles, and so on, but not with right-angled triangles, paralleograms, and so on. -Projective Geometry, Coxeter pg 3. The first question of ...
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Method of determining dimensions from photographs of multiple angles and degrees of perspective/parallax for a math newbie

I have a project that begins with some 300+ reference photos of a scale model. The only measurements I am certain of are the overall length, and the linear length of one element of one part of the ...
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36 views

Blowup of $\mathbb{P}^3$ along the ideal $(w^3 + x^3 + y^3 + z^3, w^4 + \alpha wxyz)$ for fixed $\alpha \in k$

I want to compute the blowup of $\mathbb{P}^3$ along the ideal $(w^3 + x^3 + y^3 + z^3, w^4 + \alpha wxyz)$ for fixed $\alpha \in k$. I've been working with blowups for a couple weeks, but this seems ...
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Projective transformation of a projective line is a homography

Let $K$ be a field. Prove that projective transformation of a projective line determined by an affine line $K$ is the same as homography. This means that I need to prove that it has the form: $x ...
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Fibers of $V(ax_0^2+bx_1x_2)$ in $\mathbb{P}^1\times \mathbb{P}^2$

What are the fibers of $Z=V(ax_0^2+bx_1x_2)$ for $(a:b)\in \mathbb{P}^1, (x_0:x_1,x_2)\in \mathbb{P}^2$? If we fix $(a:b)$ or $(x_0:x_1,x_2)$ and dehomogenize, we can see that $Z$ is a family of ...
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Derive 3D point array from multiple 2D projections of same point array

Let's assume that we have an array of $n$ 3D points, we don't know their coordinates (thus we have $3n$ indeterminate scalar values). We also have $m$ 2D projections with known coordinates (thus we ...
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67 views

Isomorphism between quotients of linear groups

Suppose that $n$ is even. Is it true that $$\mathrm{SL}_n (\mathbb{R})/\{ \pm I \} \times \{\pm 1\} \cong \mathrm{GL}_n(\mathbb{R})/{\sim}$$ where $A\sim B$ if and only if $A=aB$ for some $a\in ...
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89 views

Transversal and complete intersection of hypersurfaces in $\mathbb{P}^{n}$

(a) Let $k<n$ and $F_{1},\dots,F_{k}$ be homogeneous polynomials of degrees $d_{1},\dots,d_{k}$ of $n+1$ variables in generic case. Prove that the corresponding hypersurfaces in ...
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Mapping an object's projected 3D path to a pre-defined top-down 2D path.

The title of the question may be misleading and the context simpler. Please suggest more appropriate tags for this question. Consider looking at a plane from two different perspectives. Perspective ...
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80 views

Mathematical Basis for Dimetric Projection

For a school project, I need to make a program that can plot $y = f(x,z)$ using a form of dimetric projection. I was given the projection formulae $$\begin{align*} x' &= x + sz\cos(\theta)\\ y' ...
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viewing ray geometry - with multiple aerial photographs

I am working with multiple aerial images. My idea is to model 3d objects (only upper parts). I am having known orientation parameters. As I am new to this field so that, I want to clarify few general ...