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

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Trying to proof that a projective variety is the Segre embedding of $\mathbb{P}^{1}\times \mathbb{P}^{3}$ in $\mathbb{P}^{7}$.

Let $I$ and $B$ be projective varieties and $$ p:I\rightarrow B $$ $$ q:I\rightarrow \mathbb{P}^{3} $$ be morphisms such that $p$ is bijective, $q$ is surjective and the fibers of $q$ are ...
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mean-deviation form, why orthogonal?

This is from my textbook Why the column of the new design matrix are orthogonal? for example, let say $A=\begin{pmatrix} 1& 1& 4\\ 1& 2& 0\\ 1& 3& 2 \end{pmatrix}$ ...
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Non-tangent lines to lines in $\mathbb{P}^3(\mathbb{C})$.

Let $\pi:\mathbb{C}^4\setminus\{0\}\to\mathbb{P}^3(\mathbb{C})$ be the quotient map. Let $Q\subset\mathbb{P}^3(\mathbb{C})$ be a smooth quadric, let $q$ be the quadratic form such that ...
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Segre embedding $\mathbb P^1\times\mathbb P^1\to\mathbb P^3$

Let $\Psi:\mathbb P^1\times\mathbb P^1\to\mathbb P^3$ be the map $$((x_0:x_1),(y_0:y_1))\mapsto (x_0y_0:x_0y_1:x_1y_0:x_1y_1)$$ and let $Q$ be the image of $\Psi$. I have shown that $Q$ is the zero ...
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cubic surface equation

If $[1,0,0,],[0,1,0],[0,0,1],[1,1,1],[1,3,2],[1,4,3]$ are six points on $P^2$ in general position and $f_0, f_1, f_2, f_3$ are the generators of the four dimensional vector space generated by cubics ...
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28 views

Projection from high dimension to lower, for visualization

I want to project high dimensional data points onto 2D screen coordinates, for visualization purposes. I want to be able to control the angles of projection manually (eg, with the mouse). I have ...
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33 views

Equation for a circle in homogeneous coordinates

The equation for a circle in homogeneous coordinates is given by $(x - aw)^2 + (y - bw)^2 = r^2w^2$. I understand that the center of the circle, given by (a, b) in euclidian space is given by (a, b, ...
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22 views

Geometric Significance that 2D Points Form a Line

I'm reading through Multiple View Geometry in Computer Vision, by Hartley and Zisserman, and on page 2 it is stated that points at infinity in the two-dimensional projective space form a line, and in ...
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21 views

Reflection is not a collineation

Could you give me a collineation that proves that you can't construct the reflection of a line $e$ across a parallel line $f$ with a straightedge only? (That is, a collineation that maps $e$ and $f$ ...
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Construction of a finite projective plane of order $p$, for any prime $p$

I have this construction of a finite projective plane (FPP) of prime order $p$, but I am not sure what's going on. We have already proved that FPPs of order $q$ have $q^2+q+1$ lines and points (if ...
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How large can a set of pairwise disjoint 2-(7,3,1) designs (Fano planes) be?

As wikipedia defines well, the Fano plane is a small symmetric block design, specifically a 2-(7,3,1)-design. The points of the design are the points of the plane, and the blocks of the design are the ...
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40 views

Definition of Finite Projective Plane clarification

I do not understand part iii. Why can't there be four collinear points? The Fano plane is an example of a $3$-uniform configuration. What about configurations that are $4$-uniform? You must ...
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Degree of maps $\mu \colon \mathbb{P}^1\rightarrow \mathbb{P}^r$

In the book I am reading right now, it is defined that for a map $\mu \colon \mathbb{P}^1\rightarrow \mathbb{P}^r$ the degree is the degree of the direct image cycle $\mu_{*}[\mathbb{P}^1]$. We are ...
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Projective Geometry: Combinatorially, but not projectively equivalent polytopes

I have a hard time understanding Projective Geometry. My task is to Find two polytopes, that are combinatorially, but not projectively equivalent. What combinatorially equivalent means is ...
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14 views

Drawing the reciprocal of a circle through the circle of inversion.

I have a general question about drawing the reciprocals of circles through the circle of reciprocation. I understand inversion and reciprocation are two entirely different things yet somehow ...
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The subgroup of $PGL(V)$ stabilizing a projective configuration

Let $P(V)$ be a projective space and consider the natural action of $G=PGL(V)$ on it. Let $S=\{p_1,\dots, p_k\}$ be a finite set of points in $V$ where $k\geq 2$. Is there any reference about the ...
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38 views

How many Fano Planes Can We Build with the Numbers from $1$ to $35$

The Fano plane is the finite projective plane of order 2, having the smallest possible number of points and lines, 7 each, with 3 points on every line and 3 lines through every point. Assume that ...
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An example calculated in Principles of Algebraic Geometry of Griffiths and Harris'.

On page 413 they write: Example. We can now make a second computation for Chern classes of projective space. Let $X_0, \ldots , X_n$ be linear coordinates on $\mathbb{C}^{n+1}$, and let ...
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54 views

Can the equation $x+y+z=1$ describe a sphere?

I know that in a three-dimensional Euclidean space, with the Euclidean distance, $x+y+z=1$ describes a plane. In the same conditions, $x^2+y^2+z^2=1$ would be a sphere (a 2-sphere to be exact). ...
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Background for reading Atiyah's first paper on the twisted cubic

What should an undergraduate know before being able to read Michael Atiyah's A Note on the Tangents of a Twisted Cubic ? Most of the words in the paper look foreign to me, but I'm very intrigued by ...
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17 views

Point, Line Duality

I am currently studying Projective Geometry and have trouble understanding the point-line duality concept. Why is the cross product of two points a line and the cross product of two lines a point?
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35 views

Projectivised tangent bundle of 2 sphere

I'm trying to understand how rotations act on the "projectivised" tangent bundle of the sphere. Let $S^2$ be the two sphere and denote by $P(TS^2)$ the tangent bundle where each tangent space ...
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Proper name of a curve from “Vanishing Surfaces”

My question is maybe more about linguistics than maths... So, if you have a 3D surface that vanishes as a curve when projected on a 2D-plane (e.g. an axisymmetric surface projected to the r-Z plane). ...
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55 views

Computing the properties of the 3D-projection of an ellipse.

I have an ellipse that is rotated around the white axis (see image below) in 3-dimensional space by an angle α. The axis passes through the perimeter and one of the ...
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Can I always replace a 4D - 3D - 2D projection with a 4D-2D projection?

When visualizing a tesseract, we usually use a 3D projection of it. Then the computer screen projects the 3D structure into a 2D image. Is it always possible to replace these two steps with a single ...
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1answer
47 views

How many non-isomorphic Fano planes exist?

The Fano plane is the finite projective plane of order 2, having the smallest possible number of points and lines, 7 each, with 3 points on every line and 3 lines through every point. So, What i want ...
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43 views

Smooth curve of genus $1$ in $\mathbb{P}_{\mathbb{C}}^1\times \mathbb{P}_{\mathbb{C}}^1$.

This question comes from Gathmann's notes of Algebraic Geometry: Show that $$\{((x_0:x_1),(y_0:y_1)): (x_0^2+x_1^2)(y_0^2+y_1^2)=x_0x_1y_0y_1\}\subset \mathbb{P}_{\mathbb{C}}^1\times ...
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56 views

Intuitive explanation for the connection between Lie Groups and projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$

In this post John Baez states that the classical simple Lie groups "arise naturally as symmetry groups of projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$". Is there some ...
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For any four distinct points $P_{1},P_{2},P_{3},P_{4}\in P^{1}(\mathbb{R})$ we have $(P_{1}P_{3};P_{2}P_{4})=1-(P_{1}P_{2};P_{3}P_{4})$.

Show that the cross-ratio has the following property: for any four distinct points $P_{1},P_{2},P_{3},P_{4}\in P^{1}(\mathbb{R})$ we have $(P_{1}P_{3};P_{2}P_{4})=1-(P_{1}P_{2};P_{3}P_{4})$. What is ...
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Which of the following quadratic forms defines a non-singular conic?

Which of the following quadratic forms defines a non-singular conic? (1). $x_{0}^{2}-2x_{0}x_{1}+4x_{0}x_{2}-8x_{1}^{2}+2x_{1}x_{2}+4x_{2}^{2}$. (2). $x_{0}^{2}-4x_{0}x_{1}+x_{1}^{2}-2x_{0}x_{2}$. ...
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Show that a projective transformation is unique.

Find the projective transformation $\tau \left ( \left [ 0,0,1 \right ] \right )=\left [ 0,1,0 \right ], \tau ([0,1,0])=[0,1,1],\tau ([1,0,0])=[1,1,1], \tau ([1,1,2])=[1,1,0]$. And show that such a ...
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30 views

Line clipping in 2D perspective transformation

Situation I have two 2D spaces which are related one to other by a transformation matrix - 3*3 homography matrix for homogeneous coordinates: The first space is "map" and the second one is "camera ...
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Quadratic equations defining the $10$-dimensional spinor variety.

Let $S$ be the $10$-dimensional Spinor variety parametrizing one of the two families of $4$-dimensional linear subspaces of the non-singular quadric in $\mathbb{P}^{9}$. I have read that there exist ...
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finding equivalent hyperovals

If $H=D(x^k)$ is a hyperoval, then $D(x^t)$ is a hyperoval equvalent to $H$ for $t=1/k$, $1-k$, $1/(1-k)$, $k/(1-k)$ and $(k-1)/k$. If I consider the Segre Hyperoval $D(x^6)$ with $q = 32 = 2^5$, how ...
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Question regarding Geometric meaning of Noether normalization theorem for projective varieties

In Ernst Kunz's ''commutative algebra and algebraic geometry'' book, ch.2, proposition 4.5 the author states: Let $K$ be an algebraically closed field, $V\subset \mathbb{P}^n(K)$ a variety of ...
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Projecting a 3D point to a fisheye plane

I am trying to calculate if a point in 3D space is in front of my fisheye camera, so looking at the OpenCV documentation (I'm not actually using OpenCV, however), we have: $a = x/z$, $b = y/z$, $r^2 ...
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34 views

Difference between Grassmann and Projective space?

I recently read the first few chapters of Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling and I am confused about ...
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29 views

Differential of the Gauss map of an algebraic variety.

Let $X=V(F)\subset\mathbb{P}^{n}$ be a smooth irreducible hypersurface. Let us consider the morphism $$ \mathcal{G}:X\rightarrow \mathbb{P}^{N}, p\mapsto \left( \frac{\partial F}{\partial ...
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The same algebraic variety defined by different sets of polynomials

Let $\emptyset\neq X\subset\mathbb{P}^{n}$ be an algebraic variety such that $$ X=V(F_{1},\ldots,F_{m}) $$ for certain linearly independent homogeneous polynomials $F_{1},\ldots,F_{m}\in ...
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Motivating the Cross-Ratio and 'the ratio of ratio's' in $\mathbb{R}\mathbb{P}^2$

Trying to come across the idea of the cross ratio naturally by thinking about the projective plane $\mathbb{R} \mathbb{P}^2$, using ideas from Brannan's Geometry book: given 4 collinear points ...
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Question regarding projective coordinate transformation

While reading Kunz's commutative algebra book, I came across a statement I can't understand. First, let me define the notations. Let $L/K$ be extension of fields, and let $\mathbb{P}^n (L)$ denote ...
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Projectivity maps that fix three points

Please check this definition: A projectivity is a bijection $PV\to PW$ induced by an isomophism $\phi: V\to W$ given by $\phi(kv)=k\phi (v)$. Now, i have seen here Old Question that an answer says ...
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How are non-homogenous elliptic curves projective varieties?

So if I am given an elliptic curve such as $Y^2Z=X^3$ then I see how it can be realized as the projective variety $Proj(k[X,Y,Z]/(Y^2Z-X^3))$. But, given an elliptic curve like $Y^2 = X^3 + X$, then ...
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19 views

Point Selection for Cross Ratio

The cross ratio relates the positions of four co-linear points in 3d space. I understand definitions a published online, such as that from wikipedia: https://en.wikipedia.org/wiki/Cross-ratio ...
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29 views

A Deeper Understanding / Interpretation of Homographies

I currently understand that a homography matrix, which allows for a mapping between planes in 3-dimensions, is a $3\times3$ matrix of the following general form: $$\begin{bmatrix} \vert & \vert ...
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28 views

Passing light through airfold blades as a function of angle?

Consider an airfoil blade sunshade (see the diagram below). It consists of multiple flat rectangular shapes which are parallel to each other but make an angle ($v$) to the horizontal plane. The blades ...
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Line Equations $(a, b, 0)$ and $(0, 0, 1)$

I am learning about projective geometry in my machine perception class. I am struggling to fully grasp the concept of lines at infinity, and I am hoping someone would be able to shed some light on ...
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Intersection of a line and line at infinity in projective space

I understand parallel lines in Euclidean space intersect at the line at infinity in terms of projective space. My question is for a single line. A single line if extended to infinity must intersect ...
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Intersection of two circles in projective space

I have checked the existing question Intersection of two circles. and model for intersection of two circles in the complex projective plane - I do not think either of these answers my question. The ...
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23 views

Hyperplanes without Axiom of Choice

For any projective space that contains more than one point, is it possible to prove that it contains a hyperplane without using the Axiom of Choice? It's easy enough to prove that there exists a ...