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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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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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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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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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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1answer
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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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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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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2answers
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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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17 views

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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32 views

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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2answers
159 views

Why are degenerate conics not projectively equivalent to nondegenerate conics?

This is what I understand about conics being projectively equivalent. Two conics $C1=V(F)$ and $C2=V(G)$ are projectively equivalent if there is an invertible matrix $A$ such that $F(X,Y,Z)=0$ iff ...
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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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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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39 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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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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11 views

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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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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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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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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17 views

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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42 views

The associated Schubert variety of a flag of subspaces of a vector space.

Let $V$ be a vector space and $W_1 \subsetneq W_2 \subsetneq ... \subsetneq W_\ell \subsetneq V $ a flag of subspaces. The associated Schubert variety is defined as : $ \Omega ( W_{ \bullet } ) = \{ ...
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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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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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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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107 views

What is the relationship between a complex manifold being Kähler, projective, nonprojective, and nonKähler?

I was wondering if this implication is true. I read a few places that $$\text{nonprojective} \Longrightarrow \text{nonKähler}$$ but I think I maybe have misunderstood. Equivalently, this is of course ...
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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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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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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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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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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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Help to understand the proof of the Riemann Mumford relation

Here i post a file where from page 617 to 618 there is the proof of the Riemann mumford relation that is the theorem 1.13. My problem is to understand the beginning of that proof. In particular ...
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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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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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63 views

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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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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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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19 views

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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41 views

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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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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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 ...