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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Nonsingular conics are isomorphic over algebraically closed field

Let $f$ be any irreducible quadratic polynomial in $k[x,y]$, where $k$ is an algebraically closed field, and let $W$ be the conic defined by $f$. Show that the coordinate ring $A(W)$ is isomorphic to $...
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What new geometries can be made from different projections?

The starting point of projective geometry is the idea that one could do geometry viewed while viewing the world with perspective. This means that parallel lines seem to converge at the horizon and ...
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Projection of $S^3$ on $S^5$ in their fiber bundle

I'm working on some physics research involving the topology of the group $SU(3)$, which has long been known to be a non-trivial fiber bundle of the unit spheres $S^5 \times S^3$, with $S^3$ the fibers....
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Interpolation of a 2D segment using its projection

Consider the following diagram: The blue segment is projected on the orange projection screen from a specific point of view. The projection is shown at the bottom of the image. About the blue segment ...
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Polynomial Maps of Real Projective Varieties

As a preface I don't much background in this area and I think I am dealing with older (or currently non-standard) definitions. Definition: A polynomial $f \in \mathbb{R}[x_1, \ldots, x_n]$ is ...
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Finding intersection point of two lines in projective space

I am trying to get into projective geometry, and here is one task which I find hard and don't really know how to approach. Let be $$L_1:= \alpha_1 x_0 + \beta_1 x_1 + \gamma_1 x_2 $$ $$L_2:= \alpha_2 ...
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Relationship between quotients of rotations of centrally symmetric polytopes into $\mathbf{RP}^2$?

According to Wikipedia, the hemicube $H$ is an abstract polytope with 3 faces, 4 vertices, and 6 edges, which may be realized as a tesselation of $\mathbf{RP}^2$ as a quotient of the cube with a ...
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Example of Bezout's theorem and common component [closed]

What is common component Bezout's theorem in $\mathbb{P}^n$? I looked for definition, but I don't understand. I would like to have some example nie I can calculate it. Then, with this example I would ...
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Intersection of projective plane with projective line in projective space

In a projective space is the intersection of a projective line and a projective plane a projective point or a projective line. Now I am looking for examples for both cases. In case it is a projective ...
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What are the homogeneous coordinates of the point at infinity on elliptic curve?

Elliptic curve in a projective plane has a structure of a group where the point at infinity serves as the identity element. However, I have seen different definitions of this point, so I would ...
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Desingularization of the standard Cremona involution of $\mathbb P^2$.

Consider the birational map $\chi$ given by the blow up of the points $(1:0:0)$, $(0:1:0)$, and $(0:0:1)$ of $\mathbb P^2$, followed by the contraction of the strict transforms of the three lines ...
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How can I find a proportional length on a perpendicular line from a perspective?

I am working on a project where I am mapping images of buildings onto 3D meshes and I want the dimensions of the mesh to be proportional to the building in the image. I have six points for a given ...
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Question about oblique and orthographic projections

The question in case of TL;DR: Given an extremely long distance (or focal length), would the image generated of a given building or buildings (or indeed of any real world object) always specifically ...
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Line through a point and two non-intersecting lines in a projective space

I have just read an introduction chapter to projective geometry and am trying to solve a few of the problems that are listed in my book. Quickly, I stumbled upon something that I cannot solve: Suppose ...
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Reconstruct Quadric from Projections on Plane Conics

Background I have two projection images of an ellipsoid in 3d-space. My goal is to reconstruct the quadric given the parametrized projected conic outlines (ellipses). I am trying to recreate the ...
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Projective completion of an affine plane

I know that a infinite point in a projective space can be defined as a equivalence class of $||$ relation in affine space. So if i have an affine space $(\mathcal{A},\mathcal{D}, \phi)$, let $\mathcal{...
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Number of lines on a singular cubic surface

A smooth cubic surface contains 27 lines, but a singular cubic surface with rational double points contains fewer lines. Question: Why is the number of lines equal to $\binom{8-r}{2}+n-1$, where $r$ ...
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Finding a square on four lines

There are four lines A, B, C, D in 3D space, they all go through (0,0,0). There is a point Ma on a line A. I need to find points Mb, Mc, Md on lines B, C, D, such that Ma, Mb, Mc, Md forms a square in ...
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A curve that intersects $2x^3y^3-30x^2y^2+11xy^3+2x^3-38x^2y+20xy^2-13y^3+16x^2+94xy+10y^2+301x-668y+662$ with multiplicity $3$ or more

let $C=2x^3y^3-30x^2y^2+11xy^3+2x^3-38x^2y+20xy^2-13y^3+16x^2+94xy+10y^2+301x-668y+662$ be a curve in $\mathbb{C^2}$. Find a conic that intersects $C$ in $(2,2)$ with intersection multiplicity $\geq 3$...
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multiplicity of the intersections between $p=x_0x_1^2+x_1x_2^2+x_2x_0^2\,\,$ and $\,\,q=-8(x_0^3+x_1^3+x_2^3)+24x_0x_1x_2$ in $\mathbb{P^2(K)}$

in $\mathbb{P^2(K)}$ where $\mathbb{K}$ is an algebraically closed field and $[x_0,x_1,x_2]$ the homogeneous coordinates, consider the following (homogeneous) polynomials: $p=x_0x_1^2+x_1x_2^2+x_2x_0^...
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Question about projective plane of order three

How can find in the projective plane of order 3, the number of subsets of three points so that these three points are not on the same line?
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3D Rotation as a Projective Transformation

I have a 8-parameter projective transformation, which maps a unit square into a quadrilateral on the plane (green). Now, I want to "rotate the image by 120 degrees in 3D around the local Y axis&...
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Why does eyes initially on the horizon line always remain on the horizon line when moving on a ground without elevation?

In this video by Youtuber Love Life Drawing , it's shown objects having their eyes along the horizon line initially still have their eyes on the horizon line when they get closer to the camera. Photos:...
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How do we project a coordinate in 4-dimensions to that of a 3-dimensional coordinate?

So essentially what I'm asking is this: so let's say that I have a set of points that I arrange into a sphere. Now, with those same points I arrange them into that of a hypersphere. How do I convert ...
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plane as a 4D anti-vector and the distance from the origin

So, we can get a plane in 4D the following way using the wedge product - p ^ q ^ r. It encompasses information on both the normal to the plane and the distance to ...
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Does tautological line bundle $\mathcal{O}(-1)$ over complex projective space have zero section?

In Complex Geometry written by Huybrechts, it defines the tautological line bundle $\mathcal{O}(-1)$ as followed $$ \mathcal{O}(-1):=\{(\ell,z)\in \mathbb{C}P^n \times \mathbb{C}^{n+1}: z\in \ell\} $$ ...
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Projective Basis Explanation, Visualisation

I have trouble understanding the geometric idea behind a projective basis. So for example in $\mathbb{P}^2(K)$ we have $[1:0:0],[0:1:0],[0:0:1], [1:1:1]$ but why? I know that $$ [1:0:0] = k\cdot \vec{...
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Mathematically, why does moving the vanishing points correspond to rotations of a 3D figure in space?

I was watching this video on how vanishing point changes as object rotates, in it, it is shown that as we move the vanishing points of the family of extended cube sides , the new configuration of ...
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Straight line in projective plane is plane in vector space (Road to Reality , page-343)

How do we construct an n-dimensional projective space $P^n$? The most immediate way is to take an $(n + 1)$-dimensional vector space $V^{n+1}$, and regard our space $P^n$ as the space of the 1-...
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Non singular point of a projective plane curve

In problem 5.1 of Fulton's Algebraic curves, we're asked to show that a point $P\in\mathbb P^2$, $P=[P_1:P_2:P_3]$ is multiple iff $F(P)=F_X(P)=F_Y(P)=F_Z(P)=0$. Here $P$ is said to be multiple if $...
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Matrix powers up to multiplicative factor

Let $A$ be a real $n\times n$ matrix, $A_n = A^n$, and $$ \bar A_n = \lbrace\alpha A_n, \alpha\in \mathbb{R}\rbrace.$$ I am interested in characterizing the behavior of $\bar A_n$ when $n\rightarrow \...
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Understanding projective plane conceptually (page-342, Road to Reality by Roger Penrose)

In the above picture, I am a bit confused how it turns out parallel lines seems to meet in the artist's potrait. Could someone explain in simple words why the roads which don't intersect in the ...
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Order of Vanishing on Projective Variety

I know for a curve $C$ in affine space, we can define a the order of vanishing at a smooth point $p \in C $by noting that $\mathcal{O}_p$, the local ring at $p$, is a DVR with maximal ideal $\mathfrak{...
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Geometric Configurations: Product identity

Wikipedia states that A configuration in the plane is denoted by $\left(p_{\gamma} \ell_{\pi}\right)$, where $p$ is the number of points, $\ell$ the number of lines, $\gamma$ the number of lines per ...
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Projective Mapping of translated Quadric

Question I have an ellipsoid described by a covariance matrix $\Sigma \in \mathbb{R}^{3\times3}$ and its centroid location $\mu \in \mathbb{R}^{3}$ (yes; notation from statistics as I want to ...
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Non-singular Conics are equivalent to some projective line.

The proof begins on pg.22. I am having trouble understanding why ${B|}_{W_y} \not\equiv 0$. Since I couldn't understand this part of the proof (on pg. 24 of the slides), I was trying to show it ...
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A question regarding to the Wikipedia definition of homography (or projective transformation).

According to the section Definition and expression in homogeneous coordinates of this Wikipedia article, we get \begin{align} y_1 &= \frac{a_{1,0} + a_{1,1}x_1 +\dots + a_{1,n}x_n}{a_{0,0} + a_{0,...
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Automorphism group of the projective line $\mathbb{P}^1(\mathbb{Z})$ [duplicate]

Consider ``the'' projective line over the integers (which can come in various guises). I have two questions: What is the scheme-theoretic automorphism group of the scheme $\texttt{Proj}(\mathbb{Z}[x, ...
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Induced collinearities

Let $\tau$ be a field homomorphism unequal to the identity. Show that the induced affinity on $\mathbb{A}^2(K)$ is not a translation or homothety. Show that the induced collinearity on $\mathbb{P}^2(K)...
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Graph of a linear map in projective spaces

Let $f:E\to F$ be a linear map that we may consider injective at first. I wonder whether there exists a concept of its "projective graph", that is the locus $Z\subset \mathbb{P}(E)\times \...
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A degree of an embedding and a self-intersection of a hyperplane section

Let $X$ be a surface inside $\mathbb{P}^3$ of degree $d$, that is given as $f=0$ where $f(x, y, z, w)$ is a homogenous polynomial of degree $d$. Let $\tilde{H}$ be a class of a hyperplane section in $\...
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Find a rational parametrization of an affine conic section

Question: construct a rational parametrization of an affine conic $$-12x^2 - 44xy -65y^2+10y-1=0.$$ My ideas: say $y = t(x+1)$ and substitute into equation.
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Find fixed point of involution of complex projective line

Question: Find fixed points of involution g: $P_1(C) -> P_1(C)$, $g^2$ = Id, if g(2/3) = 3 and g(-2/3)= 1/4 My ideas: to use cross-ratio, maybe we can say g(3) = 2/3 and g(1/4) = -2/3 so we can ...
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How to pass from finite to infinite in PG(1,3)

I am searching an operation allowing to pass from finite to infinite in $PG(1,3)$. 
I explain:
 In $PG(1,2)$ over the Boolean field $B = \{0, 1\}$, using homogeneous coordinates, you can write: $(1,0) ...
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Vakil's proof of Bertini's theorem (Theorem 12.4.2) - where do the linear conditions on the fiber come from?

As I'm reading through Vakil's notes on algebraic geometry, I'm stumbling over some of the details in his proof of Bertini's theorem. The proof in question is discussed already on this site (1, 2 and ...
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3 votes
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Projective transformations

I am studying computer aided geometry and I have a background in mathematics. For me a (real) projective transformation is a map $f: \mathbb{RP}^n \to \mathbb{RP}^n $ induced by a linear isomorphism $...
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Rigorously working with flat limits: lines meeting a curve by specialization

I am trying to get comfortable with flat limits. This question is motivated by Section 3.5.3 of Eisenbud and Harris's '3264 And All That' and Exercises 3.35 and 3.36. This section and the surrounding ...
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Rational Bézier Curves are projectively invariant

I want to prove that a Rational Bézier Curve is not only affine invariant but also a projective invariant. By affine invariance i mean that applying an affine map to the curve is the same as applying ...
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Calibrating a pinhole camera (finding $z_0$)

A pinhole camera is a very simple theoretical device for generating perspective images on a plane that a distance $z_0$ from the pinhole (a point) and whose normal vector is the direction vector at ...
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Locating vertices of a known triangle in $3D$ from a single image

Suppose you have a labelled triangle with known side lengths, and you take one image of this triangle using a known pinhole camera (i.e. the focal length is known), from a point with known coordinates,...
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