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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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Can I find the absolute positions of a calibrated camera by taking different photos of the same scene?

Suppose that I have a pinhole camera which has been calibrated and I know that its calibration matrix is $$K=\begin{pmatrix} f_x & 0 &p_x \\0 & f_y&p_y \\ 0 & 0 & 1 \end{...
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Perspective projection plane, calculating squares on the plane

Let's say I have a road I'm looking at from the top, have a square on it. Then I have a different location from which I look at the road, the square now is a convex quadrilateral. https://i.imgur.com/...
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Show that these 3 projective lines intersect on the same point

I'm stuck with the following problem of projective geometry from an assignment: Let $P_1$, $P_2$, $P_3$ and $Q$ be four points in the projective plane (over an algebraically closed field) such that ...
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Classification of Complex Surfaces. Illustrations of the failure of genus to provide a good classification.

Suppose that $X$ is a complex projective curve, i.e., a compact Riemann surface. In this case, a very useful invariant of such objects is the genus, a topological property of the manifold which ...
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Unique projective maps preserving lines

Let $\mathbb{P}(V)$ be a projective space of dimension $3$ and let $L_i$, $i=1,2,3$ be pairwise non-intersecting lines in $\mathbb{P}(V)$. If $\phi: L_1 \to L_1$ is a projective transformation, prove ...
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Intriguing geometry problem regarding isogonal lines

A line $r$ contains the points $A,B,C,D$ in this order. Let $P\notin r$ such that $$\angle APB=\angle CPD$$ Denote furthermore by $G$ the intersection of the angle bisector of $\angle APD$ and $r$. ...
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How to find the equation of an image under a central projection

Let $\pi:\mathbb{P}^3 \to V(x_2) \cong \mathbb{P}^2$ the linear projection with center $P =(0:1:0:0)$. Find the equation for the image of $C=\{(s^3:s^2t:st^2:t^3)|~(s:t) \in \mathbb{P}^1 \}$ under $\...
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Image Analysis using cross ratios

I'm stuck trying to solve an exercise regarding an image analysis. Consider a book that measures 16 cm $\times$ 24 cm lying on a table. Let the vertices of the book be denoted by A,B,C,D and the ...
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Why does the real projective plane / Boy surface look like this?

In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901 My question is, you can see that the Boy surface is made up of three ...
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Determining the matrix of a projection

I have a question about the correctness of my ideas regarding the following exercise. Define $A_0=[(1,0,0,0)]$ $B_0=[(0,1,0,0)]$ $A_1=[(0,0,1,0)]$ $B_1=[(0,0,0,1)] \in \...
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Equivalent Definitions of Lines in Projective Space

I’ve been working with two definitions of lines in $\mathbb{P}_\mathbb{R}^2$, and tried to show their equivalence. The first is that, given two points $a=(a_0:a_1:a_2)$ and $b=(b_0:b_1:b_2)$, the ...
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Project point onto surface using yaw, pitch, roll rotation and standard trig formulas

Given the known coordinates of location A (0,0,50), at which we have a laser pointer aimed downward, I need a series of equations that calculate the projected point onto the flat surface below (x,y,0) ...
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Find a projective change of coordinates

Find a projective change of coordinates that takes the projective completion of the circumference C: $x^2 + y^2 = 1$ to the projective completion of the parabola P: $y^2=2px$, $p \geq 0$ (i.e. $...
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Structure of a projective transformation group which leaves a circle and a point on the circle unchanged.

$G$ is a group of projection transformation on real projective plane. Is $G$ a finite group? And is $G$ a abelian group? This question may make the nature of derivative more clear.
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Projective-invariant differential operator

This question has been cross-posted to MathOverflow. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = 0 \...
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Desingularisation of curves (Lorenzini-Invitiation to Arithmetic Geometry, chap 6,ex 7)

Given a nonsingular complete curve over algebraically closed $\bar{k}$, which is interpreted as a field $\bar{k}(X)$ of transcendence degree 1 and its set of valuations trivial on $\bar{k}$, we may ...
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Do all projections matrices take this form?

Do all projection matrices take the form $P = A{(A^TA)}^{-1}A^T$? If so, can you help me derive it and explain it intuitively?
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What transformations can be set by projecting a straight line onto a straight line

What transformations can be set by projecting a straight line onto a straight line (without adding an infinitely distant point)? I said that the homothety with coefficient $k \neq 1$ and the ...
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Non-classical examples for generalized quadrangles

I have been told that there is no non-classical example for n={6,8} known yet for quadrangles. Could you share some study on it?
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A composition of projections with three fixed points — is it necessarily the identity?

We are given a line $l$. The line is mapped onto itself through a series of projections that involve other lines and -- importantly! -- conics. In the end, points $A$, $B$, and $C$ on $l$ appear to be ...
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Expected value and projection of a normal random variable onto a linear span…?

I just wanted to clarify a part of a proof which used the fact a random variable has zero mean. Suppose $X, Z_{s_1},\dots,Z_{s_n}$ are all jointly normal random variables for all $s_i \leq t$ and $n \...
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Proof that a Steiner's conic is a projective conic

In class, I've defined a projective conic on $\mathbb{P}(V)$ as an element of $\mathbb{P}(Q(V))$, where $V$ is a $K$-vector space and $Q(V)$ is the field of quadratic forms on $V$. I need to prove the ...
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Ratio of “proportional to” equations equaling each other

I'm reading this paper and came across this section, which I don't completely understand. Equations (19) and (20) are "proportional to" equations, where $LHS = k RHS$. Equation (22) is formed using ...
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What are some sources to learn projective geometry?

I am looking for notes, books etc. for learning projective geometry for getting started with elliptic curves.
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Orbits of pentagon under projective transformation

The original question is: What are the $PGL(3,\mathbb{R})$ orbits on the set of pentagons (in $PG(2,\mathbb{R})$)? In the first part of the exercise we showed that $PGL(3,\mathbb{R})$ acts ...
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Homogeneous coordinates and frame change

I have a transformation T described in a frame A by an homogeneous matrix (of size 4*4). I would like to express T in another frame B. How can I do that ? Thank you.
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On an exercise in section 4 of Chapter I from Hartshorne's Algebraic Geometry

It is about exercise 4.9: Let $X$ be a projective variety of dimension $r$ in $\mathbb{P}^n$ with $n\geq r+2$. Show that for suitable choice of $P \notin X$ and a linear $\mathbb{P}^{n-1}\subseteq \...
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Have these Generalised Fermat Curves been studied before?

While trying to solve a problem in number theory, I recently came across the concept of Fermat curves. This is the set of points in the complex projective plane, defined by $X^n+Y^n=Z^n$. In the ...
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Why is the eigenvector of the smallest eigenvalue of this matrix gives the intersection of a set of lines in homogenous coordinates?

Consider $L = \{l_1 \cdots l_n\}$ be a set of lines in the plane in homogeneous coordinates. In this definition, a line $ax+by+c = 0$ is given by its direction $l^T = (a/c,b/c,1)^T$ such that it ...
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Identifying the Plane at Infinity in the World Necessitates Determining the Affine Geometry of the World?

Page 18 of my computer vision textbook, Multiple View Geometry in Computer Vision (Second Edition), by Hartley and Zisserman, states the following: 1.8 Auto Calibration $\vdots$ Generally ...
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Degree and codimension of nondegenerate varieties

A projective variety $X \subset \mathbb{P}^r$ (i.e. reduced, irreducible closed subscheme) is called nonegenerate, if it is not contained in any hypersurface $H \subset \mathbb{P}^r$. Equivalently, ...
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Plot projective curves

Do you know any website, app or program that can plot curves in the projective plane? I know the projective plane $\mathbb{P}^2 \mathbb{R}$ can be visualized as a sphere with the antipodal points ...
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Find a projective change of coordinates that maps the (projective) line L to the (projective) line G

Let $L$, $G \subset \mathbb{P}^2$ be lines. Show that there exists a projective change of coordinates $T$, such that $T(L)=G$. This is how we defined a projective change of coordinates in $\mathbb{P}...
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What is a projective range?

Wikipedia says a projective range is the dual of a pencil of lines on a point. Surely that means: all points on a line. But Wikipedia also says a projective range may be a projective line or a conic....
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Example for intersection, multiplicity and tangents for projective plane curves

We recently got introduced to projective plane curves in our class, however we just defined it and never really talked about examples. And when I am trying to come up with a just any curve, ...
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Prove that a harmonic homology preserves the conic

I came across a question in the book by Judith N. Cederberg and I’m learning about projective geometry. One of the question was “Show that a harmonic homology whose centre and axis are pole and ...
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Determine the polarities of a self polar triangle

Consider a triangle $PQR$, $P(0,2,1), Q(1,0,2), R(0,4,9)$. Determined the polarities if triangle $PQR$ is self polar. By definition of self polar triangle, point $P$ gets mapped to line $QR$, point $...
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Inverse of the adjugate operation

In projective geometry, the map between a primal and dual quadric is the adjugate: $adj(Q) = Q^*$. The map from dual to primal is then the inverse adjugate, $Q = adj^{-1}(Q^*)$, as in this paper. ...
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“line at infinity” in projective plane

("Algebraic Geometry: A Problem Solving Approach" by Thomas Garrity) I am struggling with Exercise 1.4.12.1 in the above, which I quote with some context: Here is my intuitive thinking: (a) lines ...
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Why is a pole-polar relation preserved under mappings?

If $P$ and $p$ are pole and polar with respect to a polarity with matrix $C$; then $P’$ and $p’$, their images under a collineation, will be pole and polar with respect to the polarity with matrix $C’$...
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Proving that a point is the result of only two lines intersecting and a line is the result of only two points being aligned

Let $S_0$ be a set of four points in the real projective plane such that any three points of $S_0$ are not aligned. Let $L_0 := \emptyset$. For every integer $n \ge 1$, we define the following: ...
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Projective subspace of same dimension

I'm looking at families of curves in $\mathbb{P}^2$ (over $\mathbb{C}$), specifically the set $\mathcal{L}_d$ of projective curves defined by a homogeneous polynomial $P \in \mathbb{C}[x_0,x_1,x_2]$ ...
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Projective/ Finite Geometric Basics!

I'm taking intro to coding theory and am having some trouble understanding the basics of Projective Geometry, since our text does not give it much discussion. Namely, if PG(r-1,q) is the set of all ...
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Fourth point of intersection of two conics

Five points in general position define a unique conic section. Let $Q_1$ be a conic through points $A,B,C,E_1,F_1$ and likewise $Q_2$ through $A,B,C,E_2,F_2$. Two conics (over an algebraically closed ...
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How to check for a projective space?

When I have a division ring commutative its pretty straight forward! But when its not commutative then I'm stuck, it's possible that in this case its not a projective plane? Can someone give a ...
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Any Names for Cubes of more than 4-dimensions?

So, when you look at shapes that are projected into dimensions higher than 3, the most famous example from what I've seen is the cube. The n-dimensional hypercube has been theorized in higher ...
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If All the Points Lie On a Plane, Then Why Does the Linear Mapping Reduce to …?

I previously asked a question with regards to what the matrix $\mathrm{H}_{3 \times 3}$ is/represents in the following textbook excerpt: In applying projective geometry to the imaging process, it ...
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Motivation for 2D homogeneous coordinates: ratio always determinate

I'm working through an old textbook called Algebraic Projective Geometry, by Semple and Kneebone. Early in the text, the authors write: When we introduced complex points (on p. 12) we explained ...
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Notation for (skew) Young Tableaux in Hubsch

I'm looking at the notation given for Young tableau in Hubsch's book "Calabi-Yau Manifolds: A Bestiary for Physicists" - see Chapter 3, p. 96. A Young tableau (for a $U(n)$ representation) is denoted ...
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Projective Transformations: “If all the points lie on a plane, then the linear mapping reduces to …”

Page 7 of my computer vision textbook, Multiple View Geometry in Computer Vision, says the following: In applying projective geometry to the imaging process, it is customary to model the world as a ...