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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Parameterisatio of Curve in projective space
Background: I've started reading Miles' "Undergraduate Algebraic Geometry" (Link) recently though struggling a lot.
I'm stuck at processing the following paragraph...
Sec. 1.7 ...
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Example of Line at infinity in Miles' Book
Background:
I've started reading Miles' "Undergraduate Algebraic Geometry" (Link) recently though struggling a lot. \
Please allow me to just paste the example on Page 23 as follows;
I ...
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Tools to investigate unusual algebraic structure
I will begin with a mostly motivational thought about the projective plane. In this plane, every two lines intersect at a singular point. Let's mark the lines set as $\mathcal{L}$ and the points set ...
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Prove that there is no holomorphic map $s: \mathbb{P}^n \to \mathbb{C}^{n+1} \setminus \{0\}$ with $p \circ s = \operatorname{id}$.
Let $p:\mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{P}^n$ be the natural projection. Prove that there is no holomorphic map $s: \mathbb{P}^n \to \mathbb{C}^{n+1} \setminus \{0\}$ with $p \circ s = \...
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Determine the involution when given two pairs of point on a line
I'm studying about involution on a projective line (line with point at infinity). An involution is a map from a projective line $l$ to itself that satisfied $f \circ f =$ is the identity map.
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Mathematical representation of projection plane
I've gotten the following assignment, given this specification of a screen I need to project a video onto, I want the video to appear flat.
My idea was to find a parametric equation describing this ...
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Intersection of *segments* in 3D using projective geometric algebra?
There is a relatively common technique to find the intersection of two segments in 3D, which can be found in page 304 of Graphics Gems
Note that a segment is a compact subset of a line.
The technique ...
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Show collinear points in 3d project to collinear points in 2d
Let $A,B,C$ be collinear points in 3d. Show that their projections $a,b,c$ respectively, onto 2d image plane are also collinear. If $A=[x_1,y_1,z_1], B=[x_2,y_2,z_2], C=[x_3,y_3,z_3]$ then $a=[x_1/z_1,...
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Maximal number of multiple points for an irreducible quartic
I was working on this problem, but I don't see how I can solve it. I was given a hint, but I don't know how to use it. Can anyone help me? Thanks in advance!
Let $f \in \Bbb C[x_0, x_1, x_2]$ be an ...
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Unique quadric though three disjoint lines
I read "Undergraduate algebraic geometry" by Miles Reid. I have a question about exercise 7.2 in it.
Exercise states that for any $3$ disjoint lines $l_1, l_2, l_3 \in \mathbb{P}^3(\mathbb{C}...
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Determine the ray which passes through the center of a sphere given its projection on the image plane.
I am not a mathematician, but I know that a sphere projected on the image plane becomes an ellipse under perspective transformation.
I don't know whether the ray starting from the center of projection ...
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Convex angles of (n) straight lines intersected at a point. [closed]
How many convex angles, not including straight angles, are formed by n straight lines that intersect in one point?
Example: A convex angle is an angle between zero and 180.
If 2 lines intersect at a ...
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What are projection transformations?
As much as I know a projection transformation is the mathematical conversion of a map from one projected coordinate system to another, generally used to integrate maps from two or more projected ...
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Orthogonal Projection Area of a Zonogonal Cylinder
A zonogon is a convex polygon that is made up of pairs of parallel line segments that are congruent. A zonogonal cylinder is a cylinder with identical and identically aligned zonogons as the end caps. ...
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What's the equivalent for spheres to homogeneous coordinates for projective spaces?
Projective or homogeneous coördinates are an extremely useful parameterization of projective spaces (indeed often used to define them!), but they are redundant -- a projective space of dimension $n$ ...
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Tangents to a projective plane curve.
I'm studying William Fulton's "Algebraic Curves," and I'm currently studying projective plane curves. One of the problems in the section asks us to find the tangents to $xy^4+yz^4+xz^4$ at ...
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Help understanding what is the projection in problem II.6.3 (a) Hartshorne
In algebraic geometry by Robin Hartshorne, exercise II.6.3.a is written as follows
Cones. In this exercise, we compare the class group of a projective variety $V$ to the class group of its cone (I, ...
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Problems about inversion map
Let $\tau_{\omega}$ be the inversion with respect to circle $\omega$ and $\omega$ is the unit circle in $\mathbb{E}^2$ with centre $O=(0,0)$. There are three cases:
a. Find $\tau_{\omega}(\ell)$ where ...
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Nonlinear Coordinate Transform via Intersections?
I've defined a sort of 'warp' procedure for 2d shapes, and I'm curious whether it's familiar within the math canon or even has a name. Given its simplicity I'm sure that there's a formal definition ...
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Geometric interpretation of dual Variety.
While I was reading the book, Discriminants, Resultants, and Multidimensional Determinants by Gelfand, Kapranov & Zelevinsky, I came upon the following statement:
Let $X \subseteq \mathbb{P}^n$ be ...
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Segre Embedding and lines on $x^2+y^2+z^2=1$ over $\mathbb{C}$
Let $x^2+y^2+z^2=1$ be the unit sphere over $\mathbb{C}$. Prove that the sphere has 2 rulings by straight lines.
I have learnt the Segre embedding $\mathbb{P}^1(\mathbb{C})\times \mathbb{P}^1(\mathbb{...
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2D different integer valued vertices coordinates of cube projection
On paper we, orthogonal, project a cube as in provided image. But is it always really a cube? And, in this particular example, the 8 different 2D coordinates of the vertices have integer values. But ...
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On Marsden's 'Introduction to Mechanics and Symmetry' Exercise 5.3-4. (fubini study form is closed)
In exercise 5.3-4. in Marsden's book I'm asked to
prove that $\mathbf d \Omega^{fs} = 0$ on $\mathbb P \mathcal H$ directly,
where $\mathbb P \mathcal H$ is an arbitrary projective Hilbert space (...
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Projective Geometry Interpretation In combinatorics
Find the maximum number of subsets that satisfy this trait
~ Each Subset has 4 element
~ Each two subset share 2 elements in common
~ There can't be more than 1 number that are included in all subsets
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Orthogonal Projection Area of a 3D Cuboid
This problem is asking the same as this problem, but is a cuboid instead of a cube and the independant variables are the roll, pitch, and yaw. I wrote some Mathematica code that finds the area ...
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Looking for a 5-point quotient akin to the cross-ratio.
Very sorry for the vague question.
What am looking for is a function $f:l^5 \rightarrow \mathbb{R}$ where $l $ is a line containing points (including the point at infinity of course).
Let $f(A, B, C, ...
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Why is a degenerate quadric well defined?
Let $K$ be a field with $\text{char}(K) \neq 2$ and $Q$ be a quadric in the projective space $P(K^n)$. Let $M$ be a symmetric $n×n$-matrix over $K$ such that $Q \leftrightarrow x^T M x$.
In my course ...
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projective transformation that preserves angle at a point
I try to find all projective transformations on $\mathbb RP^2$ that fix the point $[0,0,1]$, and preserve the angle between any two lines through $[0,0,1]$.
Using SageMath to calculate Jacobian I ...
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Do fixed lines of projective transformations contain a fixed point?
This post for linear transformations provides counterexample $(x,y)\mapsto(x+y,y)$, but the projective transformation $[x,y,z]\mapsto[x+y,y,z]$ does have a fixed point $[1,0,0]$ on any fixed line. Is ...
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Example of Open sets in Projective Plane
In my self-study of Algebraic geometry, I came across the following argument;
The Projective plane $\mathbb{P}^2$ comes with a standard Affine cover consisting of the three open subsets;
$\mathcal{U}...
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Determine and classify the projective conics
Determine and classify the projective conic that contain the points:
$\langle 0,0,1\rangle,\langle 0,1,1\rangle,\langle 1,0,1\rangle,\langle 1,1,1\rangle,\langle 1 / 2,2,1 \rangle$.
I used this ...
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Homogeneous coordinates in projective geometry
I am studying Projective Geometry in 3D for Computer Vision. I am confused on the high-level rationale behind our need to map from heterogeneous to homogeneous coordinates, and I would like to confirm ...
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On implicit equations of rational quadratic Bézier curves
Rational quadratic Bézier curve with control points $\boldsymbol{B}_0 = [x_0 : y_0: w_0], \boldsymbol{B}_1 = [x_1 : y_1: w_1], \boldsymbol{B}_2 = [x_2 : y_2: w_2]$
in homogeneous coordinates of $\...
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Is It using basis change? Which is the way?
Let be $F : P^4(R)\rightarrow P^4(R)$ an homography with the matrix A. (P is the projective space)
$A = [[1,1,0,0,0], [0,1,1,0,0], [0,0,1,0,0], [0,0,1,2,1],[1,0,0,0,1]]$
I have the plane $Z : x_2 = ...
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Is $K_F\cdot C\leq K_X\cdot C$ for a fibre $F\subseteq X$ containing the curve $C$?
I have put this question on Math Overflow
Let $X$ be a projective $\mathbb{Q}$-factorial variety ("variety" is irreducible and reduced over a field of characteristic zero; not necessarily ...
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Perspective sequence that maps collinear points A,B,C,D to D,C,B,A
Find the perspective sequence that maps collinear points A,B,C,D to D,C,B,A.
Attempt: If we need to find a sequence of three perspectives that (A,B,C)->(A,C,B), where A, B, C are collinear, then ...
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Inverse of Zenithal projection
Zenithal perspective projections are generated from a point P and carried through the sphere to the plane of projection as illustrated in the figure below.
By a simple geometric relationship, we can ...
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Perspective warp evaluation as a convex combination of 4 sample points
Suppose we know that $F:\mathbb{R}^2 \to \mathbb{R}^2$ is a perspective warp function; that is,
there exist real parameters $a,b,c,d,e,f,g,h,i$ such that, for all $(x,y)$:
$$
F((x,y)) = ((ax + by +...
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Different types of multiplicity in algebraic geometry
I'm following a course on algebraic geometry. We have seen different kinds of "multiplicity" and I don't understand the difference between them. My course is in dutch, so I will try to ...
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Prove Newtons trident is an algebraic curve
I was working on a problem where I have two curves $C_1 \longleftrightarrow x^2-y=0$ and $C_2 \longleftrightarrow xy-1=0$ in $\mathbb{A}^2$ (so a parabola and a hyperbola). Now consider the connection ...
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Generic fiber of the projection $V(x_0y_0^p + x_1y_1^p+x_2y_2^p) \subseteq \mathbb{P}_k^2 \times_k \mathbb{P}^2_k \rightarrow \mathbb{P}^2_k$
Let $k$ be an algebraically closed field of characteristic p and $f=x_0y_0^p + x_1y_1^p+x_2y_2^p$. Let $X=V(f) \subseteq \mathbb{P}_k^2 \times_k \mathbb{P}^2_k$, embedded as $X=V(x_0^{p-1} f,x_1^{p-1}...
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Basic Projective Plane Question
So the axioms for a projective plane are given by:
Any two “points” are contained in a unique “line.”
Any two “lines” contain a unique “point.”
There exist four “points”, no three of which are in a “...
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Pinhole camera projection of 3D multivariate Gaussian
Consider a 3D Gaussian with $3\times 1$ mean $\boldsymbol \mu$ and $3\times 3$ covariance $\boldsymbol \Sigma$ (which is symmetric positive semidefinite):
$$
p(\mathbf x) = \frac{1}{\sqrt{\...
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Euler's line : a projective geometry proof
In a given triangle $ABC$, let
$G$ be the common point to the three medians,
$H$ be the common point to the three altitudes, and
$M$ be the common point to the perpendicular bissectors of the three ...
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Intersection of an arbitrary number of projective varieties
Let $X_1,...,X_r\subseteq \mathbb{P}^n$ be projective varieties of dimensions $i_1,...,i_r$. Are there any criteria to determine if $X_1\cap ... \cap X_r\neq \varnothing$?
I know that if I have $X_1,...
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Learning roadmap from Euclidean geometry to Projective geometry
As we know that Euclidean geometry deals with triangles, quadrilateral,... and circles. Later in college we learn projective geometry but in quite modern language (vector spaces, equivalent relations)....
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Finding the dimensions of a box (cuboid) given a hexagon filled in to look like the box
Suppose I have hexagons that like the ones below but I know the area and the points of each hexagon that represent a cuboid of dimensions g,h,d. How can I find the values for g, h, and d? Any pointers ...
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Family of conics generated by two conics
Let $a,b$ be conics as shown in the graph below, let $K$ be a point on $a$ and draw the two tangents of $b$ from it, intersecting $a$ at $L,M$ respectively. Then, draw tangents of $b$ from $L,M$, ...
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Project picture frame on a wall with known height but unknow length
I'm trying to project a picture frame on a wall using using the css property matrix3D.
The idea is: we draw the four corners of the wall on any room photo and the picture frame is projected on the ...
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Proving $\mathbb{P}^2-\{[0:0:1]\}$ is not affine
I'm assuming the field to be algebrically closed
I know that there are other questions on this site that address the problem in the title, but the answers are just hints like: "you have to prove ...
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