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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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Finding the Transform matrix from 4 projected points (with Javascript)

I'm working on a project using Chrome - JS and Webkit 3D CSS3 transform matrix. The final goal is to create a tool for artistic projects using projectors and animation - somewhat far away from using ...
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1answer
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How to calculate true lengths from perspective projection?

Suppose that I have a single point perspective drawing like . and suppose also that I know some of the real horizontal distances and distances along lines converging to vanishing point. E.g if i know ...
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The intersection of two parallel lines

My teacher told me that two parallel lines have a point of intersection, it is called Point at infinity. But I I can't understand how it can be true or how was it proved, can someone explain this to ...
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Transforming $2D$ outline into $3D$ plane

I am writing a program where I would like to allow the user to draw 4 connecting lines, such as: And convert this shape into a 3D plane. Is this possible? Is there an existing algorithm to do so? If ...
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1answer
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Homography between ellipses

This is a spin-off from a comment on Stack Overflow. How can I find a homography between two ellipses in the plane?
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5answers
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Help understanding Algebraic Geometry

I while ago I started reading Hartshorne's Algebraic Geometry and it almost immediately felt like I hit a brick wall. I have some experience with category theory and abstract algebra but not with ...
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1answer
374 views

Cross-ratio relations

The way I define the cross-ratio in projectve geometry: Let $P_0,P_1,P_2,P_3$ being four points on a projective line G, such that $P_0,P_1,P_2$ are pairwise distinct. Let $\pi:\mathbb KP^1\rightarrow ...
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Difference between Projective Geometry and Affine Geometry

I recently started reading the book Multiple View Geometry by Hartley and Zisserman. In the first chapter I came across the following concepts.. Projective geometry is an extension of Euclidean ...
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1answer
308 views

How can I visualize a four-dimensional point inside a Schlegel diagram of a tesseract?

I would like to draw a Schlegel diagram of a tesseract to visualize via a Cartesian coordinate system inside the tesseract the symmetry of some four-dimensional points located in a range of integer ...
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1answer
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Visualizing tuples $(a,b,x,y)$ of the extended Euclidean algorithm in a four-dimensional tesseract. Are there hidden symmetries?

I am trying to visualize the possible symmetries in the Euclidean four-dimensional space of the $4$-tuples of points $(a,b,x,y)$ generated by the extended Euclidean algorithm, where $ax+by=gcd(a,b)$. ...
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1answer
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Rectify image from congruent planar shape objects

I am implementing an algorithm to remove projective distortions on the following image. I understand this is possible by applying the following transformation: $$ \begin{matrix} 1 &...
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1answer
111 views

Projective Co-ordinate Geometry

I am learning projective geometry in my computer vision course. So, we represent a co-ordinate point in an image as a homogeneous co-ordinate as $(x,y,1)$. My professor says that if we are given two ...
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1answer
283 views

Relationship of aspect ratio to the homography matrices between a rectangle and an arbitrary quadrilateral

I've been reading everything I can on the perspective mapping between a 2D rectangle and the projection onto the plane in 3D space of a rectangle. I've learned that any such quadrilateral resulting ...
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How do you create projective plane out of a finite field?

I have heard and read unclear mentions of links between projective planes and finite fields. Is it possible to construct a projective plane (or a Steiner system) starting out with a field? Could you, ...
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Plücker Relations

Let $K$ be a field, $1 \leq d \leq n$ integers and $V$ an $n$-dimensional vector space. The Plücker relations are quadratic forms on $\wedge^d V$ whose zero set is exactly the set of decomposable ...
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1answer
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2D Coordinates of Projection of 3D Vector onto 2D Plane

The plane $P$ is passing through the origin and has normal $n$. $u$ is a 3D vector and $u'$ its projection onto $P$: $u' = u - \langle u,n \rangle n$ (assuming $n$ has unit length). $e'_1$ and $e'_2$ ...
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1answer
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Showing positive definiteness in the projection of ellipsoid

I would like to show that the projection onto the $xoy$ plane of the centered ellipsoid given by the definition $$\mathbf{x'}\mathbf{A}\mathbf{x}=1$$ where we have a positive definite $$\mathbf{A}= \...
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2answers
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Projectivization and stereographic projection, or: Why nonlinear + nonlinear = linear?

This previous question had me thinking about something I've taken for granted. Consider $\mathrm{SL}_2(\mathbb{C})$ acting on $\mathbb{C}^2$, which descends to an action on the complex projective line ...
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homeomorphism between the real projective line and a circle

I'm currently following an introductory course in geometry and it was mentioned that the real projective line is homeomorphic to a circle. Could someone please state the topologies on both the real ...
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1answer
539 views

Resolution of Singularities: Base Point

Consider the curve $y^2=4x^3-ax-b$, where $a$ is a fixed constant and $b$ is a free constant. For each value of $b$ we get a family of curves. Part 1: Show that the family of curves intersect at ...
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2answers
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Back-projecting Pixel to 3D Rays in World Coordinates using PseudoInverse Method

For perspective projection with given camera matrices and rotation and translation we can compute the 2D pixel coordinate of a 3D point. using the projection matrix, $$ P = K [R | t] $$ where $K$ ...
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2answers
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Geometric interpretation of the Logarithm (in $\mathbb{R}$)

(Note: limited to $\mathbb{R}$) (Note: Geometric here means with straightedge and compass) Standard approaches to introducing the concept of Logarithm rely on a previous exposition of the ...
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1answer
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Problem in Deducing Perspective Projection Matrix

I understand the traditional way(use similar triangle and make depth value linear) to deduce the perspective projection matrix. But I want to try another approach after I read this text: Fundamentals ...
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1answer
615 views

How to prove the parallel projection of an ellipsoid is an ellipse?

Take the following ellipsoid in implicit form as an example: $$x^2 + 2 y^2 + 3 z^2 + x y + y z - 2 xz = 5$$ which shows: The parallel projection of the ellipsoid onto $xoy$ coordinate plane can be ...
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0answers
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Properties of quadrilaterals resulting from perspective projection of rectangle of known aspect ratio

Given list of quadrilaterals (from image processing), are there any properties/calculations that can filter out those that are not perspective projection of a target rectangle of specific aspect ratio?...
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2answers
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Show that the line $KD$ bisects $\angle{EKF}$

Let $P$ be an interior point of an acute-angled triangle $ABC$. The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$. The lines $AP$ and $EF$ intersect each other ...
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Augmented Reality Transformation Matrix Optimization

i am a software developer, i'm working on an Augmented Reality system. I'd like to receive some advice in order to optimize my math model. My program has to be slim and fast. Here's the situation: ...
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3answers
567 views

Problem in proving that $\mathbb{A}^2$ is not homeomorphic to $\mathbb{P}^2$

let $k$ be an algebraic closed field. All the spaces are equipped with the usual zariski topologies. All the proofs of this fact that I've seen rely on the fact that two lines in $\mathbb{P}^2$ ...
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1answer
424 views

How to obtain the equation of the projection/shadow of an ellipsoid into 2D plane?

Given an ellipsoid equation of the form \begin{equation}\label{eq_1}x'Ax=1\end{equation} where $A\in\mathbb{R}^{n\times n}$ is positive definite and non-diagonal and $x\in\mathbb{R}^n$. So, how can I ...
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1answer
713 views

Rational parametrization of a conic

Question: Find a rational parametrization of the conic whose equation in homogeneous coordinates is: $x^2+y^2-xy-z^2=0$ Hence find all rational numbers $x, y$ such that: $x^2+y^2-xy=1$ It seems I ...
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1answer
768 views

Extension of regular function

This is an exercise in Hartshorne's book. For a quasi projective variety $Y$ with dimension $\geq 2$ and $p \in Y$ a normal point, if $f$ is regular on $Y-\{p\}$ then $f$ can be extended to a ...
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1answer
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Prove two parallel lines intersect at infinity in $\mathbb{RP}^3$

I have to prove two parallel lines intersect at infinity in $\mathbb{RP}^3$. I have to use the direction vectors and that points at infinity have last coordinate $0$. I tried solving a system of ...
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3answers
117 views

Projective geometry general question

Can someone please help me solve this problem? I am a bit confused. Embed $R^2$ in the projective plane $RP^2$ by the map $(x, y) → [1, x, y]$. Find the point of intersection in $RP^2$ of the ...
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1answer
324 views

Line between points in projective space?

I am trying to find the line through the points $(0 : 1 : 0)$ and $(1 : 1 : 1)$ in $\mathbb P^2$ and $(0 :1 : 0: 1)$ and $(1: 1: 1: 0)$ in $\mathbb P^3.$ Would the first line be the set of points $\{(...
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2answers
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A property of projection on a convex and closed set

Let $S$ be a convex and closed set. The projection onto $S$ is defined as $P_S(x) = argmin_{y \in S} ||x-y||_2$. I want to show that if $x \in S$, then for any $y$, $$ \langle P_S(y) -x,P_S(y) -y\...
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1answer
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Meaning and types of geometry

I heard that there's several kind of geometries for instance projective geometry and non euclidean geometry besides the euclidean geometry. So the question is what do you mean by a geometry, do you ...
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1answer
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Whitehead's axioms of projective geometry and a vector space over a field

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
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1answer
297 views

What is a “harmonic quadruple”?

Can anyone tell me what a "harmonic quadruple" is? I had a problem in Australian mathematical Olympiad paper. The solution uses something like harmonic quadruple to prove that two sides are parallel. ...
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1answer
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In the affine plane, I am having trouble with these definitions

If the number of points in an affine plane is finite, then if one line of the plane contains $n$ points then: all lines contain $n$ points, every point is contained in $n + 1$ lines, there are $n^2$ ...
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1answer
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Learning projective geometry

My ultimate goal is to learn some algebraic geometry with the more concrete immediate goal of understanding things like how $\mathbb R\mathrm P^2$ is embedded in $\mathbb C\mathrm P^2$ or how $\mathbb ...
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4answers
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Geometry or topology behind the “impossible staircase”

This question on the topology of Escher games reminded me of a question I've had in my head for a little while now. Is there anything interesting geometric or topological that can be said about the ...
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1answer
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topologies of spaces in escher games

There have been a couple of games released (or in development) in the past couple of years which do some weird topological tricks: Echochrome (video), Crush (video), and Fez (video). Do the spaces ...
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1answer
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Prerequisite of Projective Geometry for Algebraic Geometry

I studied Euclidean Geometry in high-school, and I have not studied anything relates to geometry since I started studying in university. I am now intending to study Algebraic Geometry, however, I ...
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3answers
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Understanding cross ratio and harmonic conjugates

I'm studying projective geometry and I'm really having trouble with ''grokking'' what's it all about. Is there an easy/intuitive/visual way to understand cross ratio? I understand that it's ...
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1answer
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arithmetic and geometric genus for a reducible plane curve

If $C$ is an irreducible plane curve we have the well known formula relating the airthmetic (obtained via the degree-genus formula) and the geometric genus $$\frac{(d-1)(d-2)}{2} - \sum \frac{r(r-1)}{...
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Textbook for Projective Geometry

So while not actually a specific problem I'm struggling with, I was hoping for some of your insight! For a course, I'm currently reading Stillwell's Four Pillars of Geometry. While it does a nice ...
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1answer
798 views

Conic by three points and two tangent lines

With the exception of degenerate situations, a conic is uniquely determined by five points lying on it. Likewise, five lines tangent to a conic uniquely define that conic. With four points and one ...
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1answer
471 views

Distinguished open sets in $\mathbb{P}^n$

I have given a homogenous polynomial $F\in\mathbb{C}[x_0,\ldots,x_n]$ of degree $d>0$ and consider the open set $U_F=\{p\in\mathbb{P}^n; F(p)\neq 0\}$. I'd like to prove that $U_F$ is isomorphic to ...
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3answers
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Equation for non-orthogonal projection of a point onto two vectors representing the isometric axis?

Suppose I have two vectors that are not orthogonal (let's say, an isometric grid) representing the new axis. Suppose I want to project a point onto these two vectors, how would I do it? Dot product ...
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4answers
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Geometric Interpretation of the Cross-Ratio

The cross ratio of 4 points $A,B,C,D$ in the plane is defined by $$(A,B,C,D) = \frac{AC}{AD} \frac{BD}{BC}$$ And it's a ratio which is preserved under projections, inversions and in general, by ...