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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4answers
22k views

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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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 ...
5
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1answer
732 views

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 ...
7
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1answer
1k views

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 &...
5
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1answer
191 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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1answer
50 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 ...
3
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1answer
633 views

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 ...
2
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1answer
300 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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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
7k views

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 $e'_2$ ...
4
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3answers
259 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 $\mathbb{P}^2$ ...
2
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2answers
63 views

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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1answer
319 views

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?
10
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2answers
2k views

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: ...
2
votes
1answer
93 views

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 ...
0
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1answer
73 views

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 ...
0
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1answer
794 views

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 $n^2$ ...
10
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1answer
867 views

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 ...
14
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1answer
503 views

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 ...
8
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3answers
828 views

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 ...
11
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0answers
719 views

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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654 views

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 ...
11
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1answer
280 views

Is a line just an infinitely large circle?

A line is infinite, right? Well, if $-\infty = \infty$, then a line is an infinitely large circle. (Does this have something to do with $1/0$?) It seems wrong, is it? Could I disprove it? How would ...
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3answers
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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, ...
5
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1answer
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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 ...
2
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1answer
255 views

Calculating the projective closure with more than one generator

I am given a variety $X = Z(f_1,f_2)$ in affine 3-space (in $x,y,z$), and I would like to compute its projective closure $Y = Z(g_1,\dots,g_n)$ in projective 3-space (in $x,y,z,w$). I have seen this ...
2
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0answers
192 views

Projection and Pseudocontraction on Hilbert space

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let $...
11
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4answers
2k views

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
219 views

Weil does not imply Cartier on variety $X$.

Show that the divisor $D$ defined by $a = b = 0$ in the variety $X \subset \mathbb{A}^4$ defined by $ad - bc = 0$ $($the cone on a smooth quadric surface$)$ is not locally principal. My attempt at ...
7
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2answers
430 views

Perspective problem - trapezium turned square

True or false: If you draw a trapezium on the ground, there always exists a point above (but not necessarily directly above) the trapezium such that the trapezium looks like a square from that point. ...
6
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1answer
175 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 ...
5
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1answer
142 views

projective cubic

I have some difficulties to prove that the image of the function $f:\,\mathbb P^1\longrightarrow\mathbb P^3$ such that $$(u,v)\longmapsto (u^3,\,u^2v,\,uv^2,\,v^3)$$ is the algebraic projective set $$...
5
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1answer
81 views

Number of ways of coloring projected faces of any hypercube

EDIT: This is the concrete problem in its current state: How many ways are there of separating a figure consisting of $l$ layers of squares with connected vertices (as the one seen in Fig. 5, for ...
5
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2answers
537 views

Analytically flavored book in projective geometry

I am looking for a book in projective geometry, using the apparatus of linear algebra, complex analysis, and, perhaps, modern algebra, in full. The counterexample is the Hartshorne's book on ...
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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 ...
4
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1answer
412 views

The Process of Choosing Projective Axes to Put an Elliptic Curve into Weierstrass Normal Form

I'm reading the book "Rational Points on Elliptic Curves" and on page 23 the author takes an arbitrary (non-singular) elliptic curve in the projective plane and finds a rational point $O$, referring ...
4
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1answer
89 views

Description of varieties in $\mathbb{P}^2\times \mathbb{P}^1$

If $[x:y]$ are coordinates of $\mathbb{P}^1$ and $[X:Y:Z]$ are coordinates of $\mathbb{P}^2$, what do the following varieties look like? $V(x^2X+y^2Y+xyZ)\subset \mathbb{P}^2\times \mathbb{P}^1$ $V(...
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1answer
102 views

Why is $\phi:\mathbb{P}^n\rightarrow \mathbb{P}^m$ constant if dim $\phi(\mathbb{P}^n)<n$?

Let $\phi:\mathbb{P}^n\rightarrow \mathbb{P}^m$, $n\leq m$. I want to demonstrate that if dim $\phi(\mathbb{P}^n)<n$ then $\phi(\mathbb{P}^n)=pt$ (ex. 7.3(a), ch.II from Hartshorne). It's well ...
2
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0answers
52 views

line at infinity

I tried solving the following question, could you have a look at my answer and tell me whether it's right or wrong? All input is appreciated. Question: Let $ABCD$ be the vertexs of a parallelogram in ...
2
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1answer
200 views

Blow-up in Projective Space: Choosing the Appropriate Chart

Consider $x^2+y+\alpha=0$ (made-up example), where $(x,y)\in\mathbb{C}^2$ and $\alpha\in\mathbb C$ is a free parameter. The equation defines a family of curves. The curves have no common points in ...
2
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1answer
179 views

Desargues Theorem of two triangles in perspective has symmetric order 120(why?)

10 points and 10 lines construct Desargues' theorem, but since the order of the symmetric group is 120 this means we are permuting 5 elements. But I am confused to what these elements are. In total ...
2
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1answer
47 views

Natural way of looking at projective transformations.

Let $k$ be a field and let $V$ and $W$ be finite-dimensional $k$-vector spaces, where $\dim(V)\ge1$ and $\dim(W)\ge1$. Let $q:V\to\mathbb{P}(V)$, $u\to[u]$ be the quotient map. By my teacher, a map $f:...
2
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1answer
66 views

Hypersurfaces meet everything of dimension at least 1 in projective space

The following exercise is taken from ravi vakil's notes on algebraic geometry. Suppose $X$ is a closed subset of $\mathbb{P}^n_k$ of dimension at least $1$, and $H$ is a nonempty hypersurface in $...
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1answer
152 views

Birational isomorphism $\mathbb{P}^n\times \mathbb{P}^m\to \mathbb{P}^{n+m}$

One can show that $\mathbb{P}^n\times \mathbb{P}^m$ is birational to $\mathbb{P}^{n+m}$ by making note of the Zariski topologies and the canonical isomorphism between affine spaces $\mathbb{A}^n\times ...
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95 views

Plucker's $\mu$

In Jürgen Richter-Gebert's book "Perspectives on Projective Geometry", he talks about Plucker’s $\mu$ in Section 6.3. He says that this trick was used by Plucker quite often. Plucker's trick involves ...
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2answers
96 views

Manifold over a Finite Field

I'm trying to either associate a manifold with a finite field, or, ideally find a way of considering finite fields as manifolds, in a non-trivial manner. I hope to be able to use this to extend ...
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0answers
56 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 ...
0
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1answer
998 views

4D to 3D projection

Im trying to calculate the position of 4D point in 3D world. I started with 2D and tried to extend it to the 3D and then to 4D. Firstly, I found out that its easy to calculate the projected position ...
5
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1answer
188 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 ...