Projective geometry is closely related to perspective geometry. These types of geometry originated with artists around the 14th century.

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Quadratic equations defining the $10$-dimensional spinor variety.

Let $S$ be the $10$-dimensional Spinor variety parametrizing one of the two families of $4$-dimensional linear subspaces of the non-singular quadric in $\mathbb{P}^{9}$. I have read that there exist ...
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The associated Schubert variety of a flag of subspaces of a vector space.

Let $V$ be a vector space and $W_1 \subsetneq W_2 \subsetneq ... \subsetneq W_\ell \subsetneq V $ a flag of subspaces. The associated Schubert variety is defined as : $ \Omega ( W_{ \bullet } ) = \{ ...
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Show that a projective transformation is unique.

Find the projective transformation $\tau \left ( \left [ 0,0,1 \right ] \right )=\left [ 0,1,0 \right ], \tau ([0,1,0])=[0,1,1],\tau ([1,0,0])=[1,1,1], \tau ([1,1,2])=[1,1,0]$. And show that such a ...
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19 views

Line clipping in 2D perspective transformation

Situation I have two 2D spaces which are related one to other by a transformation matrix - 3*3 homography matrix for homogeneous coordinates: The first space is "map" and the second one is "camera ...
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1answer
17 views

For any four distinct points $P_{1},P_{2},P_{3},P_{4}\in P^{1}(\mathbb{R})$ we have $(P_{1}P_{3};P_{2}P_{4})=1-(P_{1}P_{2};P_{3}P_{4})$.

Show that the cross-ratio has the following property: for any four distinct points $P_{1},P_{2},P_{3},P_{4}\in P^{1}(\mathbb{R})$ we have $(P_{1}P_{3};P_{2}P_{4})=1-(P_{1}P_{2};P_{3}P_{4})$. What is ...
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15 views

Which of the following quadratic forms defines a non-singular conic?

Which of the following quadratic forms defines a non-singular conic? (1). $x_{0}^{2}-2x_{0}x_{1}+4x_{0}x_{2}-8x_{1}^{2}+2x_{1}x_{2}+4x_{2}^{2}$. (2). $x_{0}^{2}-4x_{0}x_{1}+x_{1}^{2}-2x_{0}x_{2}$. ...
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Question regarding Geometric meaning of Noether normalization theorem for projective varieties

In Ernst Kunz's ''commutative algebra and algebraic geometry'' book, ch.2, proposition 4.5 the author states: Let $K$ be an algebraically closed field, $V\subset \mathbb{P}^n(K)$ a variety of ...
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The same algebraic variety defined by different sets of polynomials

Let $\emptyset\neq X\subset\mathbb{P}^{n}$ be an algebraic variety such that $$ X=V(F_{1},\ldots,F_{m}) $$ for certain linearly independent homogeneous polynomials $F_{1},\ldots,F_{m}\in ...
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Help to understand the proof of the Riemann Mumford relation

Here i post a file where from page 617 to 618 there is the proof of the Riemann mumford relation that is the theorem 1.13. My problem is to understand the beginning of that proof. In particular ...
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1answer
18 views

finding equivalent hyperovals

If $H=D(x^k)$ is a hyperoval, then $D(x^t)$ is a hyperoval equvalent to $H$ for $t=1/k$, $1-k$, $1/(1-k)$, $k/(1-k)$ and $(k-1)/k$. If I consider the Segre Hyperoval $D(x^6)$ with $q = 32 = 2^5$, how ...
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5 views

Projecting a 3D point to a fisheye plane

I am trying to calculate if a point in 3D space is in front of my fisheye camera, so looking at the OpenCV documentation (I'm not actually using OpenCV, however), we have: $a = x/z$, $b = y/z$, $r^2 ...
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1answer
32 views

Difference between Grassmann and Projective space?

I recently read the first few chapters of Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling and I am confused about ...
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1answer
58 views

Motivating the Cross-Ratio and 'the ratio of ratio's' in $\mathbb{R}\mathbb{P}^2$

Trying to come across the idea of the cross ratio naturally by thinking about the projective plane $\mathbb{R} \mathbb{P}^2$, using ideas from Brannan's Geometry book: given 4 collinear points ...
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25 views

Differential of the Gauss map of an algebraic variety.

Let $X=V(F)\subset\mathbb{P}^{n}$ be a smooth irreducible hypersurface. Let us consider the morphism $$ \mathcal{G}:X\rightarrow \mathbb{P}^{N}, p\mapsto \left( \frac{\partial F}{\partial ...
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1answer
26 views

A Deeper Understanding / Interpretation of Homographies

I currently understand that a homography matrix, which allows for a mapping between planes in 3-dimensions, is a $3\times3$ matrix of the following general form: $$\begin{bmatrix} \vert & \vert ...
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16 views

Intersection of two circles in projective space

I have checked the existing question Intersection of two circles. and model for intersection of two circles in the complex projective plane - I do not think either of these answers my question. The ...
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39 views

Question regarding projective coordinate transformation

While reading Kunz's commutative algebra book, I came across a statement I can't understand. First, let me define the notations. Let $L/K$ be extension of fields, and let $\mathbb{P}^n (L)$ denote ...
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1answer
19 views

Point Selection for Cross Ratio

The cross ratio relates the positions of four co-linear points in 3d space. I understand definitions a published online, such as that from wikipedia: https://en.wikipedia.org/wiki/Cross-ratio ...
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1answer
5 views

Projectivity maps that fix three points

Please check this definition: A projectivity is a bijection $PV\to PW$ induced by an isomophism $\phi: V\to W$ given by $\phi(kv)=k\phi (v)$. Now, i have seen here Old Question that an answer says ...
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How are non-homogenous elliptic curves projective varieties?

So if I am given an elliptic curve such as $Y^2Z=X^3$ then I see how it can be realized as the projective variety $Proj(k[X,Y,Z]/(Y^2Z-X^3))$. But, given an elliptic curve like $Y^2 = X^3 + X$, then ...
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1answer
25 views

Passing light through airfold blades as a function of angle?

Consider an airfoil blade sunshade (see the diagram below). It consists of multiple flat rectangular shapes which are parallel to each other but make an angle ($v$) to the horizontal plane. The blades ...
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1answer
28 views

Line Equations $(a, b, 0)$ and $(0, 0, 1)$

I am learning about projective geometry in my machine perception class. I am struggling to fully grasp the concept of lines at infinity, and I am hoping someone would be able to shed some light on ...
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Intersection of a line and line at infinity in projective space

I understand parallel lines in Euclidean space intersect at the line at infinity in terms of projective space. My question is for a single line. A single line if extended to infinity must intersect ...
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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 ...
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Hyperplanes without Axiom of Choice

For any projective space that contains more than one point, is it possible to prove that it contains a hyperplane without using the Axiom of Choice? It's easy enough to prove that there exists a ...
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16 views

pencil of cubic curve passing six points

Let [1,0,0],[0,1,0],[0,0,1],[1,1,1],[1,3,2],[1,4,3] be a six points in general position. The question is how can determine the pencil of cubic curve passing through these points? Many thanks.
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Prove Ceva's Theorem for the case where D and E are ideal but F is ordinary.

A triangle is is defined as the area covered by three lines. Their points of intersection are A,B and C. Prove Ceva's Theorem for the case where D and E are ideal but F is ordinary. From my ...
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1answer
86 views

Collinearity of points in a projective setting

Let $ ABC$ be a triangle and $P$ a random point on the same plane as the triangle. Let $l$ be a line passing through $P$. Let $A_1,B_1,C_1$ be the intersection points of $BC,CA,AB$ with the ...
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Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
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Convert affine coordinates to projective coordinates?

For any rational map represented by $(\frac{x^4+3y}{x^2+1}, \frac{x+1}{y})$ in affine coordinates, write down the corresponding representation $[F_1(X, Y, Z) : F_2(X, Y, Z) : F_3(X, Y, Z)]$ in ...
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Why cannot the homogeneous coodinates be zero?

Given a point (x, y) on the Euclidean plane, for any non-zero real number Z, the triple (xZ, yZ, Z) is called a set of homogeneous coordinates for the point. Why can't Z be zero?
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Characterization of projective convexity

Let $K$ be a closed set in projective space $\mathbb P^n$. Is it true that $K$ is "projectively convex", i.e., its intersection with every line is connected, if and only if it is the projectivization ...
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1answer
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Show that if v $\in$ V is an eigenvector of T, then [v] $\in$ P(V) is a fixed point of the projective transformation $\tau$ defined by T.

Let T : V $\rightarrow$ V be an invertible transformation. Show that if v $\in$ V is an eigenvector of T, then [v] $\in$ P(V) is a fixed point of the projective transformation $\tau$ defined by T. ...
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Is there is any other method to produce a third set of collinear points rathar than the Pappus's hexagon method?

Pappus's hexagon theorem: Given one set of collinear points $A,B,C$, and another set of collinear points $a,b,c$, then the intersection points $X,Y,Z$ of line pairs $Ab$ and $aB$, $Ac$ and $aC,Bc$ and ...
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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 ...
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55 views

Degree of a morphism from a curve to $\mathbb P^1_\mathbb C$: explicit description

Let $f:X\to \mathbb P^1_{\mathbb C}$ be a non-constant (i.e. surjective) morphism (of $\mathbb C$-varieties/schemes) from a smooth complex projective curve to the projective line. The degree of the ...
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1answer
38 views

Prove that three skew (i.e. non-intersecting) lines in $P^{3}$(R) have an infinite number of transversals (i.e. lines meeting all three).

So I think I might need to use the concept of general position but not sure how to proceed. What is a good way to prove this? Thanks.
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Find the invariant points.

Let $R$ be an isometry from the $\mathbb R^2$ to $\mathbb R^2$ defined by $R(x,y)=(x',y')$ $$x'=\frac12x+\frac{\sqrt3}2y\\ y'=\frac{\sqrt3}2x-\frac12y$$ Find the invariant points. What is the ...
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10 views

Projective Geometry and Symmedian

The tangents to the circumcircle of ABC at B and C intersect at D. E is the intersection of BC and the tangent at A and is the pole of AD. F is the intersection of BC of AD. Why is it that when AE and ...
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2answers
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What is the interpretation of homogeneous line intersection?

I understand homogeneous coordinate systems. I read the intersection of lines in homogeneous coordinate can be computed by taking a cross products of lines $l_1(a_1,b_1,c_1)$ and $l_2(a_2,b_2,c_2)$. ...
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Is the cone of a manifold a manifold of dimension one higher?

I think the cone of a manifold in complex projective space (the preimage of it by projection) would be a manifold of dimension one higher, but I don't know how to show this.
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Projective curve $x^3+y^3=2z^3$ in $\mathbb P^2$ singular?

Is the projective curve $x^3+y^3=2z^3$ in $\mathbb P^2$ (defined over $\mathbb{C}$) singular or nonsingular? If singular, what are the types of these singularities? For an affine curve, one would ...
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Find the vertices of the “triangle” in $P^{2}(R)$ whose sides are the projective lines $P(U_{1})$, $P(U_{2})$, $P(U_{3})$.

So $U_{1}$, $U_{2}$ and $U_{3}$ are the 2-dimensional vector subspaces of $R^{3}$ defined by $x_{0}$=0, $x_{0}$+$x_{1}$+$x_{2}$=0, $3x_{0}$-$4x_{1}$+$5x_{2}$=0 respectively. What is a way to find the ...
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1answer
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Show that the linear subspace is the set of points obtained by joining each X and Y by a projective line.

So $U_{1}$, $U_{2}$ are vector subspaces of V. I need to show that the linear subspace P($U_{1}$+$U_{2}$)$\subseteq$ P(V) is the set of points obtained by joining each X$\in$ P($U_{1}$) and Y$\in$ ...
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1answer
32 views

Is there a projection matrix for 2D to 1D perspective projection?

I was wondering, if there is a projection matrix for a perspective projection of a 2D point to a line. E.g. a random point being projected to the line at $x=1$, parallel to the y axis in the ...
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Cross-ratio of $4$ points on a projective line in a $n$-dimensional projective space

If we want to define the cross-ratio of four points on a 1-dimensional projective line in a $n$-dimensional projective space, then we have to choose a projective frame, so three different points on ...
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Projective Geometry - Pole/Polar

A circle is inscribed in quadrilateral $ABCD$ so that it touches sides $AB, BC, CD, DA$ at $E, F, G, H$ respectively. (a) Show that lines $AC, EF, GH$ are concurrent. In fact, they concur at ...
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Clarification in a theorem statement regarding intersection of Complex Algebraic Curves in $P_2$

I have a theorem in the book Complex Algebraic Curves- Frances Kirwan : *If two projective curves $C$ and $D$ of degrees $n$ and $m$ respectively in $P_2$ intersect at exactly $n^2$ points and if ...
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Rings of Regular functions, and regular maps between Quasi Affine to Quasi Proj. Varieties.

I have studied classical algebraic geometry a while ago. I want to sum up in short as possible everything regarding their rings of regular functions. If my understanding not correct, please correct ...
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Showing a Variety is Rational?

I'm trying to show that the following varieties are rational: $V_1=V(y^2z-x^3)$ and $V_2=V(xyz-x^3-y^3)$. But I can't think of how to show they are birationally equivalent to $\mathbb{A}^n$ or ...