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

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Intuitive explanation for the connection between Lie Groups and projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$

In this post John Baez states that the classical simple Lie groups "arise naturally as symmetry groups of projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$". Is there some ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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describing proj. seurface.

I have the surface $W=Z(x_0x_1-x_2x_3)$, in $\mathbb{P^3}$ and I want to describe it as a union of an affine piece and some other piece laying in $\mathbb{P^2}$. My solution is to look at: ...
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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 ...
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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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Intersection of circle and line.

Now we can realize that intersection of two parallel lines at point of infinity in projective space. I can manage to visualize this and compute it. My issue is if we have the two varieties $ X_1= Z ...
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Show that any two distinct lines in $\Bbb P^2$ intersect in one point.

Show that any two distinct lines in $\Bbb P^2$ intersect in one point. Proof(My attempt). Let $L_1, L_2$ be any two distinct lines in $P^2$. Write $L_i = V (a_iX + b_iY + c_iZ), i = 1,2$. It ...
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Plücker matrix - Rank 2 proof

How can I proof that the Plücker matrix of the form $$ \begin{pmatrix} 0 & -L_{01} & -L_{02} & -L_{03}\\ L_{01} & 0 & -L_{12} & -L_{13}\\ L_{02} & L_{12} & 0 & ...