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

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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 ...
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Projective transformation

I need to find the function $f$ that satisfies the following: $f((1:1:0))=(0:1:1)$ $f((0:1:1))=(1:0:1)$ $f((1:0:1))=(1:1:0)$ If I let: $x=(1:1:0)$ $y=(0:1:1)$ $z=(1:0:1)$, then I get $f(x)=y$, ...
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1answer
22 views

Projective Geometry with Clifford Algebra - lost Inner Product

Projective geometry may be studied with the tools of Clifford Algebras by adding a new direction (see for example this article). But as far as I understand it, only blades and null spaces are used for ...
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16 views

Mipoint between points in projective space

Is there a way to define the midpoint between points in projective space?
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23 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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Relation of Tissot's indicatrix to meters

I have given the following function. tissot(lon_0, lat_0, radius_deg, npts, ax=None, **kwargs) Draw a polygon centered at lon_0,lat_0. The polygon approximates a circle on the surface of ...
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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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2answers
31 views

When the intersection between a sphere and a cylinder is planar?

We have a sphere and a circular cylinder. Let the sphere center be $O$ and radius $R$, and the cylinder axis $a$ and radius $r$. I solved the specific case intersection graphically on 2 planar ...
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33 views

Gradient function

Let A (red) and B (green) 2 distinct points anywhere in a 3D space. I am looking for a function which take a point P, and returns the value in blue in the picture. Each blue number in the picture ...
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28 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
47 views

Projective Geometry in $\mathbb{R}^{3}$: “Lonely lines” in source/image planes

I am reading some lecture slides about projective geometry in $\mathbb{R}^{3}$. In particular, given a source plane, $S$, an image plane, $I$, and a focal point, $f$, the issue at hand is the ...
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1answer
23 views

Perspective projection of a circle: what is the size of the semi-major axis?

It can be proven that the perspective projection (or camera projection) of a circle is an ellipse. But I also need to prove that the semi-major axis has the same size as the radius of the original ...
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63 views

Fermat's Last Theorem and the Projective Curve $C_N$

In "Silverman & Tate" on page 230 in the appendix on projective geometry, there is the remark: The $N$th Fermat curve $C_N$ is the projective curve: $$C_N: X^N + Y^N = Z^N$$ ...
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52 views

Complex projective manifolds and smooth projective varieties

Look at the following theorem: The following two categories are equivalent: The category of non-singular projective varieties over $\mathbb C$. (Where a variety is understood as in ...
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1answer
187 views

What is $\mathbb{P}^{\infty}$?

Can we look at a complex projective space $\mathbb{P}^{\infty}$? I am curious to know what would it be. What is the right intuition to think about it? I know $\mathbb{P}^{n}$ is a space of ...
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24 views

Stereographic projection of a sphere

What should have been a simple exercise in geometry has morphed into a multi-day affair with me figuratively tearing my hair out. I have no clue what's wrong. This image accompanies the problem: ...
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1answer
121 views

Ways to project arbitrary Fractals on 2D objects and 3D objects w different dimensions?

I am trying to create a house/texture in 3D and in 2D with fractals, perhaps related. My friend said that fractals can have different dimensions such as 1.74, 1, 4.71111... and pretty much anything. ...
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88 views

Find all holomorphic diffeomorphisms $f:\mathbb{CP}^1\to\mathbb{CP}^1$

The complex projective line $\mathbb{CP}^1$ is the complex manifold defined by the quotient of $\mathbb{C}^2-\{(0,0)\}$ by the relation $z\sim w$ if $z=\lambda w$ for $\lambda\in\mathbb{C}-\{0\}$. I ...
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36 views

Principal divisor of rational functions over nonsingular curves and pullback

I'm studying the theory of divisor over algebraic varieties for a seminar and I came across a problem that I think I solved almost completely except for a point that I'm missing. Let be $k$ an ...
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13 views

The Cusp $w^2 + p(z,w)=0$ is desingularizable in the origin $O \in \mathbb{C}$

I have just studied a method in projective geometry over complex numbers on how to desingularize a curve in a point but i'm a little bit confused. I don't know the name of this classical method in ...
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1answer
28 views

Book for projective geometry

I am looking for a blue colored book about projective geometry,as I remember, on sheep or goat covers. My friend suggested me two books before. I choose another one but recently I am interested in ...
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15 views

Points at infinity correspond to asymptotic slopes

Let $ P^2\mathbb{C} = \{ [a, b, c] | a,b,c \in \mathbb{C}^* \} $ the complex projective plane. So $ [a,b,c] \sim [x,y,z] $ iff $ \exists \lambda \in \mathbb{C}^* \colon \lambda(a,b,c) = (x,y,z) $. In ...
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need help understanding this proof of projecting triangle to isosceles.

The question is a problem from the book introduction of Projective Geometry. Yesterday I asked a similar question regarding right triangles and I got a proof that used vector projections. But when I ...
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1answer
17 views

External Tangents of Circles.

If we have three circles C1 C2 C3. Consider the intersection between the external tangents of each pair of circles . Show that the three points of the intersections are collinear. Is this a well ...
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18 views

External Angle Bisector

How do you prove that for any triangle three points of intersections of bisectors of its external angles with opposite sides belong to one line. I'm having a hard time with the collinearity part.
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50 views

is it possible to project any triangle on a plane as a right triangle on another plane?

I scratching my head over this problem from my projective geometry book (C. R. Wylie, Jr). Given a triangle in the plane $z = 0$, is it possible to find a viewing point, $C$, from which the triangle ...
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1answer
23 views

Dimension of the intersection between a projective variety and a hyperplane.

Suppose that $X$ is a smooth $m$-dimensional projective variety embedded in some $\mathbb P^n_k$ (we work over an algebraically closed field). Now consider a hyperplane $H\subseteq\mathbb P^n_k$ of ...
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1answer
19 views

Why is the cross ratio of $(u,v,\lambda, \omega(\lambda))$ a constant?

In the book by Beltrametti, he states that if $\omega$ is a projectivity of a line $r$, it has two fixed points, $u$ and $v$. If $u$ and $v$ are distinct, then the cross ratio $(u,v,\lambda, ...
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39 views

How to prove: Projective transformation composition of at most three involutions?

How do you show that any projective transformation of $\mathbb RP^1$ is the composition of at most three involutions of $\mathbb RP^1$? Thanks
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2answers
36 views

Clifford Algebras for Projective and Conformal Geometry

According to Clifford Algebra: A Visual Introduction, A Clifford Algebra over $\mathbb{R}^3$ may describe the rigid motions in space (namely, conjugation acts as a reflection by a plane). A ...
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1answer
20 views

Proximal operators on Balls (Projection)

I was following this tutorial, In section 21 it is given Proximal operator over a ball $B_\epsilon$ of radius $\epsilon$ as $$\text{Proj}_{B_\epsilon(y)}(u) = y + (u-y) \max({1 , ...
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Proving $FP^2$ is transitive?

Let $F$ be a field. A collineation of $F^3$, considered as the projective plane $FP^2$, is a permutation of lines and planes of $F^3$ that pass through the origin that preserves incidence. It's ...
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45 views

Find closest point in triangle given barycentric coordinates outside

Given a non-degenerate triangle ABC and an arbitrary point P in 3D space, I can project P onto the plane defined by ABC and check whether the triangle contains it as described here. I end up with ...
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21 views

The subspace sum of a point and a closed subspace is closed

In projective geometry, a polarity is a map $\ell\mapsto\ell^\perp$ on the subspaces of $\Bbb P$ satisfying the axioms: $\Bbb P^\perp=0$ $\ell\subseteq m\implies m^\perp\subseteq\ell^\perp$ If $P$ ...
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1answer
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Projective coordinates for point at infinity on elliptic curve

What is the unique characteristic of the projective coordinates of a point at infinity? I am specifically looking for a characteristic on (short) weierstrass curves. I know that the point at infinity ...
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1answer
313 views

fixed point projective geometry

I am thinking about the following: Let $\sigma:\mathbb C P^n\rightarrow\mathbb C P^n$ be a projectivity with $\sigma\circ\sigma=id_{\mathbb C P^n}$. I define the set of all fix points by ...
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1answer
23 views

In projective geometry the dual of the cross ratio dual is an angle measurement?

I am trying to get my head around angles in projective geometry. I understand (more or less) the cross ratio and that it can be seen as an distance measurement. (for example in the Beltrami Cayley ...
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Distortion in standard protection.

Assume a field of view of $2a$ radians on a screen. Half the field of view is $a$ radians. Now we can represent half of this field of view with a triangle, with the side pointing from the view being ...
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119 views

Space of inscribed $n$-gons modulo projective transformations.

Say $P \sim Q$ ($P$ and $Q$ are «projectively equivalent») iff there is a projective transformation $f$ such that $f(P) = Q$. Then $\sim$ is an equivalence relation. I read that the space of ...
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25 views

Projection of a Triangle into a Tetrahedron

I was referring to a paper to implement an algorithm in which one of the step was to project the triangle into the ...
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24 views

Cross of two n-dimensional planes.

As you all know, there is geometric place of points of cross of two planes (given as plane vectors) explicitly written simply as ...
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130 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 ...
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Why is the projection of a vector V onto a span W, independent of the orthogonal basis of W.

Very straightforward question. I have read time and again in my book that it is independent but I don't understand why? Wouldn't changing the basis mean changing the length of the projection?
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1answer
20 views

Intersection fo Projective Lines

I think I've gone wrong with my reasoning somewere here but I'm not sure why. We embed $\mathbb{R}$ into the projective plane by $(x,y)\to[1,x,y]$, and consider the projective lines corresponding to ...
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1answer
54 views

A point $\in \mathbb{P}^2(\mathbb{C})$

What is the property that should be satisfied so that a point $\in \mathbb{P}^2(\mathbb{C})$ ? I am looking at an exercise where I have to find the flexes of a curve and this information is needed.
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73 views

How does the Möbius group act on circlines?

This is a continuation of my earlier, rather vague question. I am interested in studying the action of the Möbius group $PGL(2,\mathbb{C})$, on the circlines in the extended complex plane $\mathbb{C} ...
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What do we lose in Projective Spaces?

We can think of the Complex Numbers as an extension of the Real Numbers, similarly we can think of the Projective Plane naturally as a nice extension of the Euclidean Plane. But, when we go from real ...
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Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?

Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it ...
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32 views

Central Projection - Project point $X$ on plane $\pi$

We consider the projection of three-dimensional projective space from a center $Z$ onto image plane $\pi$: \begin{align} X \longmapsto \alpha(X) = (Z \lor X) \cap \pi \end{align} since $\pi$ ...
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How do you work with the space of circles on the sphere considered as the projective line?

I'm trying to prove some things about the action of the Möbius group on the "circlines" in the extended complex plane, ie. circles on $\mathbb{C}P^1$. I find that while I have a good grip on Möbius ...