# Questions tagged [affine-geometry]

for questions about algebraic geometry that focus on affine space. For affine mappings in linear algebra (i.e. linear mappings plus translations), please use the linear-algebra tag or another appropriate tag.

760 questions
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### a circle and a parabola have 3 intersection points

Is it possible that a circle and a parabola on a euclidean plane have 3 intersection points and the center of the circle does not lie on the axis of parabola?
1answer
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### Euclidean Geometry versus Analytic Geometry versus Affine Geometry?

What are the relationships (connections) among: Euclidean (or Plane) geometry Analytic geometry Affine geometry How do these things relate? I know that this is a very general question, so I'm ...
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### What is the name of a linear combination that has coefficients that sum up to zero?

Is there a special name for a linear combination where the coefficient add up to zero?
1answer
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### Geometric interpretation of ranks of matrices gathering coefficients of 3 affine and associated vectorial planes

Given three planes $\pi_{1}$, $\pi_{2}$ and $\pi_{3}$ $\in \mathbb{R}^{3}$ with their respect cartesian equation of the form $A_{i}x + B_{i}y + C_{i}z + D_{i} = 0$, we can determine their relative ...
2answers
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### Given four points, determine a condition on a fifth point such that the conic containing all of them is an ellipse

The image of the question if you don't see all the symbols The given points $p_1,p_2,p_3,p_4$ are located at the vertices of a convex quadrilateral on the real affine plane. I am looking for an ...
0answers
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### Understanding the connection between $PGL(3, K)$ and $AGL(2, K)$

Sorry if this is a trivial question, but I'm having trouble wrapping my head around the connection between these two groups. It seems intuitively clear to me that the group of invertible affine ...
1answer
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### Explicit affine transformation between simplex and subsimplex

Take a simplex $\mathcal P_{n}$ with corner points $[00\dots00]$, $[10\dots00]$,$\dots$,$[11\dots11]$ in $\mathbb R^{n+1}$. Slicing by a hyperplane $\sum_{i=1}^{n+1}x_i=t$ where $0\leq t\leq n$ gives ...
0answers
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1answer
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### Dimension of an affine plane

In geometry, an affine plane is defined as a system of points which fullfill: 1) Any two distinct points lie on a unique line. 2) Given any line and any point not on that line there is a unique line ...
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### There cannot be a concept of parallelism in a homogeneous projective space?

Page 4 of my computer vision textbook, Multiple View Geometry in Computer Vision (Second Edition), by Hartley and Zisserman, states the following: Affine geometry. We will take the point of view ...
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### Homotheties in an affine Desarguesian space: conjugation and factors

Definition: a class of conjugated homotheties (conjugated by a translation) is called a factor. I really don't understand this definition of a factor. I believe it has something to do with the ...
1answer
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### Intersections of hyperboloids and affine maps

Consider the cartesian product $\mathbb{H}_{n}$ of $n$ $m-1$-dimensional forward hyperbolae in $\mathbb{R}^{mn}$ as given by the parametrization: \$\mathbb{H}_i: \ \ x_i=\sqrt{(\vec{x}_{i+1}^2+1)}, \...