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Questions tagged [analytic-geometry]

Questions on the use of algebraic techniques for proving geometric facts. Analytic Geometry is a branch of algebra that is used to model geometric objects - points, (straight) lines, and circles being the most basic of these. It is concerned with defining and representing geometrical shapes in a numerical way.

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Intersection of planes in a 3d space WITHOUT vectors [closed]

We are learning about finding the intersection of planes / lines in 3D space by drawing diagrams, and I don't get it at all. The problem is we aren't doing anything with vectors or graphical diagrams, ...
paintedwolf's user avatar
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17 views

How to derive that the harmonic mean of focal segments equals the semi-latus rectum in conic sections?

My teacher told me that the harmonic mean of the focal segments in any conic section (ellipse, hyperbola, or parabola) is equal to the length of the semi-latus rectum. I was able to derive this ...
Multiversal Explorers's user avatar
1 vote
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Proving the Basis of a Scaled Vector Set J

If $ E = \left\{\overrightarrow{e_1}, \overrightarrow{e_2}, \overrightarrow{e_3}\right\} $ is a basis, prove that $ F = \left\{\alpha \overrightarrow{e_1}, \beta \overrightarrow{e_2}, \gamma \...
Gjhdby5 Vjfhu's user avatar
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0 answers
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Locus of center of hyperbola that is tangent to coordinate axes

Given the generic hyperbola $ \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 $ which is centered at the origin, suppose we shift it and rotate it, such that one of its branches becomes tangent to the ...
i don't know what i am doing's user avatar
1 vote
1 answer
58 views

Ruler and compass construction of an inscribed quadrilateral

Suppose that I have a (convex) quadrilateral $ABCD$. In the interior of it I have 4 distinct points $P,Q,R,S$ in general position (i.e. are vertex of a quadrilateral). The question how to construct a (...
quantum's user avatar
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0 votes
1 answer
22 views

Proving Linear Dependence from Cross Product Sum J [duplicate]

Prove that if $\vec{u} \times \vec{v} + \vec{v} \times \vec{w} + \vec{w} \times \vec{u} = \overrightarrow{0}$, then $\{\vec{u}, \vec{v}, \vec{w}\}$ are linearly dependent. Solution: Consider the cross ...
Gjhdby5 Vjfhu's user avatar
1 vote
1 answer
59 views

To derive the equation of the director circle of a standard ellipse using the tangent lines at any two points on the ellipse

I am currently studying conic sections and have encountered the concept of the director circle (or orthoptic circle) of an ellipse"1". The director circle is defined as the locus of points ...
Multiversal Explorers's user avatar
2 votes
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70 views
+50

Shapes with simple distance functions.

Given a set $A$ in $\mathbb{R}^2$, the distance function (DF) of $A$ is defined as $$ \delta_A(\mathbf{x}) = \inf\{\|\mathbf{x}-\mathbf{y}\|: \mathbf{y} \in A \} $$ Some sets $A$ have a nice tidy ...
bubba's user avatar
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Regular octagon inscribed in an ellipse

Problem A regular octagon inscribed in an ellipse. The foci of the ellipse $F_1$ and $F_2$ are on the midpoint of a side of a regular octagon. Proof the eccentricity of the ellipse is $e=\frac{\sqrt{...
Jun Teo's user avatar
-4 votes
1 answer
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A surface passing through two different surfaces [closed]

Suppose I have two surfaces $f_1=k_1$ and $f_2=k_2$ in 3D. Then, how do I find the equation of a surface passing through (intersecting) the two surfaces $f_1,f_2$? Like, does $f_1-f_2=k_3$ help? But, ...
vidyarthi's user avatar
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Finding the z coordinate of the center of a sphere when touching a triangle in space without iteration. [closed]

I have the coordinates of three points of a triangle in space. I also have x, y coordinates - the center of the sphere. Is it possible to find the z coordinate of the center of the sphere such that ...
Agnius Igres's user avatar
0 votes
1 answer
39 views

Line through $A$ parallel to $\alpha$ and incident on $r$.

I tried to solve point $(1.)$ of this exercise, but it seems that no such line exists. I would be very happy if someone could check my work, because it seems strange to me that there isn't a solution....
Pizza's user avatar
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Writing the equation of a family of circles touching two circles

I know that the equation of a circle passing through the intersection of two circles is $S_1+\lambda S_2=0$ and I know that the equation of a circle passing through the intersection points of a line ...
Cognoscenti's user avatar
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0 answers
21 views

Obtaining aggregate rotation and translation of 2D vector field (2d matrix of 2D motion vectors) produced from optical flow

I have the result of a dense optical flow in the form of 2D matrix of 2D motion vectors. I need to isolate the rotation and translation components of the motion to estimate camera rotation and ...
Alex Bausk's user avatar
-3 votes
0 answers
34 views

Proof the "Humpty Dumpty Lemma" with Real Analytics Geometry [closed]

Please help me to find the idea to proof "Humpty Dumpty Lemma" with Real Analytics Geometry I know it will be easier to prove it with synthetic geometry or trigonometric identities, but my ...
Lim Zhao Sen's user avatar
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0 answers
15 views

How can I find the equation of an ellipse given three points on it and that the major axis is horizontal and given his length [closed]

Here is the exact problem: Find the equation and sketch the ellipse with $10$ units horizontal major axis and passes by the points $(0,0)$, $(8,0)$, and $(0,-4)$. Please if you know the answer or at ...
Mathieu Morcos's user avatar
1 vote
1 answer
63 views

What is the equation of diameter of a rectangular hyperbola?

So I was doing this question The locus of middle points of parallel chords of hyperbola :$xy=c^2$ The locus of mid-points of parallel chords of a conic is called a diameter or so I was told. I ...
Aurelius's user avatar
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1 answer
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Distance to geometric median

The geometric median of a triangle is the point that minimizes the sum of distances to the triangle vertices. Is it true that for any triangle $ABC$ with geometric median $D$, and any other point $A'$,...
user355066's user avatar
1 vote
1 answer
29 views

Geometric Interpretation of Linear Independence and Cross Product

Prove that if $\vec{u}$ and $\vec{v}$ are linearly independent, and $\vec{w} \times \vec{u}=\vec{w} \times \vec{v}=\vec{0}$ then $\vec{w}=\vec{0}$. Interpret geometrically. Solution: $$\vec w\in \left ...
Gjhdby5 Vjfhu's user avatar
-3 votes
3 answers
95 views

Find the locus of the points $\arg (z^2-2i)=\arg (z+1+i)$ [closed]

I need Find the locus of the points $M(z)$ such that $z\in \mathbb{C}$. $$\arg (z^2-2i)=\arg (z+1+i)$$ I was try $z=x+iy$ but I can't get results
Ellen Ellen's user avatar
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-2 votes
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General proof of this special triangle property

Consider any general triangle $ABC$ contained in a two dimensional plane. Three circles are drawn having the sides of triangle $ABC$ as diameters. Prove that the centre of the circle, which is ...
MathStackexchangeIsNotSoBad's user avatar
2 votes
2 answers
82 views

Problem with finding tangent lines to an hyperbola

Given the point $Q(0,6)$ and the hyperbola of equation $$\frac{(x-\frac{1}{2})^2}{a^2}-\frac{(y-1)^2}{b^2} =1$$ $$a=-1+\sqrt{29}/2,b=\sqrt{-2+\sqrt{29}}$$ In the figure it's clear that there are four ...
Manuel Ocaña's user avatar
-1 votes
0 answers
21 views

Determine MPH of a Sphere From Diameter Change Using Image Processing

I'm looking for a formula to calculate the speed of a ball coming directly at or away from you solely based on the the change in diameter over time. For example, if I video recorded a ball that was a ...
Nebb's user avatar
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0 votes
1 answer
32 views

An ellipse $\mathcal{E}$ touches a fixed ellipse $\mathcal{C}$ at $A$, prove the length of semi-major axis of $\mathcal{E}$ is constant

$\mathcal{C}$ is an ellipse with center $O$ and semi-major axis length $=a$, semi-minor axis length $|OB|=b$. $A$ is a point moving on $\mathcal{C}$. $E$ is a point on $\mathcal{C}$ such that $OE$ is ...
hbghlyj's user avatar
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1 vote
1 answer
40 views

Number of circular cylinders/cones through an ellipse

Given an ellipse $x^2 / a^2 + y^2 / b^2 = 1\;(a>b>0)$ on $xy$-plane. Then an arbitrary elliptical cylinder through the ellipse has equation $(x / a + u z)^2 + (y / b + v z)^2 = 1$ for some $u,v\...
hbghlyj's user avatar
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2 votes
0 answers
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$(C): (x-4)^2 + (y-3)^2 = 4, A(-1;0), B(-3;0), M \in (C).$ How to find min/max of $P = MA + MB?$

$(C): (x-4)^2 + (y-3)^2 = 4$ $A(-1;0)$ $B(-3;0)$ $K(-2;0)$ $M$ is a point on the circle. I wonder what the method is to find the minimum value of $MA + MB$ as $M$ is moving on $(C)$. I'd love to hear ...
ten_to_tenth's user avatar
0 votes
1 answer
33 views

Contradiction: Exact sequence of constant sheaves, but not exact when using long exact sequence?

So, I'm again trying to understand the statement made in Masaki Kashiwara's and Pierre Schapira's book "Sheaves on Manifolds" that I describe in the post: Why is $ \mathfrak{Mod}(A_{Y}/f) $ ...
Duarte Costa's user avatar
0 votes
1 answer
40 views

Clarification on Solving Two Equations to Find the Inradius of a Triangle

I am trying to find the inradius of a triangle with sides 5, 7, and 8. To approach this, I'm using the method of perpendicular segments from vertices to opposite sides and utilizing the area and equal ...
Oth S's user avatar
  • 121
0 votes
2 answers
53 views

Geometry Application for System of Linear Equations

Question : Find the intersection (if any) of the line $x=(1,0,-1)+\lambda(3,2,1)$ and the plane $x = (-1,-7,-7) + \alpha(3,5,1)+ \beta(1,-2,-5)$. My work so far: I equated both equations and ...
N.K's user avatar
  • 3
2 votes
2 answers
95 views

Find the radius of the n+1th circle if it is constructed to be internally tangent to circle number 1, externally to circle number n and number 2.

Given a circle with radius $r_1$ a circle (number 2) with radius $r_2 < r_1$ is drawn such that it is tangent to circle internally. a circle (number 3) is constructed with radius $r_3= r_1 - r_2 $ ...
pie's user avatar
  • 5,393
2 votes
1 answer
48 views

How to calculate the radius of an outer circle from two symmetric internally tangent circles? (Applied Wind Turbine Problem 🔧)

Motivation & Context The below problem is related to an open-source CAD model of a locally-manufactured small wind turbine. The generator of the turbine is made from two rotating magnetic disks ...
gbroques's user avatar
1 vote
0 answers
41 views

Example of Grauert's Division Theorem

We are trying to compute an explicit example of Grauert's Division Theorem, as it appears in De Jong and Pfister's Local Analytic Geometry book. In particular we are trying to compute the following ...
nelynx's user avatar
  • 381
1 vote
2 answers
54 views

Average distance from a point on a circle to the y-axis.

This is a simple question, but I must be making some mistakes as I don't seem to get the answer in the book. Question: Determine the average distance from a point on $x^2+y^2 = 9$ to the $y$-axis. My ...
Teodoras Paura's user avatar
1 vote
1 answer
55 views

Rotation of a function on the unit sphere

Let $u$ be a smooth function on the sphere such that $|\nabla u(0,0,1)|=|\nabla u(0,0,-1)|\neq 0$. Let $\hat{u}(x)=u(-x)$ be the reflection of $u$ with respect to the origin. I wonder if there is a ...
MathGeek1024's user avatar
7 votes
0 answers
140 views

Functions with restrictive behavior on $\mathbb{S}^2$

Let $ f $ be a smooth function defined on the sphere such that the set of points where $ f(x) - f(\tilde{x}_y) $ vanishes divides $\mathbb{S}^2$ into exactly four regions for all $y\in \mathbb{S}^2$, ...
MathGeek1024's user avatar
1 vote
0 answers
41 views

Why is the triangular inequality not applicable in this case?

In $Oxyz$ space, consider three points $A(-1;0;0), B(0;-1; 0), C(0;0;1)$ and the plane $(P): 2x - 2y + z + 7 = 0.$ Consider $M \in (P).$ What is the smallest value of $S = |\vec{MC} + \vec{MB} - \vec{...
ten_to_tenth's user avatar
1 vote
1 answer
40 views

Find the center of an ellipse such that it is tangent to a parabola at a point on it

Question: Given the parabola $ (r - V)^T R D_p R^T (r - V) + b_0^T R^T (r - V) = 0 $ where $r = [x, y]^T $, $D_p = \begin{bmatrix} a && 0 \\ 0 && 0 \end{bmatrix} $, $R$ is a $2 \times ...
i don't know what i am doing's user avatar
0 votes
1 answer
46 views

Find the center of an ellipse such that it becomes tangent to a second ellipse at a certain point on it

Question: Given the ellipse $ (r - C)^T Q_1 (r - C) = 1 \tag{1}$ where $r = [x,y]^T $ , $C $ is a variable center, and $Q_1$ is a symmetric positive definite $2 \times 2$ matrix, and given a second ...
i don't know what i am doing's user avatar
0 votes
0 answers
80 views

How to prove that a line is tangent to a conic section if and only if the line has only one intersection point with it.

I want to show that defining tangent lines of a conic section as lines that have only one intersection point with it is equivalent to defining tangent lines using the 'calculus' way, aka, lines that ...
coder114514's user avatar
0 votes
0 answers
5 views

Bounds on the volume of the image of a cube, through bounds on area of cross sections

Let $f:[0,1]^4\to\mathbb{R}^4$ be a smooth map. Assume that for every “vertical” affine plane $F=\{(a,b)\}\times [0,1]^2$, Area$(f[F])< a$. Assume also that for every “horizontal” affine plane $E= [...
JustSomeGuy's user avatar
3 votes
0 answers
16 views

Spiric sections by imaginary planes

To quote a previous question on spiric sections: The spiric section is the curve obtained by slicing a torus along a plane parallel to its axis. In comparison, Wikipedia uses an algebraic definition ...
hbghlyj's user avatar
  • 2,780
1 vote
1 answer
45 views

Locus of centre of circle which bisects two smaller circles

Consider two fixed non-intersecting circles ( not necessarily of equal radii ). A circle which intersects both the circles also bisects their circumference. What is the locus of the centre of this ...
An_Elephant's user avatar
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6 votes
4 answers
955 views

Expressing the area of an isosceles triangle as a function of one of its angles.

We are given a circle with radius $1$, its center point and an inscribed isosceles triangle with $AB=AC$ and its height (as shown in the picture below). Can we express the area $(ABC)$ as a function ...
ράτ's user avatar
  • 829
0 votes
1 answer
19 views

Validation of Assertion Regarding Parallel Vectors and Their Magnitudes

If $ \mathbf{u} \parallel \mathbf{v} $, $ |\mathbf{u}| = 2 $, and $ |\mathbf{v}| = 4 $, then prove that $ \mathbf{v} = 2\mathbf{u} $ or $ \mathbf{v} = -2\mathbf{u} $. Aimin for contradiction: $ \...
Gjhdby5 Vjfhu's user avatar
1 vote
1 answer
36 views

Analytical Method for Finding the Closest Point on a 3D Quadrilateral Polygon Face from a Line Segment

I am interested in developing an analytical method to determine the closest point on a convex quadrilateral polygon face, defined by four points (A, B, C, and D), from a given line segment connecting ...
Thinesh's user avatar
  • 13
3 votes
2 answers
153 views

Determine the length of the rod that can be inscribed in a cuboid

Question You have a cuboid of dimensions $2a \times 2b \times 2c $. I want the find the (maximum) length of the right circular cylindrical rod of radius $r$, that can inscribed in the cuboid. Use ...
i don't know what i am doing's user avatar
0 votes
3 answers
55 views

How must I find the third vertex of an equilateral / right isosceles triangle given the coordinates of 2 vertices?

This question has been asked in different ways at different points of time in the Math SE, but I'm trying to look for proof-wise simplicity here. If the ends of hypotenuse of a right isosceles ...
user1299519's user avatar
1 vote
1 answer
38 views

How to explicitely reduce the expression of the intersection of plan and sphere from 3D to 2D?

If you have for example a plan ($\mathcal{P}$) and a sphere ($\mathcal{S}$), let say : $$(\mathcal{P}) \enspace \enspace \enspace \enspace z= \frac{1}{2} $$ $$(\mathcal{S}) \enspace \enspace \enspace \...
jozinho22's user avatar
  • 127
1 vote
1 answer
63 views

Is it possible to take the RGB-space and reverse the direction of the grayscale axis while "preserving" the rest of the cube? If so, how?

In Python, I have been attempting to generate a slightly different-from-normal RGB cube with the only difference being that the grayscale is reversed, just for fun. My code generates the RGB cube and ...
DarthEwok07's user avatar
1 vote
1 answer
29 views

Prove that if a curve y(x) is concave, (dy/dx, y-intercept of tangent to curve with slope dy/dx) are the coordinates of g(t), the supremum of y-tx

The problem description is as follows: Suppose there is a concave function $y(x)$. Now, suppose that we're interested in plotting a curve $g(t) = \sup[y - tx]$. Prove that $g(dy/dx)=y_t-tx_t$, where $...
ZED's user avatar
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