# 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, ...
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### 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 ...
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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, ...
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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 ...
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### 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....
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
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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 $A'$,...
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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: $$\vec w\in \left ... -3 votes 3 answers 95 views ### Find the locus of the points \arg (z^2-2i)=\arg (z+1+i) [closed] I need Find the locus of the points M(z) such that z\in \mathbb{C}.$$\arg (z^2-2i)=\arg (z+1+i)$$I was try z=x+iy but I can't get results • 2,317 -2 votes 0 answers 42 views ### General proof of this special triangle property Consider any general triangle ABC contained in a two dimensional plane. Three circles are drawn having the sides of triangle ABC as diameters. Prove that the centre of the circle, which is ... 2 votes 2 answers 82 views ### Problem with finding tangent lines to an hyperbola Given the point Q(0,6) and the hyperbola of equation$$\frac{(x-\frac{1}{2})^2}{a^2}-\frac{(y-1)^2}{b^2} =1a=-1+\sqrt{29}/2,b=\sqrt{-2+\sqrt{29}}$$In the figure it's clear that there are four ... • 348 -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 ... • 19 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 ... • 2,780 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\... • 2,780 2 votes 0 answers 32 views ### (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 ... • 942 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)  ... • 432 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 ... • 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 ... • 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  ... • 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 ... • 23 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 ... • 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 ... 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 ... • 193 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, ... • 193 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{... • 942 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 ... 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 ... 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 ... 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= [... • 951 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 ... • 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 ... • 2,766 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 ... • 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:  \... 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 ... • 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 ... 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 ... 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 \...
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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 \$...