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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How Calculate To Interior Reflex Angle Of Concave

For example I have a concave polygon and I know all of coordinates of the points. How can I calculate interior reflex angle without knowing other angles ? Thanks in advance!
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Silverman Proposition 2.5 computation

In the proof of Proposition 2.5 in Silverman's Arithmetic of Elliptic Curves, the author defines a map $$E_{ns} \to \overline{K}^*, \quad [X,Y,Z] \mapsto 1 + \frac{AX}{Y},$$ where $E_{ns}$ is the ...
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How Can I Calculate Irregular Polygon's Internal Angle? [closed]

For example there is irregular polygon and I choose a vertex and I want to calculate it's interior angle, the angle can be more than, less than or equal 180 degrees also I have coordinates. How can I ...
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Shadow of a rod

AB is a rod which is held such that $A=(1,-2,3)$ and $B=(2,3,-4)$ . A source of light is at the origin. Find the length of the shadow of the rod on a plane screen whose equation is $x+y+2z=1$ I ...
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Finding an envelope for a moving circular sector

Preamble: I want to find the curve which bounds a moving circular sector, i.e. an envelope for the following family of plane curves. Suppose that we are given a "perspective" point $T$ and ...
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Puzzle: Area of a square based on parallel lines going through it's corners [closed]

Here is a little puzzle I got from my math teacher. I had a little trouble solving it; I tried to find the equations to lines by solving multiple systems, but to no avail. Could you guys please help ...
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2 answers
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Find volume of solid bounded by given surfaces. $z=a+x,z=-a-x,x^2+y^2=a^2$

Find volume of solid bounded by given surfaces. $$z=a+x, \qquad z=-a-x, \qquad x^2+y^2=a^2$$ This is the solid. We can find volume of solid that has positive $z$ value and multiply by $2$. And for ...
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Find general and parametric equations of the plane containing the points $A(3, 0, 0), B(0, 1, 0)$ 'perpendicular' to the $XY-$plane.

Question : Find general and parametric equations of the plane containing the points $A(3, 0, 0), B(0, 1, 0)$ 'perpendicular' to the $XY-$plane. My Try : Seeing that the plane is perpendicular to the $...
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How to find the coordinates of the vertices of an equilateral triangle inscribed in a given circle?

Let $C = (a, b)$ be any given point in the plane, and let $r$ be any given positive real number. Then how to find the coordinates of the vertices of an equilateral triangle inscribed in the circle $$ \...
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Does a combination of linearly independent vectors have a minimal value?

My question is: Let $v_1,v_2 \in \mathbb{R}^n$ be linearly independent. There exists $c > 0$ such that $$ \|v_1\cos(\beta t) - v_2\sin(\beta t)\| \geq c $$ for all $t \in \mathbb{R}$? I know $\|...
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Find location and orientation of a pinhole camera from a given image of a triangle with known sides contained in a known plane

So, I am playing with problems related to perspective images produces by a simple pinhole camera. I came up with the following problem. Suppose you have a triangle of known side lengths, that is ...
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Restoring third coordinate for triangle by its orthogonal projection and similar triangle

Suppose we have triangle $\Delta OAB$ lying on plane $z=0$ with coordinates $O(0,0,0), A(x_a,y_a,0), B(x_b,y_b,0)$ Also there is triangle $\Delta EFG$, but we know only coordinates of its orthogonal ...
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Perpendicular vector in a plane in spherical coordinates system

Let's suppose we have a unit tangent vector $\mathbf{\hat{t}}$ along a curve at point $\mathbf{P}$. We can construct a plane perpendicular to this unit vector which passes through $\mathbf{P}$. I need ...
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Determining best path for lighting up my bookshelf

I'm having an issue determining the best pathway. I have a bookshelf custom-made and here's a photo of it: And here are the measurements of the same bookshelf. Other than the topmost rectangle which ...
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Minimum distance between an ellipse and a hyperbola

An ellipse is specified in vector form as follows $P_1(t) = C_1 + V_1 \cos(t) + V_2 \sin(t) $ where $C_1$ is the center of ellipse, and $V_1, V_2$ are mutually orthogonal, and extend along the semi-...
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Locus of middle points of the chords of conicoid which are parallel to $xy$ plane and touch the given sphere

I have the following conicoid before me: $ax^2+by^2+cz^2=1$ I have to find the locus of the middle points of the chords which are parallel to the plane $z=0$ and touch the sphere $x^2+y^2+z^2=a^2$. ...
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Minimum distance between two general parabolas specified in vector form

Two parabolas are specified in vector form as follows $P_1(t) = C_1 + V_1 t + V_2 t^2 $ where $C_1$ is the vertex of the first parabola, and $V_1, V_2$ are mutually orthogonal, and extend $V_2$ along ...
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Minimum distance between an ellipse and a parabola

An ellipse is specified in vector form as follows $P_1(t) = C_1 + V_1 \cos(t) + V_2 \sin(t) $ where $C_1$ is the center of ellipse, and $V_1, V_2$ are mutually orthogonal, and extend along the semi-...
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Minimum/Maximum distance between two ellipses

Two ellipses are specified in vector form as follows $P_1(t) = C_1 + V_1 \cos(t) + V_2 \sin(t) $ where $C_1$ is the center of the first ellipse, and $V_1, V_2$ are mutually orthogonal, and extend ...
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1 vote
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Find the volume of B given an equation for A

We define a solid A by: $$\frac{19}{x^2} + \frac{14}{y^2} \leq z^4 \quad (0 \leq z \leq 1)$$ We define a solid B by: $$x^2 + y^2 \leq z^4 \quad (0 \leq z \leq 1)$$ The volume of the solid B is ? ...
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Minimum Number of Variables required to represent any 4 points in Cartesian Co-Ordinates?

The points are guaranteed to be the vertices of some orthogonally oriented rectangle. A trivial solution to problem is to use 8 variables, Two for each point. A better solution is to use 4 values, ...
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Convert Ellipsoid from Cov and Mean to Quadric Representation

Question I have an ellipsoid represented as a Covariance $\Sigma \in \mathbb{R}^{3\times3}$ and a mean centroid $\mu\in \mathbb{R}^{3}$. I want to represent it as a homogenous quadric, which is a $4\...
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Determine x-coordinate of 3rd point on CIRCLE from y-coordinate and 2 other points

I'm trying to construct an arc for a custom engineering curve. Say that the arc is on a circle having points $A: (x_1, y_1)$, $B: (x_2, y_2)$, and $C: (x_3, y_3)$, where $x_1 < x_2 < x_3$ and $...
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2 votes
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Locating a point within a triangle with given conditions on distances to vertices

Given $\triangle ABC$, with known sides, find the location of point $P$ such that $ PB = k_1 PA $ and $ PC = k_2 PA $ where $k_1 \gt 0, k_2 \gt 0$ are given constants. So for this, I translated the ...
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Find the area of the region that lies inside $r=\sin(2\theta)$ and outside of $r=\cos(2\theta)$ [closed]

Find the area of the region that lies inside $r= \sin(2\theta)$ and outside of $r=\cos(2\theta)$ using polar coordinates. Generally, I could use help setting up the integral in order to solve for the ...
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P is a point of the segment AB $\iff \vec{CP} = \alpha \vec{CA} + \beta \vec{CB}$

I am trying to prove the following statement: P is a point of the segment AB $\iff \vec{CP} = \alpha \vec{CA} + \beta \vec{CB}$ where $A,B,C$ are arbitrary points, with $A \neq B$ , and $\alpha, \beta ...
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hessian plane equation basis change with transformation matrix

I've a plane defined in hessian form in 3D by normalized direction (orthogonal vector) (x, y, z) and a signed distance. The distance is signed, because I need to have the option to change plane sides. ...
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How to prove this hypothesis regarding slopes and ellipses?

Let $a, b\in \mathbf{R}^+, \lambda >1$. $\Omega: \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, Point $M(\dfrac a{\lambda}, 0), A(-a,0),B(a, 0)$. Let line $l$ pass through $M$ and intersect with $\Omega$ at ...
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Area of a crossed diagonal quadrilateral

If four coordinates of vertices are given, the area of the first convex quadrilateral is expressed in known standard matrix form. How is the net (positive and negative sum ) area expressed for the ...
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Trying to prove the inner ball condition for $C^2$ domain

I am trying to prove the inner ball condition for a $C^2$ domain $\Omega$. Let $a\in \partial \Omega$ since $\Omega$ is $C^2$ there are $r>0$ and $f:\Bbb R^{d-1}\to\Bbb R$ a $C^2$-function such ...
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How to represent the relative geometry of two ellipses with a common focus in GeoGebra?

I'm studying an astrodynamics problem and to help my study I'd like to represent the geometry I'm dealing with. I also obtained a figure in Matlab but I need to represent many angles and so I'd like ...
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Is a usual open ball in a complex algebraic variety Zariski dense?

Let $X$ be an affine variety $\operatorname{Spec} \mathbb C[x_1,\dots, x_n]/(f_1,\dots, f_m)$. Suppose the set of closed points gives a smooth complex analytic variety in $\mathbb C^n$. Pick any $p\in ...
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What does "go through a point" actually mean?

My exercise is something like this: Let a point $O$ be the origin of $xy$-coordinate plane. We define four points $A(1,0), B(1,1), C(2,1), D(3,1)$ on the plane. Let us start from $C$, go through a ...
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How to reduce a straight line - of known equation and passing by two rotating points - to a line segment : which condition should be imposed on $x$?

Let point $P=(cos(\alpha), sin (\alpha))$ and point $Q = (cos(\alpha+ \pi), sin (\alpha+\pi))$ be two points moving on a circle ( of center $(0,0)$ and of radius $1$). The straignt line passing ...
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3 votes
2 answers
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Why are these three intersection points collinear?

This is what I found several years ago when I was in middle school: Suppose we have a circle on a plane and arbitrarily choose four different points on the circle, say $P,A,B,C$. Then draw three ...
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As a function of $a$, how many points are there in hyperboloid $x^2 − y^2 − z^2 = 2$ where the tangent plane is parallel to plane $z-ax=3$?

PROBLEM As a function of $a$, how many points are there in hyperboloid $x^2 − y^2 − z^2 = 2$ where the tangent plane is parallel to plane $z-ax=3$ ? MY APPROACH I started by finding the normal vector ...
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Chord that subtends $90°$ at the centre of an ellipse.

Consider an ellipse of the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. And draw a Chord that subtends $90°$at the centre of an ellipse. This configuration has appeared in many of the ellipse questions. ...
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Example of $C^{1,\alpha}$ domain not satisfying the interior sphere condition

I would like to prove there exists $C^{1,\alpha}$ with $0<\alpha<1$ not satisfying the interior Sphere condition. I consider $\Omega=\{(𝑥,𝑦)\in \Bbb R^2:𝑦>|𝑥|^{1+\alpha}\},$ with $0<\...
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What would be the Mean shortest distance from random points in the right angled triangle to the Hypotenuse.

The problem is to find the average shortest distance of uniformly random points from the hypotenuse in a right angled rectangle. The distance d shows the shortest distance to the hypotenuse from a ...
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A question regarding the point-slope formula : does the formula really hold for any point of the straight line?

I can see only one way to derive the point slope formula, but this derivation also seems to bring a question. Let $D$ be the straight line of slope $m$ passing through point $P=(a,b)$. Let $Q=(x,y)$ ...
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2 answers
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Where is the horizontal asymptote here?

My textbook tells me about 3 cases of how to define whether or not the function in hand has a horizontal asymptote: Here I have this function: As far as I understand, I am dealing here with case ...
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$x+2y-1=0; x^2-2y^2=n$, line should be tangent to the hyperbole, solve for n

could you help me, please? $x+2y-1=0; x^2-2y^2=n$; Solve for n. What I did (wrongly): $x+2y=1; (x^2-2y^2)/n=1; x+2y=(x^2-2y^2)/n; n(x+2y)=(x-2y)(x+2y); n=x-2y$ What does this relationship mean? The ...
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Calculate coordinates from normal vector and center

Lets say I have 2D plane, normal vector, and X,Y coordinates of center of the line. How can ...
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4 votes
1 answer
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Functions $y = x^2 + x - 1$ and $y = x^3 + 2x^2 + (a + b\sqrt{3})x - 3$ have three common points $A, B, C$ such that the circumradius is $R = 3$.

Consider two functions $y = x^2 + x - 1$ and $y = x^3 + 2x^2 + (a + b\sqrt{3})x - 3$ with $a$ and $b$ being two rational numbers such that the graphs of the aforementioned functions share three common ...
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11 votes
3 answers
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Proving that out of all the possible n-gons that exist in the unit circle, the one with the maximum possible perimeter is the regular $n$-gon.

So in the context of my Convex Analysis studies, I have come across this problem: First I have to prove that $ - \sin x $ is convex over $[0, \pi]$. That's easy enough using the second derivative ...
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Derivation of the formula for a tangent plane to a surface

I am trying to derive a formula for the tangent plane to a surface at $(x_0,y_0,z_0)$. I started with $F(x,y,z)=0$ for $(x,y,z)$ near and at $(x_0,y_0,z_0)$. It can be seen that any curve in the ...
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Offset a point on a curve in 3D space

I have a curve AB in 3D space in which I know the start A(x,y,z) and end point B (x,y,z). Now, I have a point O (x,y,z) which should be moved along the curve for some distance (D). If it's a straight ...
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3 answers
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How to find point of intersection with conic section and tangent dropped from a point not on a conic

Question: We have conic section $-12x^2 + 28xy+4x-9y^2-8y=0$ and a point not on a conic $(2/5,1/5)$, how to find an intersection point with tangent dropped from $(2/5,1/5)$ to a conic section? My ...
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-2 votes
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Is it possible to get a relationship similar to the distance between a point and its projection on a plane with the area between a graph and a plane?

This question came to me during an analytic geometry class. We were discussing the distance between a point in space and its projection on a plane. Let $P: Ax+By+Cz=D$ be a plane in space, $W$ a point ...
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1 vote
3 answers
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Locate a pinhole camera using a fiducial marker

Note: The superscript notation used refers to the frame of reference. There are three frames of reference: $w$, the world frame (in Euclidean 2-space), $c$, the camera frame (in Euclidean 2-space), ...
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