# 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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+50

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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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1 vote
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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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### 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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### 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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### 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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### 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 ...
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 ...
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), ...