Questions on the use of algebraic techniques to prove geometric theorems.

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Can a line in 3-space have all direction cosines $=\frac{1}{2}$

I immediately found that it is impossible since the squares of the direction cosines have to add to 1 and $3 \times (\frac{1}{2})^2 \neq 1$. However, the textbook asks to "interpret geometrically", ...
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30 views

Geometrical properties of tetrahedra under rotation

Consider two tetrahedra which share the same point of origin but differ in both scale and rotation over the X-axis. Can someone explain why the following points meet with these parameters? Both have ...
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1answer
28 views

Finding representations of a 3D cone [on hold]

I need some help with the following question: Find explicit, implicit and parametric representations of a 3d cone (cylindrical) with the following attributes: Its tip is located at the ...
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0answers
18 views

How to plot quadratic forms

I'm studying quadratic forms in algebra at the moment and I've been asked to plot the following curve: $$3x^2+4xy+3y^2-\sqrt{2}x+\sqrt{2}y=1$$ I have used the following transformations: ...
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2answers
20 views

Conics - required to show $SR \times S'R' = b^2$

Consider the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b > 0$. $R$ and $R'$ are the feet of the perpendiculars from the foci $S$ and $S'$ on to the tangent at ...
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0answers
10 views

Pascal and Brianchon's theorems for hyperbolic paraboloid

How would one formulate a version of these two theorems for the hyperbolic paraboloid, and what would be a simple proof? How are the classical formulations of these theorems related to this quadric?
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5 views

Prove quadrics are rational algebraic surfaces.

I have to prove that an irreducible quadric in RP^3 is a rational algebraic surface, ie, the homogeneous coordinates of any point can be expressed as polynomials in two variables. My idea was to do ...
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0answers
3 views

Quadratic singularities and local curves.

Everything is to be understood over the complex field. Assume you have two finite dimensional $\mathbb{C}$-vectorial space $V$ and $W$. You are given a bilinear form : $$\phi:V\times V\rightarrow W ...
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1answer
38 views

Finding quadric's rotation matrix

I want to know how to find the rotation matrix of a quadric in general and the eigenvalues and eigen-vectors, in particular I am given $$ Q(x,y,z)=18x^2+9y^2+14z^2+8xy+8xz−4yz−2x−6y−14z+6=0 $$ and I'm ...
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1answer
9 views

General versions for quadrics of Pascal and Brianchon theorems

I am looking for a generalization to quadrics (with proofs) of Pascal's and Brianchon's theorems. It´s for Three dimensional analytical geometry. I would be very thankful if you could point me ...
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12 views

Quadric generalizations of Pascal and Brianchon's theorems

I am looking for a generalization to quadrics (with proofs) of Pascal's and Brianchon's theorems. It´s for Three dimensional analytical geometry. I would be very thankful if you could point me ...
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1answer
18 views

Find the rotation matrix of a quadric

I want to know how to find the rotation matrix of a quadric in general and the eigenvalues and eigen-vectors, in particular I am given $Q(x,y,z)= 18x^2+9y^2+14z^2+8xy+8xz-4yz-2x-6y-14z+6=0$ and I'm ...
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0answers
30 views

Analytic geometrical properties of dodecahedron

This is a more focused question in order to hopefully get some more traction on this question: Mathematical properties of two dimensional projection of three dimensional rotated object Consider ...
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1answer
40 views

Algebraic Geometry Approach To Study The Surfaces Given The Intersection Curve

I'm NOT a Mathematician and I'm totally new to the field of Algebraic Geometry. A friend of mine told me that one thing which is studied in this field is to consider a curve as a set of points in n-D ...
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3answers
86 views

There is a unique quadric through three disjoint lines

There is a classical exercise that three disjoint lines in $\mathbb{P}^3$ are contained in a quadric surface $Q$. The existence is trivial. Every quadric in $\mathbb{P}^3$ is determined by nine ...
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2answers
19 views

Regular polygons inscribed in an ellipse

A regular $n$-gon is inscribed in an ellipse which is not a circle. what are the possible values for $n$? I know I can inscribe a square or even a equilateral triangle, but can we do it for all $n$? I ...
8
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0answers
150 views

Mathematical properties of two dimensional projection of three dimensional rotated object

Please be gentle as I do not have any degree in maths. By using a compass/straighedge method to construct Metatron's cube, a regular dodecahedron can be inferred from intersecting points. I'm looking ...
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2answers
64 views

Tangent surface of a twisted cubic curve

I am trying to describe the tangent surface to a twisted cubic curve $C$, i.e. the curve which is given parametrically by $t\mapsto(t, t^2, t^3)$. This surface $S$ is given parametrically by ...
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2answers
33 views

finding $AX\cdot BX\cdot CX\cdot DX\cdot EX\cdot FX\cdot GX\cdot HX$

Given a Regular Octagon $ABCDEFGH$, $AE=2$. On $AE$ we choose point $X$ which dividing $AE$ in the ratio of $3:1$. Need to find: $AX\cdot BX\cdot CX\cdot DX\cdot EX\cdot FX\cdot GX\cdot HX$ Any ...
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2answers
61 views

Finding axis of ellipse described by $x=a\cos t+ h\sin t$,$ y=b\sin t + g\cos t$

Hi I am in need of help here for my project. Basically I have managed to obtain this form of equation. Example: $a=-181,h=33,b=185.9$ and $g=18.3$. When I plot it on a graphing program, it looks like ...
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1answer
26 views

Intersection between a line and a plane.

A line can either lie on a plane, lie parallel to it or intersect it. Determine, if there is one, the point of intersection between the line given by the equation $$\displaystyle\frac{x−5}{2} ...
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1answer
24 views

Help Deriving the Midpoint Formula

One of the problems in one of the packets that I'm going through to review for a pre-test for an independent-study calculus class has asked me to derive the midpoint formula. I've gotten to the point ...
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0answers
40 views

Upper bound on the distance of orthogonal matrices

Dear math stackexchange users, I have a question on orthogonal matrices: suppose I have a matrix $X\in\mathbb{R}^{n\times n}$ and I consider the orbit of the orthogonal group $O(n)$ acting from the ...
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2answers
35 views

Find equation of the line through $(-4,8)$ and perpendicular to $2x-3y=1$ [closed]

I need to find equation of the line through $(-4,8)$ and perpendicular to $2x-3y=1$ How do I solve this exactly?
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1answer
21 views

Relation between a differential equation satisfied by parabolas and a formula for the slope of their tangents

Statement 1: The slope of the tangent at any point P on a parabola, whose axis is the axis of x and vertex is at the origin, is inversely proportional to the ordinate of the point P. Statement 2: The ...
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0answers
21 views

Equation of common tangent(s) to two parabolas

Suppose we are given two parabolas, described by their directices $d_1: a_1x+b_1y+c_1=0, d_2: a_2x+b_2y+c=0$ and foci $F_1(p_1,q_1), F_2(p_2,q_2)$. How does one find equations of common tangents to ...
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3answers
88 views

How can the centers of these 5 related circles be specified as a formula?

This is my first time posting in this forum, so please forgive me if my question is too involved or if I've posted it in the wrong area. I hope someone finds it interesting enough to try their hand at ...
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4answers
213 views

Tricky 3d geometry problem

We have a cube with edge length $L$, now rotate it around its major diagonal (a complete turn, that is to say, the angle is 360 degrees), which object are we gonna get? Astoundingly the answer is D. ...
4
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1answer
26 views

How to get coordinates of a point after an image is rotated? (with images)

I have a problem that involves a rotating image and finding a previously known point. Firstly, there is a sequence with the rotation. We start with an empty image. A line is drawn vertically, ...
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4answers
75 views

Find the coordinate of third point of equilateral triangle.

I have two points A and B whose coordinates are $(3,4)$ and $(-2,3)$ The third point is C. We need to calculate its coordinates. I think there will be two possible answers, as the point C could be on ...
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2answers
35 views

Finding the coordinates of a parallel line given line coordinates and a distance

I have a path defined by a list of (x, y) coordinates and I want to create two additional paths, one offset by a distance of 0.25, the other by -0.25. I think that could be done by finding parallel ...
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1answer
11 views

Find the equation of the hyperbola that satisfies this condition

Focus is at $F\equiv(−3−3√13, 1)$, asymptotes intersect at the point with coordinates $(−3, 1)$ and one asymptote passes through $(1, 7)$ I've solved some problems that involve equations of ...
3
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1answer
210 views

Simplifying an equation (circle)

I'm trying to work on a problem, and I'm stuck at simplifying this equation. I do not why I cannot see it: so the book gives the following equation: $$\frac{ax}{x^2+y^2}+\frac{by}{x^2+y^2}+c=0$$ ...
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1answer
36 views

Why 9 points determine a quadric

The books I have state this redult as obvious from the definitions, but it is not clear to me why this is so.
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0answers
27 views

Find the direction from which the projected area of a loop is maximal

How do I find the direction from which the projected area of a loop is maximum? Should I try to use intuition or is there a simple mathematical way to find it? The problem given was the following: ...
0
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1answer
13 views

How do I find the symmetrical point B given the centre of symmetry C and another point A?

I have a point $A (-2k; 3)$ and a point $B$ that is symmetrical to the point A given the centre of symmetry $C (-1; 0)$. I tried applying the following formula, where $x_o$ and $y_0$ are the ...
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0answers
45 views

A problem about the intersection of a plane and a sphere

So I have been trying to solve this (rather basic) geometry problem but don't know how to parametrize the functions and get an answer (please check below). Ok, so my problem reads like this: prove ...
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1answer
23 views

Given one endpoint and midpoint in (x,y) of a line segment, explain how to find the other end point.

A line segment with one end at C(6,5)has midpoint M(4,2). Determine the coordinates of the other endpoint, D. Explain your solution and describe a method to check your answer.
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6answers
199 views

Area enclosed by the graph of $13x^2-20xy+52y^2+52y-10x=563$.

Find the area enclosed by the graph of $13x^2-20xy+52y^2+52y-10x=563$. First I saw that this cannot be a circle ($xy$ term), and it cannot be an ellipse with axes parallel to the coordinate axes. But ...
5
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6answers
89 views

Why is the equations for a perpendicular line $-\frac{1}{m}$?

Why is it just $-m$? Lets say $y=x$ and the $y$ intercept is at $0$ if we created another line that was $y=-x$, wouldn't that make it perpendicular? Note: Don't exactly know what tags to use, feel ...
3
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1answer
42 views

How homogenization of line and curve works?

I am given a curve $$C_1:2x^2 +3y^2 =5$$ and a line $$L_1: 3x-4y=5$$ and I needed to find curve joining the origin and the points of intersection of $C_1$ and $L_1$ so I was told to "homogenize" ...
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1answer
62 views

Rigorous books on geometry

I am looking for a rigorous book on both 2d and 3d euclidean geometry, and also how analytic geometry can be developed from synthetic geometry. I haven't really found such a book yet. I would be very ...
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1answer
17 views

How do I determine k so that the line of the beam is parallel to a $60^\circ$ angle?

I have the equation of a beam that looks like this: $$(x + y - 5) + k(2x - 3y) = 0$$ I know that the angular coefficient of a $60^\circ$ angle is equivalent to the root of 3. $$m = \sqrt3$$ Though, ...
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1answer
62 views

formula for a sphere?

is there such a thing as a formula for a sphere? Is it $x^2+y^2+z^2=1$? if so, does the $1$ denotes a radius of $1$ for said sphere? what are the possible alterations for such a formula?
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37 views

Proving That Two Paths of Different Lengths Are Adjoined

In the section on 'Adjoining Paths' of its 'Topology' book's page on 'Path Connectedness,' WikiBooks shows that, for any topological space $X$ with members $a$, $b$, and $c$, the following…: ...
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1answer
10 views

Derivation of the Parametric Form of the Equation of a Line From Its Two-Point Form?

Wikipedia's documentation on the parametric form of a linear equation states in the paragraph between two different sets of equations available for use in determining a line's parametric equations ...
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3answers
42 views

What is the equation of the line that is parallel to the y-axis?

I have a line, parallel to the $y$-axis, that passes through a point, P: $$P(1/2,-3/5)$$ What is the equation of the line? What I tried: $$(y−y_0)=m(x−x_0)$$ $$(y+3/5)=m(x−1/2)$$ ...
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1answer
36 views

How small can an external angle of a circumference be if made of tangents?

Lets imagine the angle ABC where the lines AB and CB are tangents to a circumference which center is C. Lets assume that the points where the line AB touches the circumference is P and the point where ...
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2answers
73 views

Show that $PF.PG=b^2$ in a hyperbola

If the normal at P to the hyperbola $\frac {x^2}{a^2}-\frac {y^2}{b^2}=1$ meets the transverse axis in G and the conjugate axis in G' and CF be the perpendicular to the normal from the center C then ...
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1answer
31 views

What are the coordinates of your position?

Suppose you start at the origin, move along the x-axis 3 units. Then face downwards and move forward 4 units. Then turn right and move 7 units. Then (relative to your current position) face downwards ...