Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [analytic-geometry]

Questions on the use of algebraic techniques for proving geometric facts.

0
votes
1answer
11 views

Derive the implicit cone equation from the implicit circle equation

Is it possible to derive the implicit equation of a cone $x^2+y^2-z^2=0$ from the circle equation $x^2+y^2=1$, which is the intersection between the cone and the hyperplane $\{(x,y,z)\in\Bbb R^3\,|\,z=...
0
votes
0answers
29 views

General form of quadric surfaces

The general form of quadric surfaces is $$Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0$$ I want to classify all of the possibilities including degenerate cases with the help of ...
1
vote
1answer
16 views

Decompose a unit ball into 3 convex disjoint parts parts with common boundary

I guess that it is impossible to decompose the open unit ball $B(0,1)$ of $R^n$ for some $n\in\mathbb{N}$ in for disjoint sets $A$,$B$,$C$,$D$, such that $A$,$B$,$C$ be convex open subsets and the ...
0
votes
0answers
12 views

Tangents to different branch of hyperbola Mathematical proof

We can draw two tangents from an external point to a hyperbola The two tangents can be made on; ...
1
vote
2answers
52 views

How to find the minimum distance from origin to locus of P?

A straight line through $A(6,8)$ meets the curve$2x^2+y^2=2$ at $B$ and $C$. $P$ is such a point on $BC$ that the distances $AB, AP, AC$ are in Harmonic Progression. Find minimum distance from origin ...
1
vote
0answers
26 views

Partition of a rectangle into two subrectangles

Let $R=[a_1, b_1] \times ... \times [a_n , b_n]$ be a rectangle in $\mathbb{R} ^n$. Then: $R_1 , R_2$ are rectangles in $\mathbb{R} ^n$ such that $R=R_1 \cup R_2$ and $int(R_1)\cap int(R_2) = \...
2
votes
1answer
27 views

Prove the lines from the foci to a point on an ellipse form equal angles with any tangent vector at that point

Suppose we have an ellipse $\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$ with parametrization $\gamma(t)=(p\cos(t),q\sin(t))$. Let $\vec{p}= \gamma(t_0)=(p\cos(t_0),q\sin(t_0))$ be any point on the ellipse. Let ...
1
vote
2answers
20 views

Equation of the line perpendicular to the asymptote of the graph

Find the equation of the line perpendicular to the asymptote of the graph of the function $f(x)=\displaystyle\frac{-3x+1}{x^2-2x+1}$.
3
votes
0answers
70 views

How can I eliminate $m$ and $n$ the following equations?

To solve the problem: Lines through vertices of $\triangle ABC$ and a point $Q$ meet opposite sides at $M$, $N$, $P$. When is $Q$ the orthocenter of $\triangle MNP$?, I tried to use methods of ...
1
vote
2answers
41 views

Finding coordinates of a point in a $2d$ space

I have the coordinates of point $A$ and the angle $\alpha$ (starting from $y$ axis going clockwise) and the distance from $A$ to $B$ called $s$. $B$ lies on a line that is perpendicular to the angle's ...
-1
votes
1answer
31 views

Find the equation of ellipse: Focus at $(-\sqrt{13}, 0)$, vertex at $(0,2)$ [closed]

The equation has its centers on the origin and their major axes on OX. I want to find the equation of the given Focus at $(-\sqrt{13}, 0)$ and vertex at $(0,2)$ .
5
votes
1answer
30 views

Prove That Area of Isoceles Triangle in an Ellipse is Maximum When Vertex On The Major Axis Lies On The Line Of Symmetry of the Triangle?

This is one of 101 classes questions whose solutions can be easily found on google, but most of the solutions assume without giving any proper line of reasoning that to maximize area (unique)vertex on ...
0
votes
0answers
55 views

Rounding a trapezoidal function

I need to round the profile of a trapezoidal wave function. In case of no rounding, the function $f\left(x\right)$ is defined as: $$ f\left(x\right)= \begin{cases} \frac{A}{l_x}\cdot x & 0\...
3
votes
0answers
46 views

generic parabola in polar coordinates

Starting from the equation $y=ax^2+bx+c$ substituting I get the next equation in polar coordinates: $$a\cos^2 \theta\ \rho^ 2 + (b \cos \theta - \sin \theta)\ \rho + c = 0$$ in case C was $0$ we could ...
0
votes
0answers
13 views

Calculate new X & Y coordinate based on compressed or enlarged rectangle

I have two Rectangles as Rect1 ___________ x'', y'' | |dy | | .x',y'| | | | | 0,0------------- Here the value of x''...
1
vote
0answers
24 views

Finding the whole triangle information by one point.

I wonder if there's any way to blend two (or more) RGB colors in a reversible way? I mean, imagine we have an RGB pixel (R: 55, G:35, B:255), and we need to extract the two other RGB pixels that ...
0
votes
0answers
42 views

Distributing numbers on a spiral. [closed]

How can I distribute 65,536 numbers (from 0 to 65,535) in an Archimedean Spiral (or any other kind of Spiral or geometric shape), in a way that I can read the position of each number (as a point on ...
2
votes
1answer
58 views

Topology of $x^2+y^2 = 1$ over $\mathbb{C}^2$

I am trying to prove that $$V=\{(x,y) \in \mathbb{C}^2| \ x^2+y^2 = 1\} \simeq \mathbb{C}^* = \mathbb{C}\setminus \{0\}$$ where $\simeq$ is to be intended as homeomorphic. Fix $y\neq \pm 1$ then $x^...
0
votes
1answer
47 views

Are the directrix line and focus unique for a conic?

How to prove of disprove the uniqueness of pair of (or single for parabola) directrix line and pair of focal points for a conics? (Ignore the degenerate cases and the circle.)
2
votes
1answer
47 views

Average area of a rectangle inside the unit square

I recently came across this problem while toying with the problem of the average distance between 2 points in the unit square. These two points also define a rectangle, so I was wondering: What is the ...
0
votes
1answer
42 views

A tricky question on circles - Loney Exercise XVIII, problem 14

I am brushing up some plane and solid analytic geometry before taking a course on multivariable calculus. I am deriving important results and solving through the book, Co-ordinate Geometry by SL Loney....
0
votes
2answers
80 views

How to solve this ellipse problem?

Assume that ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$, its left focus is $F$, the line $l$ passing $F$ intersects the ellipse $C$ at $A$ and $B$. The angle of inclination of $l$ is $\...
0
votes
1answer
27 views

A straight line is fit to a data set (ln x, y). This line intercepts the abscissa at ln x = 0.1 and with slope of −0.02. Find the value of y at x = 5?

A straight line is fit to a data set $(ln x, y)$. This line intercepts the abscissa at $ln x = 0.1$ and has a slope of $−0.02$. What is the value of $y$ at $x = 5$ from the fit? I can not understand ...
0
votes
0answers
21 views

An equation of tangential axis to a curve

I have a following mathematical problem: Let's consider a curve: $$\alpha \left(t \right) = \left(\frac{1}{4}t^4 , \frac{1}{3}t^3 , \frac{1}{2}t^2 \right) , t \in R$$ Find an equation of ...
2
votes
0answers
21 views

Book reference for Analytical 3D geometry

I'm looking for a book which covers the following topics: Generating lines, Cone, Cylinder, Paraboloid, Ellipsoid, Hyperboloid of one and two sheets. Their tangent planes, normal lines, director ...
7
votes
3answers
2k views

Why does my textbook say that equation $y = 7-3x$ has infinite number of solutions while it has only one root?

Anywhere I've looked, the definition of solution of equation is root(s) of that equation. But why does a textbook say that equation $y=7-3x$ has infinite number of solutions? Thanks
1
vote
1answer
40 views

Finding mid points after rotating two lines

If $A(a,0)$ and $B(-a,0)$ are two fixed points,$(a>0)$.$P(x,y)$ is any point in the upper half of the $x,y$ plane.The join $AP$ is turned about $A$ clockwise through 90 degree and $BP$ is turned ...
0
votes
0answers
20 views

Generalised method for finding the common normal of any two graphs in 2D

I've always had trouble finding the common normals of two conic sections... In examinations, when I need to find the minimum distance between two graphs quickly, I find it very very difficult. I've ...
1
vote
1answer
20 views

The limit about the line connecting the intersection of a circle and the $y$-axis and the intersection of the shrinking circle and a fixed circle

There is a fixed circle $C_1$ with equation $(x - 1)^2 + y^2 = 1$ and a shrinking circle $C_2$ with radius $r$ and center the origin. $P$ is the point $(0, r)$, $Q$ is the upper point of intersection ...
1
vote
1answer
14 views

Showing the co-ordinates of the mid point of a line segment (from Stewart Analytic Geometry Review).

This is Question 43 from Stewart's review of analytic geometry, which can be found in PDF form here: https://www.stewartcalculus.com/data/CALCULUS_8E_ET/upfiles/6et_reviewofanalgeom.pdf There is no ...
0
votes
2answers
28 views

Are there functions with a constant output regardless of input, or functions whose input is limited to a single number? (Cartesian coordinates)

I was wondering whether a function $y=f(x)$ can be defined such that (1) its value $y$ is always constant, no matter what number substitutes $x$, (2) its argument $x$ is limited to a single number? ...
0
votes
1answer
15 views

Intuitive questions about the shape of an $\ell_1$ ball in dimensions $n \geq 4$?

It is easy to intuitively see that the shape of a two-dimensional $\ell_1$ ball is a sort of diamond, and that the three-dimensional generalization of it will be a similar shape, i.e., a shape where ...
0
votes
0answers
18 views

About Elements of a Vector Space, and Line Segments between them

I need complementary explanations for some basic definitions I encountered during introductory Geometry works. My first question: Given $V$ any vector space over a field $\mathbb{F}$, when we cite ...
1
vote
2answers
56 views

Simplifying Radicals In Heron’s Formula

When I sometimes use Heron’s formula and the Pythagorean Theorem in the coordinate system to find the area of a triangle, I get stumped at the last step: simplifying the radical. Is there a general ...
11
votes
4answers
919 views

Is there a possible geometric method to find length of this equilateral triangle?

Problem Given that $AD \parallel BC$, $|AB| = |AD|$, $\angle A=120^{\circ}$, $E$ is the midpoint of $AD$, point $F$ lies on $BD$, $\triangle EFC$ is a equilateral triangle and $|AB|=4$, find the ...
0
votes
1answer
48 views

How do you find the equation of an ellipse given the foci and and an arbitrary point on it? [closed]

Find the equation of the ellipse passing through the point (-2,-1) and with foci (1,-1) and (-2,-4).
3
votes
3answers
45 views

Find the area of parallelogram and its missing vertex

Given three radius-vectors: $OA(5; 1; 4), OB(6;2;3), OC(4;2;4)$, find the missing vertex $D$ and calculate the area of obtained parallelogram. My attempt: Firstly, we are to find the vectors which ...
0
votes
1answer
30 views

How to calculate the co-ordinates of a quadrilateral knowing the side lengths

Hi I've come across a mathematical problem, which I can't seem to solve with my limited geometry and trigonometry knowledge. I need to draw a quadrilateral (may not be rectangle always). To draw this ...
3
votes
1answer
61 views

Proving $ 2 $ angles are equal.

Hi, so I am doing a proof but I need some help proving one part of it. I'm having trouble proving that angle $ D'Bi = $ angle $ D'iB $. point $ i $ is the incenter of triangle ABC and D' is the point ...
2
votes
2answers
62 views

A cone with guiding curve $x^2+y^2+2ax+2by=0$ contains $(0,0,c)$. Its section by $y=0$ is a rectangular hyperbola. Prove its vertex lies on a circle.

A cone has its guiding curve to the circle $x^2+y^2+2ax+2by=0$ and passes through a fixed point $(0,0,c)$. If the section of the cone by plane $y=0$ is a rectangular hyperbola. Prove that the vertex ...
-2
votes
4answers
58 views

Distance between the origin (0,0) and a line y = ax +b [closed]

Derive a formula for the Euclidean distance between the origin $(0,0)$ and a line $y = ax + b$, where $a$ and $b$ are arbitrary constants.
0
votes
2answers
46 views

Find the locus of the middle point of the intercept on the line y=x+c made by the lines 2x+3y=5 & 2x+3y=8, c being a parameter?

Here's my shot: since the two lines are parallel, I figured that the middle point should be equidistant from the parallel lines,so using distance formula:- $ \frac{2x+3y-5}{\sqrt{13}}=\frac{2x+3y-8}{\...
1
vote
2answers
22 views

Corresponding Point for a Glide Reflection

I was wondering if there was an efficient method that could solve these types of problems. Here is the problem: Plot the points K = (0,0), L = (7,-1), M = (9,3), P = (6,7), Q = (10,5), and R = (1,2)...
1
vote
1answer
21 views

Why does a unit vector point in the same direction? [duplicate]

I know how to compute the unit vector $$\hat{\textbf{u}} = \left( \frac{u_1}{||u||} , \dots , \frac{u_n}{||u||} \right)$$ and I also know how to show that this will have length 1 by using the ...
0
votes
0answers
26 views

Mirror image around a curved line (say a parabola $y^2 = 4x$)

I wanted to know how we can find the mirror of an object across a curved line in the $XY$ plane, like a parabola. Along a straight line in the $XY$ plane the procedure is intuitive enough. But I ...
2
votes
0answers
43 views

How to define the complement of a “region” in $\mathbb{R}^d$ using boxes.

I have a question related to set theory, of which I am a beginner. Please add/change tags if you have better references. Other than describing my main question, I'm also highlighting with the symbol [?...
2
votes
1answer
29 views

Sum of vectors in any linear space

Suppose that vectors $x_1,x_2,…,x_n$ have the following property: for each $i$ the sum of all vectors except $x_i$ is parallel to $x_i$. If at least two of the vectors $x_1,x_2,\dots ,x_n$ are not ...
0
votes
0answers
8 views

Analytic sets in a Manifold

For a holomorphic mapping $ f:M\to N$ of manifolds, where $k\in\mathbb{N}$ is fixed. The set $\{x\in M:rk_xf\le k\}$ is a analytic by the Rank Theorem. I dont see why this result holds trivially, ...
0
votes
1answer
64 views

How to calculate ray

In ray-tracing technique critical point is to calculate rays which came out from eye $E$ to target $T$ through pixel $P_{ij}$ on viewport. The "viewport" is represented as rectangle divided to square ...
0
votes
1answer
18 views

Orientation of a normal vector of a plane

I found this question: vector normal to a plane It seems to be related to a problem I'm struggling with, but I need to know what is the rule for the normal vector's orientation. Assuming we want a ...