Tagged Questions

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

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0
votes
2answers
233 views

Help me with Cylinder -coordinates problem, back to Cartesian or not? How to do it fast?

Source of the problem, 3b here. Problem Question Electricity density in cylinder coordinates is $\bar{J}=e^{-r^2}\bar{e}_z$. Current creates magnetic field of the form ...
1
vote
0answers
62 views

geometry of points in $\mathbf{Z^3} $ and center of mass

Given the set of $37$ points in $\mathbf{Z^3} $ in which no 4 points are on the same plane. Show that there exist 3 point A,B,C in this set such that $ (\frac{x_A+x_B+x_C}{3}, ...
2
votes
3answers
463 views

How to fill up the gap between a typical advanced undergraduate algebraic curve course and High school basic geometry/precalculus course?

Based on this question i asked recently: A question about geometry of plane curve books, i think it is too advance for a HS student/ typical second or third year undergraduate math majors to read on ...
3
votes
2answers
1k views

How to calculate angles and X,Y coordinates for drawing a hand of playing cards on a canvas

Scenario: I'm programming a module to draw a deck of cards on a canvas as though they were being held by a person. Edit: I've cleaned up the question as best I can to be clearer. What I'm looking ...
1
vote
0answers
2k views

Direction ratio of a vector in 3d?

Suppose I have a vector whose starting end is the origin & the other end can be in any of the 8 quadrants (in 3d). I can easily get direction ratios for the vector & also know in which ...
0
votes
1answer
137 views

“Way” to decide if points are in a rectangle.

Suppose $P_1=(x_1,y_1)$, $P_2=(x_2,y_2)$ are two points. Also suppose that we have a rectangle which we just know the value of its sides $a$ and $b$. I am looking for some kind of formulation which ...
2
votes
0answers
44 views

elipsoid surface intersection in $\mathbb{R}^3$

Is there an explicit parametric solution describing the curve result of the intersection of two elipsoid surfaces with abitrary position and orientation in $\mathbb{R}^3$?
0
votes
1answer
34 views

Not very clear about “parametric form”

For example, let the surface $S$ in $\mathbb R^3$ be formed by taking the union of the straight lines joining pairs of points on the lines $$\{x=t,y=0,z=1\},\qquad \{x=0,y=1,z=t\}$$ with the same ...
1
vote
3answers
384 views

Constructing a line with a known line, intersection point and angle.

I am creating a Java game with collisions. I found myself stuck on the following problem. I have got two known lines: $y$ and $i.$ $i$ is the inbound direction and $o$ the outbound direction, ...
0
votes
1answer
2k views

Given two vertices, how to find the other two vertices of a rhombus?

$A\;(-3,-4) $ and $ C \; (5,4)$ are the ends of the diagonal of a rhombus $ABCD$. Given that the side BC has gradient $\frac{5}{3}$; How could we find the coordinates of $B$ and hence of $D$? ...
2
votes
1answer
297 views

Slope of a line perpendicular to a line of slope $0$.

I have three points: $P = (2,5)$ $Q = (12,5)$ $R = (8,-7)$ I need to find the equation of the line thru $R$ which is perpendicular to $PQ$. How do I do this? $PQ$'s gradient is ...
2
votes
1answer
49 views

Finding unknown 3D vector given 2 known vectors and 2 known angles

So I have a 3D vector math problem that I'm having difficulty solving. Basically I have two known vectors in the form (x,y,z), let's call them C and P, and I want to find a third unknown vector, let's ...
5
votes
1answer
366 views

What is the path equation that is created with the middle point of a fixed length line segment that touching both ends to an ellipse.

Ellipse equation is $(\frac{x}{a})^2+(\frac{y}{b})^2=1$ and the length of line segment is $2k$, if we move the line segment all around of the ellipse while touching both ends to the ellipse. What is ...
-1
votes
1answer
106 views

Properties of lines in the Pixel + Zoom geometry

Problems Prove that in PZ geometry, every PZ line has an equation of the form ax+by=c, where a, b, c are all rational numbers and a and b are not both zero. Prove that every equation of the form ...
11
votes
5answers
3k views

Calculating the area of an irregular polygon

Given the length of the sides of an irregular polygon (no coordinates provided) how do you compute the area of the maximum area of the polygon? Thanks in advance
1
vote
1answer
381 views

Finding the coordinates of a point

I have the $x$ and $y$ values to $2$ points $A(x_a,y_a)$ and $B(x_b,y_b)$. I need to determine the coordinates of a third point $S(x_s,y_s)$ that is at $r$ distance away from both A and B points and ...
0
votes
2answers
88 views

Existence of n-dimensional polyhedron given edges

The following assertion is true in $2$ and $3$ dimensions: Given $\sigma_{ij},\ 1\leq i\neq j\leq n$ with $\sigma_{ij}=\sigma_{ji}$ and $\sigma_{ij} \leq \sigma_{ik}+\sigma_{kj}$, then there exist ...
0
votes
2answers
2k views

Three-circle intersection for circles of unbounded integer radius

I have three circles. One is at $(0,0)$ and has radius $n$, another has is at $(1,0)$ and has a radius $m$, and the third is at $(0.5, \sqrt{0.75}))$ and has a radius of $o$. All of the radius values ...
3
votes
4answers
3k views

How to calculate the two tangent points to a circle with radius R from two lines given by three points

I need to calculate the two tangent points of a circle with the radius $r$ and two lines given by three points $Q(x_0,y_0)$, $P(x_1,y_1)$ and $R(x_2,y_2)$. Sketch would explain the problem more. I ...
2
votes
2answers
758 views

Problem Solving Question Relating to Directions and finding Burger Jack

I stopped at a street corner and asked for directions to Burger Jack. Unfortunately, the person I wasked was Larry Longway, whose directions are guaranteed to be too complicated. He said,"You are now ...
0
votes
2answers
1k views

Perpendicular Vectors

Find the equation of the line passing through a point $B$, with position vector $ \vec b$ relative to an origin $O$, which is perpendicular to and intersects the line $\vec r= a+ \lambda \cdot c$, ...
4
votes
2answers
577 views

Applied ODEs in trajectory problem

I'm having a hard time solving this problem: Let there be a town $A$ in a shore of a river. Let $x=0$ be the shore. Let $(0,0)$ be the location of the town. Let $B$ be another town, in the ...
3
votes
1answer
519 views

finding one circles radius so that it tangentially touches two other set circles

I am designing a water fountain on google sketchup and have run into a problem. I am designing the contours of the stone in the fountain. I would attach a picture of the problem but i need 10 ...
0
votes
3answers
186 views

Finding the $Y$-intercept of a line, given two points

I'm not sure how best to ask this, so I'll try to explain. Say I have a line drawn between the points $(-1,50)$ and $(2,30)$. How can I figure out the $Y$-value when the line crosses the $X$-value of ...
1
vote
0answers
74 views

Relationship between Number of circles required to surround a circles and the distance function?

In Why is a circle in a plane surrounded by 6 other circles, the implicit assupmtion is the distance is Euclidean, my question is: Are there any relation between the distance function being used and ...
1
vote
1answer
2k views

how can I obtain enclosed area between two circles in cartesian coordinates?

In the diagram below (from here fig.2, page.5) the enclosed area between two circles (shaded area) has been indicated $a_{t+\delta_{t}}$. Can anyone help me how can I compute this? is it true? ...
3
votes
1answer
129 views

What volume does $2x \le x^2+y^2+z^2 \le 4x$ represent?

I'm evaluating $\iiint_V f(x,y,z) dV$ where V is defined by $$2x \le x^2+y^2+z^2 \le 4x $$ To simplify things I swapped x and z, and moved to spherical coordinates: $$ 0 \le \theta \le 2\pi, 2 \cos ...
5
votes
1answer
124 views

An equivalence relation on regions of the plane.

Let $R\subseteq\mathbb{R}^2$. Consider the set of all "horizontal sections" $H_R =${$Rb|b\in\mathbb{R}$}, where $Rb=${$a\in\mathbb{R} | (a,b)\in R$}. Similarly consider the set of "vertical sections" ...
3
votes
1answer
152 views

How can one use the logarithm function to define angles?

In dealing with the complex logarithm function, I read that the imaginary part of $\log w$, is also called the argument of $w$, $\operatorname{arg }w$, and it is interpreted geometrically as the angle ...
1
vote
4answers
587 views

Finding the equation of a circle

$A=(3,1)$ and $B=(-1,-1)$ are points on a circle of center $(k, -3k)$ find the value of $k$ I begin by assinging the values $\ g = -k $ and $\ f=3k $. I then substitute $(3, 1)$ and $ g= -k, f= ...
6
votes
1answer
334 views

Why is $m$ used to denote slope?

What is the reason, historically, that the letter $m$ is used to denote the slope of a line?
2
votes
0answers
43 views

degeneracy loci of dimension $2$

Let $X$ be a smooth complex projective variety of dimension $n \ge 4$ and let $F$ and $E$ be two (holomorphic) vector bundles of rank $f$ and $e$ over $X$. Given a morphism $\varphi: F \to E$ of ...
3
votes
1answer
1k views

Use Pappus' theorem to find the moment of a region limited by a semi-circunference.

This is part of self-study; I found this question in the book "The Calculus with Analytic Geometry" (Leithold). $R$ is the region limited by the semi-circumference $\sqrt{r^2 - x^2}$ and the ...
3
votes
1answer
325 views

Maximum cosine for angle between 2 vectors when 1 vector is partially unknown

assuming I have two vectors $A$ and $B$, where $A$ is completely known and from $B$ I know only that the first k components are 0. What is the maximum possible cosine value for the angle between the ...
5
votes
4answers
630 views

Why do all circles passing through $a$ and $1/\bar{a}$ meet $|z|=1$ are right angles?

In the complex plane, I write the equation for a circle centered at $z$ by $|z-x|=r$, so $(z-x)(\bar{z}-\bar{x})=r^2$. I suppose that both $a$ and $1/\bar{a}$ lie on this circle, so I get the equation ...
7
votes
3answers
831 views

How does this equality on vertices in the complex plane imply they are vertices of an equilateral triangle?

I've read that if the complex numbers $a_1$, $a_2$ and $a_3$ are the vertices of a triangle in the complex plane such that $$ a_1^2+a_2^2+a_3^2=a_1a_2+a_2a_3+a_1a_3 $$ then the vertices are actually ...
1
vote
0answers
74 views

Translate group definition into geometry system

I need to reword the definition of group (the four axioms: closure, associativity, identity and invertibility) to be lines and points of non-Euclidean geometry (the axiom system defined as geometry). ...
3
votes
2answers
215 views

Find $DF$ in a triangle $DEF$

Consider we have a triangle ABC where there are 3 points, D, E, F such as point D lies on the segment AE, point E lies on BF, point F lies on CD. We also know that center of a circle over ABC is also ...
2
votes
4answers
7k views

How to Determine an Equation of a Circle using a Line and Two Points on a Circle

My question goes like this: Determine the equation of a circle tangent to the $x$-axis and passing through $(5,1)$ and $(12,8)$. I need not only the answers, but also the steps on how you did it so ...
0
votes
1answer
165 views

Vectors problem

can anyone help me with this problem: Is it possible to construct three vectors (a,b,c) in 3D, such that angle between a and b is 30 degrees, between a and c is 150 degrees, and between b and c is 30 ...
1
vote
1answer
112 views

Circle locus, how to satisfy the equation.

$A(-3,1), B(0,-5), P(X,Y)$ If $|AP| = 2|BP|$ prove that $x$ and $y$ satisfy the equation: \begin{aligned} \ x^2+y^2-2x+14y+30 =0 \end{aligned} I get as far as determining the ...
1
vote
2answers
898 views

Finding & Plotting equation of hyperbola given foci, and difference in distances between them.

I have to plot the hyperbola (3 of them actually) in MATLAB, and so it'd be good if I could find some sort of general formula. The foci do not necessarily have to be on the axes (e.g. $(5,3)$ and ...
1
vote
2answers
533 views

Stereographic projection of a regular tetrahedron inscribed in the Riemann sphere?

I've been reading about stereographic projections. I did a problem about finding the stereographic projection of a cube inscribed inside the Riemann sphere with edges parallel to the coordinate axes. ...
1
vote
1answer
245 views

How to scale a polyhedron contained a 3-sphere?

In the 3-sphere simulator I am building, the viewpoint is contained in the space of a 3-sphere (the surface of a 4-D hypersphere), and the user is able to navigate through it. There are some ...
3
votes
1answer
921 views

Equation of a sphere as the determinant of its variables and sampled points

Searching for an equation to find the center of a sphere given 4 points, one finds that taking the determinant of the four (non-coplanar) points together with the variables $x$, $y$, and $z$ arranged ...
3
votes
1answer
178 views

Algebra question about Triangle Interiors

I was reading about Triangle Interiors on Wolfram Alpha: http://mathworld.wolfram.com/TriangleInterior.html and they have a simple equation: $$\mathbf{v} = \mathbf{v}_0 + a\mathbf{v}_1 + ...
0
votes
2answers
662 views

Given the cartesian coordinates of four points, how to calculate the interection of two lines they form?

Given four complex numbers $A, B, C, D$ interpreted as points on the plane, how can I calculate the number that represents the intersection of the lines formed by $A, B$ and $C, D$?
0
votes
2answers
136 views

is this equation solvable?

Can someone please solve these 2 equations to get values of h and k? I know the values of h and k but not sure how to solved these equations to get h and k 's values $(20.01 - h)^2 + (17.94 - k)^2 = ...
1
vote
3answers
111 views

calculating a point on circumference

See the diagram Known values are A: (-87.91, 41.98) B: (-104.67, 39.85) C: (-96.29, 40.92) L: 14.63 // L is OC Known angles ...
2
votes
5answers
2k views

finding center of circle

How can I calculate center of a circle $x,y$? I have 2 points on the circumference of the circle and the angle between them. The 2 points on the circle are $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$. The ...