For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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Parametrizing regions of complex plane

Let $\Omega=\mathbb{C}\setminus \lbrace t e^{it} \ \vert t \in \mathbb{R}_{\geq0} \rbrace$ I need to write $\Omega= \coprod_{i=0}^{\infty} R_i$ where each $R_i$ is the region bounded by from $t=2k ...
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1answer
9 views

What separates the dot product from the scalar projection?

Just a little problem with geometric intuition here (or perhaps I just haven't slept in far too long!). I know that the scalar projection of vectors $ \vec{u} $ and $ \vec{v} $ is defined as $ ...
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1answer
19 views

Defining the equation of an ellipse in the complex plane

Usually the equation for an ellipse in the complex plane is defined as $\lvert z-a\rvert + \lvert z-b\rvert = c$ where $c>\lvert a-b\rvert$. If we start with a real ellipse, can we define it in ...
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Sum of a convex set and a complement of a convex set

In $\mathbb R^n$ equipped with the Euclidean norm, let $B$ be a convex set and $A$ be a subset such that $\mathbb R^n\setminus A$ is a convex set. Is it true that $A+B$ is a complement of a convex ...
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1answer
21 views

Find equation of ellipse given two tangent lines at given points and a point on ellipse

I'm attempting to generate an ellipse for a stair simulation game of mine, and the inputs are: - A point on the ellipse - The slope of the tangent line to the ellipse at that point - Another point on ...
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2answers
27 views

Gradient of a line using Pythagora's theorem

I've been trying to solve a question but I am having some difficulties. See the below diagram. I also know that the gradient of line c is twice as high as b i.e. c rises twice as fast as b. ...
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1answer
49 views

does the volume of a ball remain constant under deformation?

I'm a psychology student and was reading Piaget, he says that the volume of a sphere (ball of clay) remains constant if we deform the sphere into a roll for example, If you take the limit case of the ...
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3answers
33 views

Prove that if $l$ is a line in the classical euclidean plane, then there is a point $p$ that lies on $l$

Suppose that $\mathbb{P}$ is a Classical Euclidean Plane (satisfies all five of Euclid's postulates). Can you prove that if $L$ is a line in $\mathbb{P}$, then there is at least one point $p$ in ...
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1answer
23 views

show that two angles in a circumcircle are equal

We have the following circumcircle to ARP where PR = PQ are tangents to the smaller circle I need to show that the angle a = the angle b, which is equivalent to show that RP = AP', or show that the ...
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2answers
72 views

Euclidean Geometry challenge.

Can someone help me on this one? I have found that $\frac{1}{(x+1)^2}+1=\frac{1}{x^2}$, but I can't solve the fourth degree equation that comes with it. There must be a easier way!
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1answer
23 views

Explanation for relationship between hypotenuse segments and leg lengths?

|` | ` x | ` | ` c a | z /` | / ` y |_/ ` |/|_____` b I'm new to this SE, but I have an SO account, so hello! Can ...
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1answer
29 views

How to understand sinus?

In $\Delta PQR$ we have $\angle PQR=60^\circ$, $QR=4$ and $PR=a$. For which values of $a$ are there 0, 1 and 2 triangles matching the description? I think I'm supposed to use the law of sines, ...
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1answer
43 views

Construction of a triangle given some special points ($O,H,I$)

I'm a newbie in this site. I tried to search if this question was already answered but I'm not sure on how to do it. The problem is: given three distincts points $O,H,I$ namely the circumcenter, the ...
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0answers
14 views

Finding a plane where a vector lies (and generating its normal)

I have a vector in homogeneous coordinates $l = (a,b,1)$ and want to calculate a plane from it (for later doing intersections in 3D coordinates and calculate collision coordinates) with regards to the ...
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1answer
16 views

I don't understand this from my lecture note. What is described by $x=X(\xi)$? Why $\xi(t)$ is a curve since $\xi \in R^{m-1}$?

I don't understand this from my lecture note. What is described by $x=X(\xi)$? Why $\xi(t)$ is a curve since $\xi \in R^{m-1}$?
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0answers
23 views

Geometric proof concerning circles

From a point P outside a circle draw two tangents to the circle touching at points A and B. draw a secant line intersecting the circle at points C and D. choose point Q on chord CD such that angle DAQ ...
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2answers
40 views

Triangle in parabola

I have a problem. In my triangle one vertex is in the vertex of the parabola and two others are in parabola. This is a isosceles triangle and I know one angle in this triangle : 120 grades. The ...
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0answers
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Difference in (the concept of the distance between a line and a point) between Euclidean and non-Euclidean Geometry

What is the difference in (the concept of the distance between a line and a point on a graph) between Euclidean and non-Euclidean Geometry. Is this concept the same in all kinds of geometry??
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2answers
23 views

Can ellipse equation be transformed through one of its foci?

Can we transform ellipse equation to represent an ellipse transformed by tilting it through its focus such that its center point moves in circular manner and one of its focus stays at constant ...
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1answer
37 views

Weil divisors fail over singular varieties

Let be $k$ an algebraically closed field. We know that if $X$ is an irreducibile, normal variety, one can associate to every rational function $(f)\in k(X)^*$ a Weil principal divisor $$(f)=\sum_{Y} ...
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0answers
70 views

Crossed Ladders Problem

Two ladders, one 10 meters long and the other 8 meters [long], have been placed in a trench as indicated in the opposite figure. Their point of intersection, M, is 3 meters from the base of the ...
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0answers
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Rational representation of conics

Currently I'm beginning my study of rational curves (Rational Bezier and NURBS) all books that I've read tell me that is "well known" that conics can't be represented by Bezier or even a B-Spline. ...
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2answers
23 views

How can I use Menelaus' theorem here (Simson line)?

Given 4 points on a circle A, B, C, and P. Draw the orthogonal projections of P onto triangle ABC and call them $P_1, P_2,P_3$. Show that $P_1, P_2,P_3$ are collinear. After drawing this out, I ...
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1answer
11 views

Proving that $2$-D parabolic coordinates are orthogonal

How can we prove that the parabolic coordinate system in two dimensions is orthogonal? I tried using the dot product, but don't know where to start or what basis vectors can be used in two dimensions. ...
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1answer
35 views

How would I use vectors for this geometry problem?

Consider a quadrilateral ABCD. K, L, M, N are the midpoints of the segments AB, BC, CD, DA respectively. O is the intersection point of LN, KM. Let P and Q be the middle points of the diagonals AC ...
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2answers
38 views

Trisecting the sides of a triangle.

Consider the hexagon formed by the six points which trisect the sides of a triangle(two on each side). Is is true that when we connect opposite points in this hexagon, the lines intersect at a single ...
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2answers
70 views

Distance between four points

I have four points as shown in this figure: I want to calculate one vector for all these points. So, what would be the correct way: 1) I take the vector between $A-B, B-C, C-D$ and add them $(A-B ...
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1answer
29 views

Is there a name for this special, “most parallel” ultraparallel line in hyperbolic geometry?

Suppose you're in the hyperbolic plane, and you have a line L and a point P not through L. There are an infinite number of lines parallel to L that go through P. However, there's one line M which is ...
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6answers
408 views

Is it possible to find the area of a shape from its perimeter?

Is it possible to find the area of a free form shape knowing the perimeter? An example would be a clover leaf shape. If the perimeter is 96 how would I know what the area would be?
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26 views

a valuable question about differential geometry, the curve$\beta(s)=\alpha(s)-rn(s)$

Let $\alpha(s)$,s$\in [0,l]$ be a closed convex plane curve positively oriented. The curve $$\beta(s)=\alpha(s)-rn(s)$$,where r is a positive constant and n is the normal vector, is called a parallel ...
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38 views

27 Lines on a Cubic

In Ravi Vakil's notes, there is a proof (in section 27, of course) of the famous result that every nonsingular cubic hypersurface in $\mathbb{P}^3_k$ over an algebraically closed field $k$ has exactly ...
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20 views

Intercept planet following an elliptical path (i.e. interplanetary space travel)

So, just as in this question (Intercept path to object following an elliptical path) I have a simple game where I want spaceships to intercept planets, which follow elliptical paths (in my case ...
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3answers
38 views

The lines with equations $y = 5x − 6$ and $10x + cy = 8$ are perpendicular, find c [on hold]

The lines with equations $y = 5x − 6$ and $10x + cy = 8$ are perpendicular. Find the value of c. Well, I am not sure even where to start
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1answer
60 views

How can I find the common axis of 2 cones in space that have the same base radius but different heights?

How do I find the 3D vector describing the axis of 2 overlapping cones, like this: If I have only the following information: Coordinates of the common tip Coordinates of a point on the yellow ...
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0answers
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Showing there is a cartesian coordinate system on EG.

I'm pretty sure I should just show there is a bijection between the points in EG and elements of R^2. How do I do this? note: EG=Euclidean Geometry
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19 views

Questions on curvilinear asymptotes

I just saw curvilinear asymptote which sort of fascinated me. A little bit of thinking raised two questions for which I couldn't get the the answer by googling. Is there a general method to find a ...
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1answer
23 views

Line between points in projective space?

I am trying to find the line through the points $(0 : 1 : 0)$ and $(1 : 1 : 1)$ in $\mathbb P^2$ and $(0 :1 : 0: 1)$ and $(1: 1: 1: 0)$ in $\mathbb P^3.$ Would the first line be the set of points ...
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1answer
37 views

Formula to calculate angle on a fan or semicircle

How do I calculate the angle shown in the picture given the height, width, and the arc deduction of $2$? I had applied the Right Triangles formula to calculate the hypotenuse: $h^2 = a^2 + ...
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1answer
11 views

find a point such that the distance between this point and a plane is equal to the distance between another point and a line [on hold]

I have: $r:(2,1,0)+\lambda(0,4,-3) \qquad \pi:4y-3z-4=0 \qquad A=(2,4,4)$. I have to find a point $C$ such that $d(C,\pi)=d(A,r)$ where $d$ is the distance
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0answers
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Can Magnification/Scaling (transformation) be prepresented by a vector?

Vectors represent three bits of information: Magnitude, Line of Action, and Direction. A Translation (transformation) can be represented by a vector: object is moved By so much (magnitude) Along a ...
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0answers
17 views

Parametric equations for hypocycloid and epicycloid

Suppose that the small circle rolls inside the larger circle and that the point $P$ we follow lies on the circumference of the small circle. If the initial configuration is such that $P$ is at ...
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0answers
16 views

For each point on a line there exists a unique perpendicular line through that point

I'm trying to show that in an absolute plane (only the first four axioms without the parallel axiom hold) for each Point $P\in l$ there exists exactly one perpendicular line through $P$. My idea was ...
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1answer
12 views

Given a number of vertices , a radius, and rotation calculate vertices' coordinates for regular polygons

So, I know half the answer to this but I don't know how to adjust it for rotation. I believe formula the below is correct if I did not have to take into account rotation. $r \cos(2 \pi i / n) = y$ ...
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0answers
15 views

Point on ellipse after walking a distance on the perimeter [duplicate]

I've the equation of an ellipse. Given a point (x,y) on the ellipse and a length L , I want to find the coordinates (x1,y1) of the point where I'd end up after taking a walk of length L from (x,y), ...
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Find Formal Proof on Loci Theorems

Please help me to prove this theorems of loci Theorem 12 The locus of a point at a given distance from a given line is two lines parallel to the given line and at the given distance from it. Theorem ...
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8 views

$\frac{4}{3}(1,-1,0)^T+\textrm{vect} \left((-1,1,-1)^T\right)$

Here I have a passage on the conclusion, but I do not understand it. the conclusion say : $f$ is the oriented axis of screw.. how I can from $\frac{4}{3} \begin{pmatrix}1\\1\\0 ...
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Can ${n \choose 2}$ points be covered by lines determined by $n$ points?

Let $S=\{(a_1,b_1),\ldots,(a_{n \choose 2},b_{n \choose 2})\}$ be ${n \choose 2}$ points on the plane. Does there exist $n$ points, such that the lines determined by the $n$ points cover all the ...
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0answers
8 views

union of two Geodesics

Let $X$ be a metric space. $a,b,c \in X$ and $\sigma_1 , \sigma_2$ geodesics from $a$ to $b$. Let $\sigma$ geodesic from $b$ to $c$. Want to show : $\sigma_1 \cup \sigma $ geodesic $\Rightarrow ...
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1answer
20 views

Length of tendon in circle [on hold]

What is the length of chord that pass on two specific point. For example I have circle ( r=1) point1 :(x1,y1) point2(x2,y2); length of chord?
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20 views

a question about differential geometry, the relation between osculating plane and the points of $\alpha(s)$

question:please prove the limit position of the circle passing through $\alpha(s)$,$\alpha(s+h_{1})$,$\alpha(s+h_{2})$ when $h_{1}$ and $h_{2}$ approaches 0 is a circle in the osculating plane at s, ...