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

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2
votes
3answers
38 views

Expand polygon to grid in $x-y$ plane

Given a polygon in the $x-y$ plane, what is the simplest formula for expanding the polygon so that all sides lie on a grid? The image below demonstrates the problem I am trying to solve. The filled ...
2
votes
1answer
69 views
+50

A connectivity-preserving function from a connected set onto an interval

Let $C$ be a connected set in the plane and $I$ the unit interval interval. Call a function $f$ from $C$ onto $I$ Connectivity-preserving if the following is true for every subset $I'\subseteq I$: ...
11
votes
1answer
225 views

What is the geometry behind $\frac{\tan 10^\circ}{\tan 20^\circ}=\frac{\tan 30^\circ}{\tan 50^\circ}$?

This identity is solvable by help of trigonometry identities , but I think there is an interesting and simple geometry interpretation behind this identity and I can't find it. I found it when I ...
19
votes
7answers
2k views

Is it possible to solve any Euclidean geometry problem using a computer?

By "problem", I mean a high-school type geometry problem. If no, is there other set of axioms that allows that? If yes, are there any software that does that? I did a search, but was not able to ...
0
votes
1answer
19 views

Question on Geometry and cyclic circles

Two circles intersect at $P$ and $Q$.Through $P$ two lines $APB$ and $CPD$ are drawn to intersect circles at $A,B,C,D$. $AC$ and $DB$ when produced meet at $O$. How do I prove that $OAQB$ is cyclic ...
0
votes
0answers
16 views

approximate projection into eigenvector space

Given a matrix A, $3 \times 3$, that is symmetric I calculate a matrix V, $3 \times 3$, whose columns are the corresponding right eigenvectors and a diagonal matrix D, $3 \times 3$. of eigenvalues so ...
0
votes
1answer
19 views

Find the height $h$ of a circular segment based on the Radius $R$ and length $c$

I have found the formula for calculating the R of a circle, based on a circular segment, which is: $$ R=h/2+c^2/(8h)$$ where $R$ is radius, $c$ is the length of the segment, and $h$ is the height of ...
-2
votes
0answers
22 views

The volume of a specific rectangular prism is represented by $V(x) = -2x^3 + 10x^2 + 300x$. How do roots, vertices, and end behavior apply?

The volume of a specific rectangular prism is represented by $V(x) = -2x^3 + 10x^2 + 300x$, where $x$ is the height of the prism. How do roots, vertices, and end behavior apply? How is the graph ...
0
votes
0answers
24 views

disk-disk intersection area

I have two disks of radii $R_1, R_2$ with distance between centers, $d < R_1 + R_2$. How can I find the surface area common to the two disks? Rationale: Solar irradiation / energy input in ...
0
votes
1answer
23 views

How many points among them should we take to ensure that some two of them are less than the distance $1/5$ apart?

We are given a fixed point on a circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0,1,2,\ldots$ from it we obtain points with ...
0
votes
1answer
26 views

Jordan curve of infinite length

I was thinking about Jordan curve with infinite length and Koch snowflake seems to be a valid answer intutively. Can anyone give mathematical proof for this?
0
votes
0answers
34 views

Having trouble interpreting the geometry of this setup.

A circular conductor, with cross section given by $(x-d)^2+y^2=b^2$, i.e. radius $b$ and centered on $x=d$, has a circular core, made up of the interior of the circle $x^2+y^2=a^2$, with ...
0
votes
2answers
24 views

Find the projection of the line $x+y+z-3=0=2x+3y+4z-6$ on the plane $z=0$

Find the projection of the line $x+y+z-3=0=2x+3y+4z-6$ on the plane $z=0$ The equation represents the line of intersection of two planes. Using augmented matrix $$ \begin{bmatrix} 1 & 1 ...
1
vote
3answers
36 views

How to prove that $x^2 + 3y^2 = 1$ is contained inside of the unit ball?

What is the best way to show that $S = \{(x,y) | x^2 + 3y^2 = 1\}$ is contained in the unit ball without graphing the set?
1
vote
0answers
23 views

Given $5$ points on a sphere, divide the surface into $5$ congruent connected regions containing one point.

There are $5$ points on the surface of a sphere. Is it always possible to divide the surface into $5$ connected congruent regions such that each region contains one of the $5$ points?
0
votes
1answer
22 views

Constructible numbers defined over the rationals

If $z$ is constructible, then its minimal irreducible polynomial has a degree a power of $2$. Does the polynomial have to be defined over the rationals? I am asking this because we can ...
0
votes
0answers
12 views

Hyperplane of an mn-dimensional space [on hold]

Can someone explain to me why the hyperplane of a $mn$-dimensional space would have dimension $(m-1)n$?
0
votes
1answer
8 views

Computation of minimal axis-aligned bounding box of an arc segment.

I'm trying to compute the minimal bounding box of an arc segment so when it's time to render it, I only have to examine pixel coordinates within a minimal rectangular region. The code below covers ...
-4
votes
2answers
65 views

The area of square [on hold]

What is the area of the square versus a,b and c ? Thanks Edit to clarify the question, based on the OP's response to comments. The segments with lengths $a$ and $c$ are parallel, joined by $b$ ...
2
votes
4answers
180 views

Proving algebraic equations with circle theorems

I got as far as stating that OBP=90˚ (as angle between tangent and radius is always 90˚), and thus CBO=90˚- 2x. CBO=OCB as they are bases in a isosceles. COB=180-90-2x-90-2x. But after this, i am ...
3
votes
2answers
83 views

How to determine the reflection point on an ellipse

Here is my problem. There are two points P and Q outside an ellipse, where the coordinates of the P and Q are known. The shape of the ellipse is also known. A ray comming from point A is reflected by ...
0
votes
1answer
20 views

What does equally oriented mean

What does it mean for two triangles to be equally oriented? I have heard this term a lot but I haven't seen a definition of it. I know that in $3$-space two triangles are considered to be equally ...
0
votes
1answer
16 views

Conformal curvature line parametrization

While reading a paper I found a definition which is confusing me. Def: A conformal curvature line parametrization $(x,y) \to F(x,y)$ is called isothermic. I know what a conformal ...
2
votes
1answer
105 views

Which of the $43,380$ possible nets for a dodecahedron is the narrowest?

I want to fit multiple regular dodecahedron nets on to an infinitely long roll of paper. I want this to result in the largest possible dodecahedrons, for a roll of a given width. My hunch is that the ...
4
votes
3answers
275 views

Topological boundary vs geometric boundary

Let $M_1=B((0,0),1)=\{(x,y) \mid x^2+y^2<1\}$ $M_2=\{(x,y) \mid x^2+y^2\le1\}$ What are the interior of $M_1$ and $M_2$ ? And what are the boundary of $M_1$ and $M_2$ ? How do I find them? ...
0
votes
1answer
10 views

Modulo angle (rotation) in $2D$ space

Input parameters: space dividor (number), vector (vec2) Desired result: Divide space in $X$ sectors, then move all vectors to one sector. (Angle of any vector wont be larger then $360/X$.) Example ...
3
votes
1answer
62 views

Circle is similar to a polygon with infinite number of sides

It is know from the time of Euclid, that a circle is similar to a polygon with infinite number of sides. But this ^^ is informal. Do you know any formalization where it appears that a circle is a ...
0
votes
3answers
28 views

calculate radius of circle that by given length of square that is inside it

in this picture a length of square edge is 8 cm. I want to calculate the radius of circle. i try to calculate it, but i don't know how. I calculate this:
2
votes
4answers
17k views

Finding parametric equations for the tangent line at a point on a curve

Find parametric equations for the tangent line at the point $(\cos(-\frac{4 \pi}{6}), \sin(-\frac{4 \pi}{6}), -\frac{4 \pi}{6}))$ on the curve $x = \cos(t), y = \sin(t), z=t$ I understand that in ...
5
votes
2answers
260 views

Looking for a (nonlinear) map from $n$-dimensional cube to an $n$-dimensional simplex

I am looking for a (nonlinear) map from $n$-dimensional cube to an $n$-dimensional simplex; to make it simple, assume the following figure which is showing a sample transformation for the case when ...
1
vote
2answers
452 views

Possible areas of an isosceles triangle

I've got the following GRE question. I would like to know how to do this using only the knowledge that is required for the GRE test. In triangle ABC , AB=AC=2. Which of the following could be the ...
3
votes
2answers
360 views

Find parametric expression of an arc given its start point, end point and central angle in 3D cartesian coordinate system

In a 3D cartesian coordinate system, the coordinates of start point and end point have been given as $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$. If the central angle of the two points (the one smaller ...
1
vote
1answer
32 views

Showing that $\alpha$ satisfies the equation $\sin 2x=x$

This is an A level question. For better understanding, I will attach a screenshot of the question and the mark scheme. Question: Here's what I have done: $$A(OBA) = \frac 12r^2α$$ [basic ...
2
votes
3answers
54 views

Basic question about angles and measurement in degrees

I have a doubt related to angles which I am a bit embarrassaed to ask since I know is something of basic geometry, but nevertheless my question is the following: As I understand it, an angle between ...
2
votes
1answer
43 views

Quarter Circle packing

Just today, I was making tortilla chips, and I began to wonder, what is the most efficient way to pack circular quarters onto the plane? This sort of circle packing is most efficient for circles, ...
4
votes
0answers
33 views
+100

Point set where each point has unity distance to all other points ($L_1$ metric)

I want to construct a point set where each point has the same (w.l.o.g., unit) distance to all other points in the $L_1$ metric. Example: The points $\left(\frac{1}{2},0\right)$, ...
0
votes
1answer
389 views

How to get perpendicular line to an edge of a polygon.

This is a pretty basic geometry question, but I couldn't find an answer clear enough for me on Google (I don't know much about math). Let's say I have a rectangle. I have the coordinates for the four ...
1
vote
3answers
42 views

Distance of centroid to incenter

Suppose there is a right triangle where all side-lengths are integers. The distance from the circumcenter to the centroid of the triangle is 6.5. Find the distance from the centroid to the incenter ...
0
votes
1answer
24 views

How do you calculate the change in thickness of a cylinder, if you shave off a flat section?

I have a piece of steel, cylindrical (hollow), 200mm outside diameter with 160mm inside diameter (...
-3
votes
2answers
33 views

How to determine that the 3 points given in homogeneous coordinates are collinear? [on hold]

How do I prove that the 3 points given in homogeneous coordinates are collinear? $$A=(1,3,2)^T, B=(0,6,8)^T, C=(3,3,-2)^T$$
1
vote
1answer
18 views

Rotation matrix between two similar cuboids using their upper sides ( and the planes defined by these sides)

I have two different images and with them an estimation of two planes ( defined in the same system). I would like to get the rotation matrix, quaternion or euler angles of a surface within this ...
2
votes
1answer
61 views

Does a convex hull solution in 3 dimensions result in a minimum-area or maximum-volume solution?

The wikipedia entry for convex hull shows a 2-d example of a random set of points on x-y plane, and the "elastic band" solution that bounds the points with the convex hull solution. The definition of ...
0
votes
1answer
26 views

length of radius of circles between their tangents

In this question, we have five circle that touch each other. we draw their tangents. If we know that smallest circle radius is 8 and biggest circle radius is 18, then what is the length of PF? Note: ...
0
votes
1answer
18 views

Translate a Rectangle Position from 1 Image to another [on hold]

I have a Large Size Image.Since its too large for processing within a small time, i need to resize it.I have the coordinates of a rectangle in the resized image.Is there a way i can translate this ...
0
votes
3answers
22 views

Area of rectangle and triangle derivation

I was wondering about the derivation for the area of a triangle and the area of a rectangle. Of course, we all know them to be $\dfrac{1}{2}bh$ and $bh$ respectively, but where is the derivation of ...
4
votes
2answers
391 views

Axis aligned rectangle inscribed in rotated rectangle

I start with an axis aligned rectangle, $R$, that I rotate by the angle $\theta$ to get $R'$. Afterwards I'd like to identify another axis aligned rectangle, $P$ with the following additional ...
0
votes
0answers
21 views

locus of a variable straight line [on hold]

Geometry: A variable straight line always intersects the lines x=c,y=0; y=c,z=0; z=c,x=0. find the equation to its locus. taking the equation of a line in parametric form and substitute the given ...
0
votes
0answers
12 views

How to compute homography matrix H from four corresponding points [duplicate]

I am using 4 point correspondence to compute elements in Homography matrix $H$. \begin{align*} [x']={}& [h_1 h_2 h_3] [x] \\ [y']={}& [h_4 h_5 h_6] [y] \\ [(1)]={}&[h_7 h_8 h_9] [(1)] ...
0
votes
1answer
32 views

How many non-congruent triangles with perimeter 11 have integer side lengths? [on hold]

How many non-congruent triangles with perimeter 11 have integer side lengths? I failed to solve it. Can anyone help?
3
votes
1answer
772 views

Prove that if Triangles ABC = DEF in a metric geometry, then line AB contains exactly two of the points D, E, and F.

Prove that if Triangles ABC = DEF in a metric geometry, then line AB contains exactly two of the points D, E, and F. We are not allowed to use the facts: In a metric geometry, if triangles ABC=DEF, ...