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

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1answer
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Matrix for rotation around a vector

I'm trying to figure out the general form for the matrix (let's say in $\mathbb R^3$ for simplicity) of a rotation of $\theta$ around an arbitrary vector $v$ passing through the origin (look towards ...
2
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3answers
673 views

“World's Hardest Easy Geometry Problem”

This question is a "corollary" (if you will) to the World's Hardest Easy Geometry Problem (external website). Formally, this is called Langley's Problem. The objective of that problem was to solve for ...
2
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2answers
180 views

Finding a curve that intersects any line on the plane

Question Is there a curve on plane such that any line on the plane meets it (a non zero ) finite times ? What are the bounds on the number of such intersections. My question was itself ...
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0answers
132 views

Equation of an intersection of two cones when the intersection is an ellipse

The two cones with vertex $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ and generating angle of two cones is $\alpha$ given. I need to write the equation of the intersection of two cones ...
2
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2answers
304 views

Proof: Invariant angle measure - same result for any circle drawn.

Below I have quoted Wikipedia. I am particular interested in the statement: The value of $\theta$ thus defined is independent of the size of the circle: if the length of the radius is changed ...
2
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1answer
262 views

Obtaining Least square adjusted single line by intersecting many 3D planes

I am working with many 3D planes and looking for a Least square solution for below case. IF I am having many number of 3D planes knowing only one point and the normal vector (for eg. O1 and N1), ...
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1answer
1k views

Is a sphere a closed set?

The unit sphere in $\mathbb{R}^3$ is $\{(x,y,z) : x^2 + y^2 + z^2 = 1 \}$. I always hear people say that this is closed and that it has no boundary. But isn't every point on the sphere a boundary ...
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3answers
579 views

Is there a geometric realization of Quaternion group?

Is there a geometric realization of the Quaternion group: $$Q = \langle i,j,k \mid i^2 = j^2 = k^2 = ijk \rangle$$ I dont think it can be realized as the symmetries/rotations of a 3D shape so could ...
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3answers
980 views

Construction of a right triangle

It's a high school level question which we can't seem to solve. Here it is: Given 2 lines, one of the length of the hypotenuse and the other with the length of the sum of the 2 legs, construct ...
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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
9
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1answer
313 views

Calculating $\sin(10^\circ)$ with a geometric method

Excuse me if this is a simple question: What is a simple geometric method for calculating $\sin(10^\circ)$ using only the sines of $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$? Generally, is ...
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3answers
8k views

Equation of angle bisector, given the equations of two lines in 2D

I have two lines in 2D expressed with general equation (or implicit equation): First line: $a_1x+b_1y=c_1 \qquad(1)$ Second line: $a_2x+b_2y=c_2 \qquad(2)$ If the two lines are intersecting I will ...
8
votes
3answers
167 views

Proof by induction using Fubini's Theorem

I am asked for the volume of the region $x_1+\cdots+x_n\leq 1$ where $x_1,...,x_n\geq 0$. I am proposing that the volume $V(n)$, is given by $$ V(n) = ...
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votes
4answers
4k views

equilateral triangle with integer coordinates

Is it possible to construct an equilateral triangle with coordinates on a grid of integers? I think the answer is no, but how can I prove this? I started with a triangle with coordinates (0,0) (a,b) ...
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votes
2answers
2k views

Numbers of circles around a circle

"When you draw a circle in a plane of radius 1 you can perfectly surround it with 6 other circles of the same radius." BUT when you draw a circle in a plane of radius 1 and try to perfectly surround ...
6
votes
3answers
166 views

Error measurement between given perfect 2D shape and freeform shape drawn by user

What method should I use to calculate the error between a given perfect shape (e.g. circle, triangle, rectangle etc.) and a freeform shape drawn by the user, which more or less closely matches the ...
5
votes
1answer
375 views

What is “general position” of hyperplanes?

A brief question. I was reading some mathematical writing in which the author makes the following statement: Consider $S$ hyperplanes in general position... What is "general position"? ...
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5answers
648 views

What is a parallel line?

We are learning vectors in class and I have a question about parallel lines and coincident lines. According to wikipedia a parallel line is: Two lines in a plane that do not intersect or touch at ...
5
votes
1answer
357 views

Ellipse Question

I have only worked with ellipses aligned with the x or y axis. However, how can I approach the following: Suppose we have an ellipse centered at the origin of the following form $$ax^2 + b xy +c y^2 ...
5
votes
2answers
163 views

Warp-like pattern in a closed curve

Given a closed curve in 2D space that intersects itself (transversally, and there's no point in which three paths or more meet), is it possible to look at it as a Celtic knot so when you follow it, ...
4
votes
3answers
287 views

What does the secant value represent?

What does the secant value represent? I know that $$\sec = 1/\cos(\theta)$$ but really I do not know what this value represents, so I need your help. A clear example with images would be appreciated. ...
4
votes
3answers
276 views

There isn't a product operation that is commmutative on $ \mathbb{R}^{n} $ that satisfies all the field axioms for $ n \geq 3 $.

This proof is broken down into simple easy algebra and vector questions. I would like to discuss different answers and approaches. Please see pg 162-163 on books.google.ca/books?isbn=0387290524 ...
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3answers
893 views

Determinants and volume of parallelotopes

The absolute value of a 2 by 2 matrix determinant is the area of a corresponding parallelogram with the 2 row vectors as sides. The absolute value of a 3 by 3 matrix determinant is the volume of a ...
3
votes
2answers
77 views

Drawing a triangle from medians

Is it possible to draw a triangle, if the length of its medians $(m_1, m_2, m_3)$ are given only? Someone asked me this question, but I can not see it. Is it really possible? UPDATE Apart from the ...
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votes
2answers
93 views

Area of a triangle in terms of areas of certain subtriangles

In triangle $ABC$ , $X$ and $Y$ are points on sides $AC$ and $BC$ respectively . If $Z$ is on the segment $XY$ such that $\frac{AX}{XC} = \frac{CY}{YB} = \frac{XZ}{ZY}$ , then how to prove that the ...
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2answers
153 views

Check Points are line, triangle, circle or rectangle

How to determine geometric properties of four distinct points in a plane (x1,y1), (x2,y2), (x3,y3), (x4,y4) represented in the 2-D Cartesian coordinate system, whether these four points are on a ...
3
votes
1answer
2k views

Calculate Camera Pitch & Yaw To Face Point

How do you calculate pitch & yaw for a camera so that it faces a certain 3D point? Variables Camera X, Y, Z Point X, Y, Z Current Half Solution Currently I know how to calculate the pitch, ...
3
votes
4answers
3k views

Deriving the Area of a Sector of an Ellipse

A sector $P_1OP_2$ of an ellipse is given by angles $\theta_1$ and $\theta_2$. Could you please explain me how to find the area of a sector of an ellipse?
3
votes
1answer
186 views

Triangle and Incircle

Today in class me and my friend were discussing cool problems that we've done. And he asked me to.find with proof something interesting. Triangle ABC has right angle at B and we drop a perpindicular ...
3
votes
2answers
254 views

Why can any affine transformaton be constructed from a sequence of rotations, translations, and scalings?

A book on CG says: ... we can construct any affine transformation from a sequence of rotations, translations, and scalings. But I don't know how to prove it. Even in a particular case, I found ...
3
votes
6answers
19k views

Showing three points are collinear

Show that (-1,8), (1, -2) and (2,1) lie on a common line. Any help understanding how to go about doing this is greatly appreciated.
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3answers
1k views

A parallelogram and a line joining a vertex to the midpoint of opposite side

In a parallelogram ABCD. M is the midpoint of CD. Line BM intersects AC at L and it also intersects AD extended at E. Prove that EL=2BL PS: This is not a homework problem. I was solving geometry for ...
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3answers
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How to find where $3$ lines intersect.

I've got a programming exercise I need to do, but I just can't figure out the math part. I need to check if $3$ of $6$ lines intersect in the same point. I am given the equation $ax+by=c$, and I ...
2
votes
2answers
72 views

Iterativ centroid-triangle sequence

Three points that are not on a straight line - $A_1$, $A_2$ and $A_3$ - are given; for $n = 4, 5, 6, ...$ $A_n$ is the centroid of the triangle $A_{n−3},A_{n−2},A_{n−1}$. Q:Is there a point that ...
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1answer
156 views

Why are two definitions of ellipses equivalent?

In classical geometry an ellipse is usually defined as the locus of points in the plane such that the distances from each point to the two foci have a given sum. When we speak of an ellipse ...
2
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1answer
2k views

Calculating circle radius from two points on circumference (for game movement)

I'm designing a game where objects have to move along a series of waypoints. The object has a speed and a maximum turn rate. When moving between points p1 and p2 it will move in a circular curve ...
2
votes
3answers
978 views

3D to 2D rotation matrix

I have been trawling through this forum but am struggling to understand the maths a bit. Currently I have a 2D plane within a 3D space and I have the coordinates for them. I want to work on this 2D ...
2
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0answers
87 views

Generalizations of equi-oscillation criterion

When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ...
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2answers
2k views

Counting number of distinct regions with intersecting circles

Given $n$ circles of possibly different radii, how many distinct regions can there be? For small $n$, I can work it out with pictures. (I'm pretty sure $n=4$ can yield 13 distinct regions, but not ...
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2answers
46 views

Geometrical place with circles…

How to find the geometrical place of all centers of a circles that tangent from inside to the circle $x^2+y^2=R^2$ and the $y$-axis? (Suppose that $x,y\geq 0$)
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1answer
77 views

Diophantine quartic equation in four variables, part deux

A recent Question asked for all positive integer solutions of a simple quartic in four unknowns: $$ wxyz = (w+x+y+z)^2 \tag{1}$$ whose satisfaction is necessary for the integer side lengths ...
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2answers
826 views

How to find the intersection of three spheres (full solutions)?

The three equations of spheres are given $(x-x_{1})^2+(y-y_{1})^2+(z-z_{1})^2=a^2$ $(x-x_{2})^2+(y-y_{2})^2+(z-z_{2})^2=b^2$ $(x-x_{3})^2+(y-y_{3})^2+(z-z_{3})^2=c^2$ How do I find $(x,y,z)$ ...
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vote
1answer
3k views

Converting Lat/Long coords to Cartesian X/Y, then calculating shortest distance between point & line segment

I'm having an issue with accuracy when converting Lat/Long coordinates to X,Y and then finding the shortest distance from a Point to a Line with said coordinates. The distance is off by around 40-50% ...
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1answer
325 views

Sum of Angles in a Triangle.

Can anyone please explain how to form a better idea in understanding Sum of measures of angles in a triangle are 180 degrees.
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1answer
51 views

What is the angle $<(BDE,ADH)$?

What are ways to determine the angle $<(BDE,ADH)$ (the angle between the two planes passing through the points B,D,E and the points A,D,H respectively)?
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1answer
83 views

Area of quadrangle

In the quadrangle $ABCD$, the points $E,F,G,H$ are the midpoints of respectively $AB, BC, CD, DA$. We know that area $\triangle AHL=a$, $\triangle DIG=b$, $\triangle FJC=c$, $\triangle EBK=d$. Prove ...
0
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1answer
103 views

A curve such that all lines on the plane intersect it : cont..

Further to this question (which appears more or less settled); "Is there a curve on plane such that any line on the plane meets it (a non zero ) finite times ?" I ask now the upper bounds of the ...
0
votes
1answer
224 views

$n$ Lines in the Plane

How am I to "[u]se induction to show that $n$ straight lines in the plane divide the plane into $\frac{n^2+n+2}{2}$ regions"? It is assumed here that no two lines are parallel and that no three lines ...
0
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1answer
72 views

Square coloring

There are 3 red axis-aligned interior-disjoint squares. There are 3 blue axis-aligned interior-disjoint squares. Is it always possible to find a pair of 1 red square and 1 blue square, such that ...
0
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3answers
2k views

How do you find the distance from a point to a plane?

I am having trouble with this: Find the distance from the point $(1,1,1)$ to the plane $2x+2y+z=0$. Any ideas? Thanks.