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
888 views

How to place objects equidistantly on an Archimedean spiral?

To place objects equidistantly on an Archimedean (arithmetic) spiral, the arc length of the spiral has to increase linearly between the objects. This is what I have so far: The length of a spiral is ...
6
votes
2answers
3k views

algorithm to calculate the control points of a cubic Bezier curve

I have all points where my curve pass through, but I need to get the coordinates of the control points to be able to draw the curve. How can I do to calculate this points?
6
votes
1answer
1k views

How to calculate a heading on the earths surface?

Given an initial position and a subsequent position, each given by latitude and longitude in the WGS-84 system. How do you determine the heading in degrees clockwise from true north of movement?
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4answers
2k views

How to partition area of an ellipse into odd number of regions?

Is it possible to divide an ellipse into 3,5 or 7 etc. parts of equal area? If yes then how? Describe a circle around the ellipse and the circle of an equilateral triangle we construct. Projection ...
3
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1answer
220 views

Which Has a Larger Volume a Cylinder or a Truncated Cone?

The following link describes a programming problem. However, I am unable to work out the maths for the problem. We are given $r$ – radius of lower base and $s$ – slant height. The figure can be ...
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vote
1answer
295 views

Hyperbolic geometry

Post Number: 45 Posted on Friday, 22 March, 2013 - 04:48 pm: I was asked the following question and i do not have any clue on these. Could anyone help me in the beginning of this? Show that there ...
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14answers
8k views

Do two right triangles with the same long hypotenuse have the same area?

I watched computer monitors and I asked myself, do two monitors with the same display diagonal have the same display area? I managed to find out that the answer is yes, if two right triangles with ...
12
votes
1answer
7k views

Algorithm to get the maximum size of n squares that fit into a rectangle with a given width and height

I am looking for an algorithm that can return the number of size of n squares that fit into a a rectangle of a given width and height, maximizing the use of space (thus, leaving the least amount of ...
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3answers
7k views

The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true?

On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true?
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4answers
478 views

Visualising extra dimensions

What is the : most useful prettiest way to visualise extra dimensions in shapes and charts?
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3answers
2k views

How to compute the volume of intersection between two hyperspheres

Let's say I have two n-spheres and I've no prior knowledge about the spheres (such as one of the sphere might be inside the other one) and I need to compute the volume of the intersection of the two ...
6
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4answers
1k views

Why a tesselation of the plane by a convex polygon of 7 or more sides is not possible?

I read in several places, including Wikipedia, that a tessellation of the plane by a single, convex, $n-$sided polygon is not possible for $n\geq7$. I was not able to locate a proof, or a paper that ...
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1answer
7k views

How to find an angle in range(0, 360) between 2 vectors?

I know that the common approach in order to find an angle is to calculate the dot product between 2 vectors and then calculate arcus cos of it. But in this solution I can get an angle only in the ...
5
votes
1answer
882 views

Integer coordinate set of points that is a member of sphere surface

I have a graphic application to develop which involve many spheres. I should determine then on run time. Supposing that I have a sphere of radius r, how can I determine the sub set of the sphere ...
4
votes
2answers
444 views

Inside an equilateral triangle $ABC$,an arbitrary point $P$ is taken from which the perpendiculars $PD,PE$ and $PF$ are dropped onto the sides…

Inside an equilateral triangle $ABC$,an arbitrary point $P$ is taken from which the perpendiculars $PD,PE$ and $PF$ are dropped onto the sides $BC,CA$ and $AB$,respectively.Show that the ratio ...
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3answers
10k views

How to find the largest rectangle inside an ellipse [duplicate]

I have an ellipse that is defined by center, width and height. The axes of the ellipse parallel to the x and y. I want to find the largest rectangle that completely fits inside this ellipse. Is there ...
3
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2answers
3k 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 ...
2
votes
1answer
172 views

Creating an ellipsoidal 3D surface

I am trying to find the equation of a 3D ellipsoidal surface. I have thought of two approaches which are schematically shown below: By revolving an elliptical arc over a 3D elliptical path: Or by ...
2
votes
3answers
4k views

construct circle tangent to two given circles and a straight line

How to construct such a circle? Only straight edge and compasses are allowed.How could we draw if we were to draw the circle tangent to two straight lines and a circle.
2
votes
3answers
923 views

Determine the centre of a circle

Given a circle $O$ on a paper, we do not know the centre point. Can we draw the centre only using a ruler (by which we can only draw straight lines)? One fact I know: we can draw the tangent line at ...
1
vote
1answer
1k views

clock related challenge

A person leaves his house between 4.00 and 5.00 pm. He carefully notes the position of the minute hand and hour hand when he leaves the house. He returns back between 7.00 and 8.00 pm.He notices that ...
26
votes
5answers
4k views

Why is $\sin(d\Phi) = d\Phi$ where $d\Phi$ is very small?

I haven't touched Physics and Math (especially continuous Math) for a long time, so please bear with me. In essence, I'm going over a few Physics lectures, one which tries to calculate the Force ...
16
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1answer
350 views

Expected size of subset forming convex polygon.

If there are $4$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$, what is the expected largest size (or ...
12
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2answers
1k views

Do infinitely many points in a plane with integer distances lie on a line?

Someone posted a question on the notice board at my University's library. I've been thinking about it for a while, but fail to see how it is possible. Could someone verify that this is a valid ...
9
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3answers
778 views

Cube stack problem

How many distinct patterns are possible if you omit (a) 1 piece, (b) 2 pieces and (c) 3 pieces from a cube originally consisting of 27 smaller and equally sized cubes?
8
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2answers
115 views

New very simple golden ratio construction incorporating a triangle, square, and pentagon all with sides of equal length. Is there any prior art?

Consider three regular polygons with 3, 4, and 5 sides wherein all the polygons have sides of equal length X throughout, as illustrated below. The ratio of the red line segment a to the blue line ...
7
votes
6answers
44k views

Methods for showing three points in $\mathbb{R}^2$ are colinear (or not) [closed]

A common question is to prove/disprove that three points in $\mathbb{R}^2$ are colinear. For, example Show that $(-1, 8)$, $(1, -2)$ and $(2, 1)$ lie on a common line. What are some methods ...
6
votes
1answer
1k views

Four turtles/bugs puzzle

I was reading about the the four turtles/bugs math puzzle Four bugs are at the four corners of a square of side length D. They start walking at constant speed in an anticlockwise direction at all ...
6
votes
2answers
1k views

Aren't asteroids contradicting Euler's rotation theorem?

I am totally confused about Euler's rotation theorem. Normally I would think that an asteroid could rotate around two axes simultaneously. But Euler's rotation theorem states that: In geometry, ...
5
votes
5answers
1k 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
3answers
7k views

What is the formula for a 3D line?

Just like we have the formula $y=mx+b$ for $\mathbb{R}^{2}$, what would be a formula for $\mathbb{R}^{3}$? Thanks.
5
votes
3answers
10k views

Convert coordinates from Cartesian system to non-orthogonal axes

I have a 2D coordinate system defined by two non-perpendicular axes. I wish to convert from a standard Cartesian (rectangular) coordinate system into mine. Any tips on how to go about it?
4
votes
1answer
133 views

Average distance to a random point in a rectangle from an arbitrary point

I'm interested in the mean distance between an arbitrary 2D point, $(p, q)$, and a uniformly distributed point inside a rectangle defined by the lower left and upper right vertices $(x_0, y_0)$ and ...
4
votes
1answer
202 views

Volume of a hypersphere

We know that the area of a circle (2-D) =$\pi r^{2}$ and the volume of a sphere (3-D)= $\dfrac{4}{3}\pi r^{3}$. Question:What is the "volume"(or whatever that is called) of a n-dimensional sphere? ...
4
votes
3answers
30k views

Find the volume of the largest right circular cone that can be inscribed in a sphere of radius r?

I checked this question but didn't fully understand it. I know that the volume of a right circular cone is $V = \frac{1}{3}\pi x^2h$ I know that I must take the first derivative and set it equal to ...
4
votes
3answers
1k views

Determinants and volume of parallelotopes

The absolute value of a $2 \times 2$ matrix determinant is the area of a corresponding parallelogram with the $2$ row vectors as sides. The absolute value of a $3 \times 3$ matrix determinant is the ...
3
votes
1answer
8k views

2D rotation of point about origin

I'm in the process of learning game development and have a question regarding a simple rotation. So far, I'm visualizing the rotation as such: I've read this similar question but I'm struggling to ...
3
votes
5answers
3k views

How do I determine if a point is within a rhombus?

I know the coordinates of the 4 rhombus' vertices. I also have the coordinates of another arbitrary point (the result of a click on the screen). How do I determine if that point is within the ...
2
votes
1answer
295 views

Compute center, axes and rotation from equation of ellipse

Suppose I have the equation of an ellipse, in its implicit form $$Ax^2 + By^2 + Cxy + Dx + Ey + F = 0$$ For example the following: $$4.36\,x^2 + 2.89\,y^2 - 5.04\,xy + 30.8\,x - 0.6\,y + 81 = 0$$ ...
2
votes
3answers
58 views

How to determine the equation and length of this curve consistently formed by the intersection of Circles

Consider a Point $A$ that moves linearly on the positive $x$-axis with the velocity $1$ m/s and another Point $B$ at a distance $L$ from $A$ with position $(L,0)$. With each forward motion of point ...
2
votes
1answer
417 views

Intersection of nested closed bounded convex sets in Euclidean space

I read that in a complete Euclidean space - i.e. a normed real space with the norm induced by the scalar product - any sequence of nested bounded non-empty closed convex sets has a non-empty ...
2
votes
4answers
190 views

How to find $n+1$ equidistant vectors on an $n$-sphere?

Following this question, Proving the existence of a set of vectors, I'm looking for a way to find $n+1$ equidistant vectors on a Euclidean $n$-sphere. For $n=2$, you can pick the vertices of any ...
2
votes
1answer
672 views

Need help with this geometry problem on proving three points are collinear

Here's the figure Let A B C D be any four points. Then the angles angle ABC + angle CDA = if and only if the four points lie on a circle. As a corollary, you may conclude that if the angles of the ...
2
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1answer
452 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 is $180^\circ$ ?
2
votes
1answer
281 views

Probabilities of Non-Regular Dice

Thinking about dice: for all the Platonic solids, it's very easy to figure out the odds of a particular face landing face-up in a roll of the die. If I have an arbitrary 6-sided solid, how do you ...
2
votes
4answers
2k views

Find control point on piecewise quadratic Bézier curve

I need to write an OpenGL program to generate and display a piecewise quadratic Bézier curve that interpolates each set of data points: $$(0.1, 0), (0, 0), (0, 5), (0.25, 5), (0.25, 0), (5, 0), (5, ...
2
votes
1answer
1k views

Why is the ratio of the circumference of a circle to its diameter independent of the circle? [duplicate]

Possible Duplicates: Why is Euclidean geometry scale-invariant? Proof that Pi is constant (the same for all circles), without using limits The answer with the most up votes will be ...
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vote
2answers
52 views

How to to a better approach for this :?

If, $$x\cos A+y\sin A=k=x\cos B+y\sin B$$ Then find $(\cos A)(\cos B)$, $(\sin A)(\sin B)$ and $\cos A+\cos B$ and express them in terms of $x,y,k$ I found a solution but it included a really ...
0
votes
1answer
146 views

What is the minimum number of blocks to build this?

A rectangular solid is built using $N$ cubes of a side length of 1cm. When viewed from such an angle such that only 3 of the sides of the rectangular solid are visible, there are 231 $cm^2$ of area ...
0
votes
2answers
214 views

$A$ convex subset of a set has 'smaller' boundary than the set?

Let $A$ & $B$ be subsets of the real plane. Show that if $A$ is convex and is contained in $B$, which is a bounded set, then the length of the border of $A$ is $\leq$ than that of $B$.