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
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1answer
380 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
178 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 ...
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 ...
0
votes
1answer
141 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 ...
24
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 ...
12
votes
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 ...
11
votes
1answer
6k 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 ...
9
votes
3answers
740 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?
7
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3answers
2k views

Equation of Cone vs Elliptic Paraboloid

I can't understand why $$\frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2} \tag{*}$$ corresponds to an elliptic paraboloid and $$\frac{z^2}{c^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2} \tag{**}$$ to a cone, ...
7
votes
6answers
42k 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 ...
5
votes
1answer
275 views

On minimizing the area of an enclosing surface subject to nonnegative Gaussian curvature

This is inspired by this previous question on physical processes that might give rise to convex hulls. Consider the problem of gift-wrapping a three-dimensional object using an inextensible material, ...
5
votes
1answer
826 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 ...
5
votes
3answers
9k 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
118 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 ...
3
votes
1answer
103 views

geometric proof of $2\cos{A}\cos{B}=\cos{(A+B)}+\cos{(A-B)}$

I have seen geometric proof of identities $$\cos{(A+B)}=\cos{A}\cos{B}-\sin{A}\sin{B}$$ and $$\cos{(A-B)}=\cos{A}\cos{B}+\sin{A}\sin{B}$$ By adding two equation, ...
3
votes
1answer
7k 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
2k 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
215 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
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
1answer
595 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
votes
3answers
9k 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 ...
2
votes
1answer
439 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
247 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 ...
1
vote
2answers
51 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
380 views

The double cone is not a surface.

My question is that A double cone ( also named as "circular cone") is not a surface. I know its reason. But I cannot show this mathematically. Suppose $\sigma : U \to S\cap W$ Is a surface ...
0
votes
2answers
200 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 $B$.
12
votes
2answers
5k views

Finding the intersection point of many lines in 3D (point closest to all lines)

I have many lines (let's say 4) which are supposed to be intersected. (Please consider lines are represented as a pair of points). I want to find the point in space which minimizes the sum of the ...
9
votes
1answer
548 views

Navigating though the surface of a hypersphere in a computer game

People in StackOverflow seems not so into this theme, so I thought I could have better luck in here. I had the idea of an spaceship game where the world is confined in the surface of an 4-D ...
7
votes
4answers
5k views

Given a solid sphere of radius R, remove a cylinder whose central axis goes through the center of the sphere.

Given a solid sphere of radius R, remove a cylinder whose central axis goes through the center of the sphere. Let h denote the height of the remaining solid. Calculate the volume of the remaining ...
6
votes
3answers
4k views

Mean distance between 2 points within a sphere

I have found an answer on this site to the question of determining the mean straight-line distance between 2 randomly chosen points in a disc of radius r. I'm now trying to find an answer to the same ...
3
votes
1answer
715 views

Calculate area of a figure based on vertices [duplicate]

Possible Duplicate: How quickly we forget - basic trig. Calculate the area of a polygon How to calculate the area of a polygon? If I know all the vertices of a particular polygon/figure, ...
2
votes
2answers
132 views

What mathematical areas lie at the interface of analysis, algebra and geometry? [closed]

Would it be some area that draws on many fields such as algebraic geometry? Is there some sort of unification of these three fields?
2
votes
1answer
280 views

Steiner symmetrization preserves area?

I just finished reading (and understanding) Steiner's proof of the isoperimetric inequality. His proof (which is sadly incomplete) seems to rely much on the fact Steiner symmetrization preserves area ...
2
votes
1answer
378 views

Calculating closest and furthest possible diagonal intersections.

Calculating closest and furthest possible diagonal intersections. Please refer to the image attached. It represents a $2D$ grid with the following properties: The grid origin is $(1,1)$ at the ...
2
votes
2answers
4k views

Ellipse fitting methods.

I have set of points and want to fit ellipse to this set. I have found only function which fits ellipse in least squares sense. In this set of points there are some noise points which should not be ...
2
votes
2answers
1k views

How to get the cardinal direction from one location to another?

Given are two geo locations, each with latitude and longitude. One is the current location, the other is a target location. is there a formula for calculating the target's cardinal direction for 0 ...
2
votes
2answers
2k views

Finding the mean distance between n points evenly distributed in a disc of radius r

In reading this article about updated estimates for the number of exoplanets in the Milky Way, I am curious how to get an estimate of the mean distance between them. The Milky Way is ~50,000 light ...
1
vote
1answer
276 views

Project point onto line in Latitude/Longitude

Given line AB made from two Latitude/Longitude co-ordinates, and point C, how can I calculate the position of D, which is C projected onto D. Diagram:
1
vote
1answer
2k views

Determine if projection of 3D point onto plane is within a triangle

In 3D, given three points $P_1$, $P_2$, and $P_3$ spanning a non-degenerate triangle $T$. How to determine if the projection of a point $P$ onto the plane of $T$ lies within $T$?
0
votes
2answers
133 views

Geometry Problem about Circles and Tangents

It is the second problem from my maths notebooks, which is still unsolved. I translated it from Russian, so their may be some discrepancies in translation. So, I also added image. First problem was ...
0
votes
2answers
305 views

Medians of a triangle and similar triangle properties

Prove using similar triangle properties that "any two medians of a triangle divide each other in the ratio $2:1$. I do not know which criteria of similar triangle must be used
0
votes
4answers
5k views

Finding out an arc's radius by arc length and endpoints

I have two points. I need to draw an arc ($<180$°) between them, and I know how long it should be, but nothing else about it. Knowing either the radius length or the coordinates of the center ...
173
votes
7answers
21k views

V.I. Arnold says Russian students can't solve this problem, but American students can — why?

In a book of word problems by V.I Arnold, the following appears: The hypotenuse of a right-angled triangle (in a standard American examination) is 10 inches, the altitude dropped onto it ...
166
votes
14answers
11k views

Identification of a quadrilateral as a trapezoid, rectangle, or square

Yesterday I was tutoring a student, and the following question arose (number 76): My student believed the answer to be J: square. I reasoned with her that the information given only allows us to ...
81
votes
5answers
5k views

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif ...
99
votes
4answers
2k views

Hyperbolic critters studying Euclidean geometry

You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise. ...