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

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8
votes
3answers
905 views

How to prove that a torus has the same volume as a cylinder (with the height equal to the torus' perimeter)

I want to find the volume of a torus with a given thickness and a given radius. Let r be the radius of a circle with its midpoint at $M(0|b)$ ($b \geq r$). Now I want to rotate this circle about the ...
7
votes
1answer
2k views

Calculating Distance of a Point from an Ellipse Border

I'm thinking about using oriented ellipses to represent curves (dents/bumps etc.) in my physics engine, and have a few questions about working with them: What methods are there to finding the ...
2
votes
3answers
192 views

Vector path length of a hypotenuse

Consider the red path from A that zigzags to B, which takes $n$ even steps of length $w$. The path length of the route $P_n$ will be equal to: $ P_n = P_x + P_y = \frac{n}{2}\times w + ...
0
votes
1answer
238 views

How to calculate the area of a polygon? [duplicate]

Possible Duplicate: How quickly we forget - basic trig. Calculate the area of a polygon For a program I want to calculate the area of a given polygon. It may have every form, the sides even ...
107
votes
13answers
17k views

What's the intuition behind Pythagoras' theorem?

Today we learned about Pythagoras' theorem. Sadly, I can't understand the logic behind it. $A^{2} + B^{2} = C^{2}$ $C^{2} = (5 \text{ cm})^2 + (7 \text{ cm})^2$ $C^{2} = 25 \text{ cm}^2 + 49 ...
23
votes
4answers
1k views

Volume of Region in 5D Space

I need to find the volume of the region defined by $$\begin{align*} a^2+b^2+c^2+d^2&\leq1,\\ a^2+b^2+c^2+e^2&\leq1,\\ a^2+b^2+d^2+e^2&\leq1,\\ a^2+c^2+d^2+e^2&\leq1 &\text{ ...
22
votes
9answers
6k views

What is the (mathematical) point of geometric constructions?

The ancient discipline of construction by straightedge and compass is both fascinating and entertaining. But what is its significance in a mathematical sense? It is still taught in high school ...
10
votes
5answers
173 views

Intuitive/direct proof that a rectangle partitioned into rectangles each with at least one integer side must itself have an integer side

A challenge problem asked to show that if rectangle $R$ with axis-parallel sides is partitioned into finitely many subrectangles $R_1,R_2,\ldots,R_n$ (also with axis-parallel sides), such that each ...
13
votes
5answers
1k 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 ...
28
votes
6answers
3k views

Pythagorean Theorem Proof Without Words (request for words)

I was intrigued by a book I saw called Proofs without Words. So I bought it, and discovered that the entire book doesn't have any words in it. I figured at least it would have some words explaining ...
21
votes
6answers
2k views

Why does volume go to zero?

The volume of a $d$ dimensional hypersphere of radius $r$ is given by: $$V(r,d)=\frac{(\pi r^2)^{d/2}}{\Gamma(\frac{d}{2}+1)}$$ What intrigues me about this, is that $V\to 0$ as $d\to\infty$ for any ...
14
votes
5answers
828 views

Polynomial approximation of circle or ellipse

Trying again, with a somewhat simpler sounding question, since my previous one (Generalizations of equi-oscillation criterion) got zero response: Let $F:[0,1] \to R^2$ be a parametric polynomial ...
12
votes
5answers
824 views

Why the interest in locally Euclidean spaces?

A lot of mathematics as far as I know is interested in the study of Euclidean and locally Euclidean spaces (manifolds). What is the special feature of Euclidean spaces that makes them interesting? ...
4
votes
3answers
697 views

Constructing the midpoint of a segment by compass

When I am working with my child, I am stuck in this geometry problem. "We have two different points $M, N$ in the plane. Using only compass to construct the midpoint $I$ of the segment $MN$." Thank ...
4
votes
6answers
8k views

Equation of a rectangle

I need to graph a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? I can't find it anywhere.
3
votes
3answers
346 views

Prove that CX and CY are perpendicular

There is given convex quadrilateral ABCD. And internal bisectors of angle $\angle A$ and $\angle C$ intersect in point X. And internal bisectors of angle $\angle B$ and $\angle D$ intersect in point ...
25
votes
5answers
2k views

How to find a random axis or unit vector in 3D?

I would like to generate a random axis or unit vector in 3D. In 2D it would be easy, I could just pick an angle between 0 and 2*Pi and use the unit vector pointing in that direction. But in 3D I ...
13
votes
2answers
9k views

How many circles of a given radius can be packed into a given rectangular box?

I've just came back from my Mathematics of Packing and Shipping lecture, and I've run into a problem I've been trying to figure out. Let's say I have a rectangle of length $l$ and width $w$. Is ...
12
votes
4answers
402 views

Why do we use the Euclidean metric on $\mathbb{R}^2$?

On the train home, I thought I would try to prove $\pi$ is irrational. I needed a definition, so I used: $\pi$ is the area of the unit circle. But what is a circle? A circle is the set of tuples ...
11
votes
1answer
565 views

Nested sequences of balls in a Banach space

This seems to be a fairly easy question but I'm looking for new points of view on it and was wondering if anyone might be able to help (by the way- this question does come from home-work, but I've ...
6
votes
1answer
808 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?
5
votes
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 ...
24
votes
2answers
644 views

On nonintersecting loxodromes

The (spherical) loxodrome, or the rhumb line, is the curve of constant bearing on the sphere; that is, it is the spherical curve that cuts the meridians of the sphere at a constant angle. A more ...
11
votes
4answers
2k views

How to understand dot product is the angle's cosine?

How can one see that a dot product gives the angle's cosine between two vectors. (assuming they are normalized) Thinking about how to prove this in the most intuitive way resulted in proving a ...
10
votes
4answers
694 views

Sheaves and complex analysis

A complex analysis professor once told me that "sheaves are all over the place" in complex analysis. Of course one can define the sheaf of holomorphic functions: if $U\subset \mathbf{C}$ (or ...
10
votes
3answers
2k views

Simulating uniformly on $S^1=\{x \in \mathbb{R}^n \mid \|x\|_1=1\}$

A scheme to generate random variates distributed uniformly in $S^2=\{x\in \mathbb{R}^n \mid \|x\|_2=1\}$ is well known: generate a standard normal variate in $R^n$ and normalize it to unit norm. Is ...
6
votes
4answers
342 views

Visualising extra dimensions

What is the : most useful prettiest way to visualise extra dimensions in shapes and charts?
4
votes
1answer
388 views

Geometric construction of hyperbolic trigonometric functions

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the ...
4
votes
1answer
643 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
2k 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?
3
votes
2answers
2k views

Implementation of Monotone Cubic Interpolation

I'm in need to implement Monotone Cubic Interpolation for interpolate a sequence of points. The information I have about the points are x,y and timestamp. I'm much more an IT guy rather than a ...
1
vote
1answer
322 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 ...
13
votes
5answers
10k views

A circle with infinite radius is a line

I am curious about the following diagram: The image implies a circle of infinite radius is a line. Intuitively, I understand this, but I was wondering whether this problem could be stated and ...
8
votes
3answers
243 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?
6
votes
1answer
515 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
3answers
2k views

Euler angles and gimbal lock

Can someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading ...
4
votes
2answers
6k views

Finding a point along a line a certain distance away from another point!

Let's say you have two points, $(x_0, y_0)$ and $(x_1, y_1)$. The gradient of the line between them is: $$m = (y_1 - y_0)/(x_1 - x_0)$$ And therefore the equation of the line between them is: $$y ...
3
votes
3answers
220 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 to find them? ...
2
votes
1answer
117 views

Trapezoid Root Mean Square

I'm trying to prove that length of the line $AB$, parallel to both bases of a trapezoid, that cuts a trapezoid into two trapezoids of equal area is the Root Mean Square of the bases. In other words, ...
2
votes
2answers
949 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 ...
2
votes
2answers
3k views

A line which bisects two sides of a triangle is parallel to the third.

The answers to this question imply that a line which bisects two sides of a triangle must be parallel to the third side. Why is this true? There must be a simple proof. More generally: Let D and E be ...
2
votes
3answers
490 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 ...
2
votes
4answers
955 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
821 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 ...
0
votes
1answer
102 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 ...
8
votes
1answer
365 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 ...
5
votes
1answer
193 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, ...
4
votes
1answer
464 views

Do projections onto convex sets always decrease distances?

Suppose $(M, d)$ is some $\ell_p$ metric space (not necessarily Euclidean), and $C \subseteq M$ is a closed convex set. Consider the projection function $f_C:M\rightarrow C$ defined such that: ...
4
votes
4answers
2k 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 ...
3
votes
1answer
498 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, ...