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

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2answers
758 views

What's wrong with this solution of Tarski's circle-squaring problem?

Tarski's circle-squaring problem asks whether it is possible to cut up a circle into a finite number of pieces and reassemble it into a square of the same area. Note that this is different from the ...
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3answers
260 views

Visualizing why a right-angle rotation formula works in polar coordinates

I am trying to get a solid and intuitive handle on polar and spherical coordinates, and I'm getting stuck with what I think should be simple geometry: To find the unit vector in Cartesian coordinates ...
2
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1answer
323 views

Mathematics of Rectangles

I'm looking for all stuff relative to Rectangles Set (specialty rectangles with edges parallel to axes of orthonormal 2d space: lets note it $RS$. I found this interesting article A new tractable ...
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0answers
26 views

Isomorphims on homologies induced by cylindrical structure

This question is related to my previous question Cylindrical structure and homology. Let $S$ be a oriented compact (topological) 2-manifold. We consider a cylinder $M=S\times I$ over $S$, here $I=[0, ...
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1answer
675 views

line projection on top of a plane

If I have a horizontal line (a 3d point and 3d vector with zero z component) and another plane (could be an oblique or a horizontal; i have normal vector of the plane); then how do we get the ...
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4answers
2k views

Given two parallel line segments, how do I tell if and where they overlap?

To find if two line segments intersect I am this code The problem is this code: ...
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1answer
360 views

Generating a random point in the volume of a finite cylinder

I have a finite cylinder in three-dimensions with a long-axis defined by the endpoints $p_1$ and $p_2$, and radius $R$. What is an easy method of picking a random point in this cylinder with uniform ...
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4answers
187 views

Square on the curve

A square is drawn on the curve $y = x^3 + 27\cdot x^2 + 8\cdot x + 91$. What would be the area of the square. All the points of the square should lie on the curve. All four points lie on the curve ...
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1answer
343 views

find shortest distance from origin to parabolic form

suppose that,we have given following parabolic arc $\sqrt{x}+\sqrt{y}=\sqrt{a}$ we are trying to find shortest distance from origin to this line, i think that if we rewrite it as ...
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3answers
4k views

Area of a trapezoid from given the two bases and diagonals

Find the area of trapezoid with bases $7$ cm and $20$ cm and diagonals $13$ cm and $5\sqrt{10} $ cm. My approach: Assuming that the bases of the trapezoid are the parallel sides, the solution I can ...
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2answers
239 views

The area problem!

We have to find area of the quadrilateral formed by joining the point of intersection of the four quarter circles that are drawn from each vertex in a unit square. $\hspace{4cm}$ The challenge is ...
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2answers
5k views

How can I determine the scale factor of a pantograph from the ratio of the arms?

I know this is probably simple but I just can't see it. How can I determine the scale factor of a pantograph from the ratio of the arms?
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1answer
390 views

Is a linear conformal mapping same as a similarity transformation?

For a mapping between two Euclidean spaces, is it a linear conformal mapping if and only if it is a similarity transformation? My answer is yes, because the Jacobian matrix of a conformal ...
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2answers
188 views

How to decide that a curve segment is not an ellipse line segment?

Let me ask a question , given any short curve segment , how can you decide that it is not an ellipse line segment by a finite calculations? Thank you in advance.
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2answers
2k views

Three-circle intersection for circles of unbounded integer radius

I have three circles. One is at $(0,0)$ and has radius $n$, another has is at $(1,0)$ and has a radius $m$, and the third is at $(0.5, \sqrt{0.75}))$ and has a radius of $o$. All of the radius values ...
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3answers
3k views

Area of a parallelogram given one side and a perpendicular

This came up while I was doing some tutoring yesterday. Maybe it's simple, but it has been about 7 years since I last took geometry and I can't figure it out. Given parallelogram $ABCD$ along with ...
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2answers
263 views

Shortest way to achieve target angle

Suppose I am moving in a certain compass bearing (e.g. $270^\circ$) and I want to be going in a different direction (e.g. $120^\circ$). Is there a formula or series of math operations that I can use ...
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2answers
2k views

Deriving the Surface Area of a Spherical Triangle

A triangle on a sphere is composed of points $A$, $B$ and $C$. The $\alpha$, $\beta$ and $\gamma$ denote the angles at the corresponding points of the triangle: The Girard's theorem states that the ...
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1answer
108 views

Are two ellipse arcs always almost identical if they have the same end points and the same center of ellipses?

Edit : This question is better to be ignored until the following related question will be discussed enough. This question relates to I know "almost identical " is not mathematics. But if you have ...
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4answers
4k 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 ...
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6answers
4k views

How to find an ellipse , given 2 passing points and the tangents at them?

Please answer to a question , how to find an ellipse which passes the 2 given points and has the given tangents at them. And one related question is that the given condition can decide just one ...
3
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2answers
93 views

How do continuity, distance and irrationals arise from discreteness?

Consider a square as rendered on a computer screen: its width and height are $N$ pixels each, and its area is $N^2$ pixels. Its diagonal, when measured in pixels, is also $N$ pixels long. If you ...
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3answers
281 views

Integrating $\sqrt{1-x^2}$ by interpreting it geometrically as an area within a circle

For the integral $$\int \sqrt{1-x^2} dx = \frac{1}{2} \left ( \arcsin(x) + x \sqrt{1-x^2} \right)$$ Now it was explained to me that geometrically I could take part of the integral as an area sector ...
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1answer
184 views

Triangle given its three perpendicular bisectors and a point of an edge

Is it possible to determine a triangle given its three perpendicular bisectors (meeting at a point which will be the circumcenter) and, say, a point of an edge, or any condition that can make the ...
2
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1answer
191 views

Question about Kepler's second law.

I was watching a Fehnman lecture on YouTube, where he used Kepler's second law as an example of something he was explaining. He was showing geometrically why a line joining a planet and a Sun sweeps ...
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2answers
326 views

Can a finite number of squares, with total area at most 1, be fitted into a square with area 2?

It seems to be a theorem that a finite number of squares, with total area at most 1, can be fitted into a square with area 2 without overlaps. I am looking for a proof of this. Google led me to this ...
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4answers
4k views

How to calculate the two tangent points to a circle with radius R from two lines given by three points

I need to calculate the two tangent points of a circle with the radius $r$ and two lines given by three points $Q(x_0,y_0)$, $P(x_1,y_1)$ and $R(x_2,y_2)$. Sketch would explain the problem more. I ...
3
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1answer
431 views

Can there be multiple local minima for the sum of distances in the plane?

Let $f$ map a point in the plane to the sum of the distances to each element in a given set of points. Can $f$ have multiple local minima? For example, there is only one relative minimum when the ...
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2answers
2k views

How do I calculate the new x,y coordinates and width/height of a re-sized group of objects?

I'd like to resize a group of n-objects -- either circles or rectangles -- all at once and fit them into the newly resized area with newly calculated x,y coordinates and width/height dimensions. The ...
7
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1answer
549 views

Traversing the infinite square grid

Suppose we start at $(0.5,0.5)$ in an infinite unit square grid, and our goal is to traverse every square on the board. At move $n$ one must take $a_n$ steps in one of the directions, north,south, ...
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0answers
251 views

Gimbal lock and zero jacobian determinant

I'm having a hard time visualizing the gimbal lock problem. Suppose $p=(a,b,c)$ is a point where the euler angle $f:\mathbb R^3\to SO_3$ has zero jacobian determinant. Then to say that $p$ is where ...
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1answer
784 views

Intersection of a line with an Elliptic Curve

I am trying to show that if a line given by $y = mx + b$ intersects an Elliptic Curve given by $E(\mathbb{K}): y^2 = x^3 + Ax + B$ in three points then the line is not tangent to the curve. Given ...
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1answer
170 views

Euclidean Geometry a triangle problem

In the three dimensional figure below, is there a way to prove that $$ \angle MNK = 90^ \circ $$ $\hspace{2.8in}$
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1answer
607 views

Optimal approximation of square area with identical circles

There are 86 pages on this site alone under a search for: area, square, circle, convergent, approximation. I found one that arguably asks the same question here, but I am not sure. The question is, ...
4
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1answer
223 views

Symmetry group of the dodecahedron and its subsets

Let $G$ be the symmetry group of the dodecahedron. Indicate subsets of the dodecahedron on which $G$ acts by all possible permutations. I know that $G \simeq A_5 \times \mathbb{Z}_2$. Its order ...
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1answer
502 views

Calculate perimeter from parametric form with an ellipse?

Suppose I have a thing such as an ellipse: $$\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1$$ now we can define it so that $\frac{x}{a}=cos(\theta)$ and $\frac{y}{b}=sin(\theta)$. I ...
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0answers
112 views

The orientation of a closed discrete curve embedded in a triangle.

The two triangles $xyz$ and $x^{\prime}y^{\prime}z^{\prime}$, shown below, have opposite orientations. A closed curve $abcd$ is embedded in the first triangle ($abcd$). The vertices of the ...
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2answers
14k views

What is the equation of an ellipse that is not aligned with the axis?

I have the an ellipse with its semi-minor axis length $x$, and semi major axis $4x$. However, it is oriented $45$ degrees from the axis (but is still centred at the origin). I want to do some work ...
3
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1answer
262 views

Avoiding algebraic integration by geometric arguments

Is there a geometric way of seeing why the integral $\int\limits_{-\infty}^\infty (x^2+y^2+z^2)^{-{3\over 2}}dz={2\over x^2+y^2}$? Otherwise what is a good way of evaluating it algebraically?
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1answer
94 views

How do you find all hexes within an arbitrary arc?

Suppose you have a grid of hexagons oriented with the left and right segments of each hexagon perpendicular to the x axis. How do you find all hexes that fall wholly into an arbitrary arc of the ...
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1answer
498 views

Constructing $\pi^2$ and e with straightedge and compass

Is there a way to construct two curves with lengthratios $\pi^2$, or two areas of ratio $\pi^2$, on a plane surface, with a straightedge and compass? And is e=2.71... possible?
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1answer
744 views

Find third 3d coordinate given two other coordinates

Given the 3d coordinates of the 2 spheres (see image below) and the length of the Box, how can find the 3d coordinates at the end of the box using an equation?
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1answer
255 views

Convert a 'world' point to a 'local' point relative to a plane

I'm trying to find a way of converting a point relative to the world into a point relative to an arbitrary plane (and a way to convert back). This will be coded into C++ so I'll have to able to write ...
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1answer
295 views

Find edge between two boxes

I am building prototype tool to draw simple diagrams. I need to draw an arrow between two boxes, the problem is I have to find edges of two boxes so that the arrow line does not intersect with the ...
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1answer
182 views

Metric on the unit cube

Let $X$ be $\mathbb{R}^3$ with the sup norm $\|\cdot\|_{\infty}$. Let $Y=\{x\in X: \|x\|_{\infty}=1\}$. For $x,y\in Y,y\neq -x$ define $d(x,y)$ to be the arc length of the path $$Y\cap \{\lambda ...
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2answers
1k views

How do I map the torus to a plane?

Please see my answer on Perlin noise first. A bit of background. Imagine a solid texture, like an actual block of sky and cloud. If you "cut a sheet" of sky and display it as an image, you'd get ...
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5answers
5k views

Geometry Book Recommendation?

Can someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to ...
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2answers
4k views

Simplest equation for drawing a cube based on its center and/or other vertices

I'm looking for the simplest equation to draw a cube based on its center and/or other verticies specified. For example, let's say I have the 3D column vector (as I'm using OpenGL to do this): ...
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1answer
39 views

Terminology clarification: ***exchanges***

I need help with a terminology definition. If we say "R is a reflection that exchanges the sides a and b in some triangle", does it mean sides a and b have the same length and the reflection maps one ...
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0answers
322 views

Isomorphic triangles?

From my previous post I have learnt that spherical triangles can have different interior angle sums. Is this enough to argue that the triangles are not isomorphic? I am not sure how isomorphism works ...