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

learn more… | top users | synonyms

2
votes
1answer
267 views

Converting points on Plane in Perspective-space to 2D Coordinate points

Augmented Reality scenario: I have a piece of paper on the table and a camera pointed at it, the piece of paper appear to be slanted with two pair of edges going towards two vanishing points. Say I ...
4
votes
1answer
596 views

Orthogonal Matrices and Symplectic Matrices and Preservation of Forms

I would like to know the properties of orthogonal matrices and symplectic matrices in terms of the forms they preserve. Could someone please add and/or correct, maybe give some refs/examples? ...
1
vote
2answers
637 views

Does $Ax + By = C$ pass through any lattice point?

Given an equation of a straight line of form $Ax + By = C$. where $A,B,C$ are integers. How could we check if this passes through any lattice point or not? Please suggest me a suitable algorithm.
2
votes
0answers
154 views

Geometry vector

I have a problem geometry: " Let (O) is circumcircle of triangle XYZ. G is centroid of XYZ. I,H,K are circumcircles of triangle GYZ, GZX, GXY. Prove O is the centroid of IHK". I'm wanting use method ...
2
votes
2answers
758 views

subvolume area under the intersection of a plane (line) and a cone

I am implementing a conical filter for Gupta-Sproull anti-aliased line algorithm. Given a cone with the total volume of 1 and a radius of 1. Find the subvolume of the intersection of a line. The ...
1
vote
1answer
181 views

support function and curvature

Is it true that for centrally symmetric(symmetric with respect to the origin), strictly convex closed curves, if the support function at a point is minimum then curvature is minimum at that point and ...
2
votes
2answers
871 views

Equation for getting the length of the minor axis (of an ellipse)

I'm looking for an equation that can help me determine the length of the minor axis. I know the length of the major axis and have the Cartesian coordinates of a point somewhere on the ellipse. How ...
2
votes
1answer
98 views

Is there any invariance under the inversion mapping?

In geometry, there is a transformation called the inversion mapping which maps nonorthogonal circles into nonorthogonal lines and vice versa.(If I make a mistake, inform me, since I am not very ...
109
votes
7answers
155k views

How many sides does a circle have?

My son is in 2nd grade. His math teacher gave the class a quiz, and one question was this: If a triangle has 3 sides, and a rectangle has 4 sides, how many sides does a circle have? My first ...
4
votes
2answers
2k views

Find equation for hyperbola

Just taking (failing) a simple algebra class, can't figure this one out and no one can explain it to me and the book just tells me to do it. Find an equation for the hyperbola described: foci ...
0
votes
1answer
1k views

compute a vector which is perpendicular to another given vector (all are in 3D)

I have a vector $v_1$ (suppose $v_1= \langle a_1, b_1,c_1\rangle$) and this $v_1$ passes through the point $(x_1,y_1,z_1)$. Now I need a second vector, $v_2$ which is perpendicular to $v_1$. Suppose ...
6
votes
2answers
715 views

well separated points on sphere

Is there a way to generate k points on a n-sphere, say, $x_1,\dots,x_k$ such that $\min_{ i \neq j } \| x_i - x_j \| $ is as large as possible? Approximate solutions are also OK, I just need well ...
4
votes
2answers
650 views

About the Three Reflections Theorem

I recently solved this exercise from Hartshorne's Classical Geometry. Given three lines $a$, $b$, $c$ through a point $O$, show that there exists a unique fourth line $d$ such that ...
5
votes
1answer
337 views

Center of a polygon inside the polygon

What is the name of the point(s) in a polygon, calculated by "shrinking" the polygon until there's no surface left? Example (the light areas): Also, of possible, it would be cool to have an ...
0
votes
2answers
527 views

Applications of circumcircles?

Are there examples where the circumcircle of a triangle is useful in everyday life (not unrealistic or far-fetched ones)? Are there other interesting applications (physics, technology, mathematics)?
5
votes
2answers
735 views

Find equation of quadratic when given tangents?

I know the equations of 4 lines which are tangents to a quadratic: $y=2x-10$ $y=x-4$ $y=-x-4$ $y=-2x-10$ If I know that all of these equations are tangents, how do I find the equation of the ...
4
votes
1answer
1k views

given a set of points in cartesian plane find the point which has shortest sum of distance from all points

This I have reduced to Given a set of n points find out a point X,Y such that the $\sum_{i=1,n} (x_{i}-X)^2 + (y_{i}-Y)^2$ is minimum. Now as per the comments I found out that this is wrong. Can ...
0
votes
1answer
184 views

Probability of intersection of two geometrical figures in bounded space?

I'd like to find a closed form (if possible) expression of the probability of interesection of two geometrical figures $F_1$ and $F_2$ of area $A_1$ and $A_2$, respectively, that are have a random ...
3
votes
1answer
166 views

What is this shape?

$C = \{(c_1,c_2):c_1^2 + c_2^2 \leq 1 \}$ $G = \{(g_1,g_2): g_1 = a_1 + d_1, g_2 = a_2 + d_2, d_1^2 + d_2^2 \leq 1 \}$ C is a unit circle centered at the origin, and G is a unit circle centered at ...
19
votes
6answers
2k 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 ...
2
votes
1answer
272 views

Point Correspondence in 2D in two image

I have taken two image from two different camera and lens from 3 fixed point which makes a 90 degree angle on a plane. the position of camera when taking photo is also changed but the 3 fixed points ...
1
vote
1answer
2k views

Distance from a point to circle's closest point

So let's assume I have a point $P$ in $3d$ space $(x_0, y_0, z_0)$. And I have a circle $C$ that is centered at $(x_1, y_1, z_1)$ with a radius $r$. I need to find the distance from $P$ to the nearest ...
3
votes
2answers
544 views

Intersection of 2 spheres and a cube

In the Cartesian coordinate system, given 3 geometrical solid objects (interior plus boundary): spheres S1(x1,y1,z1, R1), S2(x2,y2,z2, R2) and a cube (which is orthogonal with coordinate system) at ...
9
votes
5answers
520 views

Does $a^2+b^2=1$ have infinitely many solutions for $a,b\in\mathbb{Q}$?

Does $a^2+b^2=1$ have infinitely many solutions for $a,b\in\mathbb{Q}$? I'm fairly certain it does, but I'm hoping to see a rigorous proof of this statement. Thanks. Here is my motivation. I'm ...
4
votes
0answers
404 views

Will two convex hulls overlap?

I ran into the following problem while working in neural nets. Given natural numbers $b$ and $r$, uniformly randomly choose $b+r$ points within a unit square. Call the $b$ points the blue points and ...
0
votes
1answer
86 views

All Embeddings of a 3-D g-Handlebody Hg in S^4 is Trivial (i.e., any two embeddings are isotopic)

I think we can argue that Sg --the genus-g surface -- has only a trivial embedding in S4 , since Sg is topologically a wedge of g S1's, and there are no knotted S1's in S4 (meaning that any two ...
3
votes
2answers
209 views

A “natural” Borel probability measure on a projective space $P R^{N}$?

Is there a simple way to construct such a measure? Preferably, one invariant under rotations and reflections of $R^N$.
1
vote
1answer
219 views

Cylinder silhouette

I have a parametric representation of a cylindrical shape (well, it's like a cone, but its spike is trimmed). I would like to have an analytic expression for its silhouette lines in terms of the ...
1
vote
0answers
235 views

Calculating the Epsilon Neighborhood of line segments in 3d

I am working on a trajectory clustering algorithm (in C++) and one of the steps required in this algorithm is to take a set of 3d line segments (D), and for each line segment (L) in D, to calculate an ...
1
vote
2answers
244 views

Algorithm for shifted center of mass calculation

Imagine that we have a set of points with defined x coordinate and mass for each of them: {($x_i$, $m_i$)}. Canonical center of mass is a point $x_0$ that divides points to the left ones and the ...
14
votes
3answers
4k views

Why do 4 circles cover the surface of a sphere?

Is there a geometric explanation for why a sphere has surface area $4 \pi r^2$ ? Ie equal to 4 times its cross-section (a circle of radius r).
4
votes
4answers
9k views

How do I convert the distance between two lat/long points into feet/meters?

I've been reading around the net and everything I find is really confusing. I just need a formula that will get me 95% there. I have a tool that outputs the distance between two lat/long points. ...
10
votes
1answer
389 views

What is a Structured Polyhedron?

In my work on lattice point enumeration of polytopes, I stumbled upon the following sequence: \begin{eqnarray} 1, 120, 579, 1600, 3405, 6216, 10255, 15744, 22905, 31960, 43131, ... \end{eqnarray} ...
2
votes
3answers
480 views

How to prove the following inequalities?

thanks for your time. i am interested in various ways/techniques/tricks/methods (induction, convexity, concavity, maximum, minimum, geometry, trigonometry, ...) for proving the inequalities and their ...
9
votes
3answers
1k views

Is there a geometric interpretation of the exponential function of real numbers?

I can visualize the exponential function with the graph $y = e^x$, but I can do that for almost any function. In addition to its graph, the function $f(x) = x^n$ can be visualized as the volume of a ...
3
votes
2answers
536 views

Third point of a triangle from only two points and all three edge lengths

I want a triangle composed of points A, B and C in Cartesian 3D space. I currently know the positions of points A and B, but I need point C. I have the line segment AB, and thus its magnitude. I have ...
1
vote
1answer
111 views

Ratio of division

In what ratio is the line joining $(4,5)$ and $(1,2)$ is divided by the y-axis? The solution given in my module assumed the ratio to be $K:1$,this might be very trivial but I can't convinced ...
6
votes
3answers
532 views

How is it that this shape can converge to what looks like a triangle but has a different perimeter?

I had this strange notion some time ago, and I recently wrote a blog post about it, as a mere curiosity. I don't really consider it a "serious" mathematical question; but out of interest, I wondered ...
5
votes
1answer
261 views

compact symplectic manifolds

Why there is no compact symplectic submanifold with dimension greater than 2 in $\mathbb{R}^{2n}$ ?
4
votes
1answer
336 views

How to improve linear interpolation in 3D

in one physics problem, there is a cube. A computationally expensive function can be calculated inside the cube. But one needs to do the calculation faster and to know for a given point inside the ...
2
votes
1answer
710 views

Intersection of a sector and a rectangle

I have a rectangle and a circular sector in a Cartesian plane. What is the easiest way (algorithmically) to tell if they intersect? Edit: I'm looking to see if the areas intersect, not just the ...
1
vote
2answers
630 views

A cylinder is not always can-shaped?

I am reading: A right cylinder with cross section $\Omega$ is a solid that is formed by translating $\Omega$ along a line, or _axis_, that is perpendicular to it. So my understanding now is a ...
7
votes
3answers
20k views

How to find intersection of two lines in 3D?

Given two lines joining A,B and C, D in 3D how do I figure out if they intersect, where they intersect and the ratio along AB at which the intersection happens? I can quite hapilly work out the ...
4
votes
1answer
281 views

Why is the identity map never equal to the product of an odd number of reflections?

Suppose I have an some plane and an identity mapping on the points of the plane. I see that the identity can be expressed as a product of an even number of reflections, since any reflection has itself ...
1
vote
1answer
171 views

formulas for rectangles and areas

I have a certain side ratio of small rectangle, and a side ratio for a large rectangle. How can I write an equation that gives me the number of small rectangles that will make the large rectangle? Is ...
6
votes
1answer
252 views

Prove that 2 of 3 triangles sharing one side overlap

Let $C, D, E$ be three non-degenerate triangles in $\mathbb R^2$. Let $c, a, b$ be the vertices of $C$, let $d, a, b$ be the vertices of $D$, and let $e, a, b$ be the vertices of $E$. I want to show ...
6
votes
3answers
1k views

Isosceles trapezoid

I was solving an exercise on Isosceles trapezoid whose diagonal was given, and I note that If I draw a diagonal in the isosceles trapezoid I got two triangles To determine the area of the triangles I ...
1
vote
1answer
341 views

Principal Curvatures of a surface

Suppose that a 3-D surface has the property that $|k_1|\leq 1$ and $|k_2|\leq 1$ everywhere, where $k_1$ and $k_2$ are the principal curvatures. Prove or disprove that the curvature $k$ of a curve on ...
2
votes
2answers
371 views

Finding points on two linear lines which are a particular distance apart

I have two linear, skew, 3D lines, and I was wondering how I could find a points on each of the lines whereby the distance between the two points are a particular distance apart? I'm not after the ...
6
votes
13answers
11k views

how to find center of an arc given start point, end point, radius, and arc direction?

Given an arbitrary arc, where you know the following values: start point (x0,y0), end point (x1,y1), radius (r) and arc direction (e.g. clockwise or counterclockwise from start to end), how can I ...