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

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3
votes
3answers
510 views

point lying inside a square

There is a square lying on a plane in 3d space. I have: N: normal of the square S: vector normal to N determining the orientation of a pair of the square sides C: coordinates of the center of the ...
3
votes
2answers
253 views

Tracing the edges of a cube with the minimum pencil lifts.

I have a cube. I want to trace all the edges of the cube only once, lifting my pencil as few times as possible. Look at the top of a cube, and label the top left vertex as a and travel Clockwise ...
1
vote
1answer
161 views

Find a segment dividing a triangle in equal areas.

In Euclid's geometry context, I have the following problem: Let ABC a triangle and P laying in AB. We need to find a point Q in AC or BC such that the triangle APQ has the half of the area of ABC. ...
1
vote
1answer
734 views

Draw a parallelogram given its sides and the angle between diagonals

I'm having trouble with this one: Draw a parallelogram knowing the lengths of its sides and the angle between the diagonals. Bonus points if the answer uses a translation, because that's where this ...
1
vote
4answers
192 views

Defining/constructing an ellipse

Years ago I was confronted with a (self imposed) problem, which unexpectedly resurfaced just recently... I don't know whether it makes sense to explain the background or not, so I'll be brief. If I ...
0
votes
0answers
23 views

Prove that CY is the external bisector of C [duplicate]

Possible Duplicate: Prove that CX and CY are perpendicular There is given convex quadrilateral ABCD. And internal bisectors of angle ∠A and ∠C intersect in point X. And internal bisectors ...
3
votes
3answers
82 views

Definition of a tangent

I've been involved in a discussion on definition of a tangent and would appreciate a bit of help. At my high school and at my college I was taught that a definition of a tangent is 'a line that ...
2
votes
1answer
63 views

Calculate area from the solid's volume

I'm trying to prove that area of some solid figures (for example a cube and a sphere) can be found using its volume. To say it I take the solid, for example a cube, I've got it's volume: $$ V_c = l^3 ...
4
votes
2answers
133 views

Version of Jordan’s theorem for unbounded curves

Let $I$ denote the open interval $]0,1[$. Let $\gamma$ be a countinous map $I \to {\mathbb R}^2$. We say that $\gamma$ is stretched if it contains points that are arbitrarily close to the origin, ...
0
votes
1answer
184 views

Geodesics (3): A projective space based on the torus (instead of the sphere)

From the Wikipedia article on projective planes: [...] consider the unit sphere centered at the origin in $\mathbb{R}^3$. Each of the $\mathbb{R}^3$ lines in this construction intersects the ...
0
votes
0answers
141 views

Geodesics (2): Is the real projective plane intended to make shortest paths unique?

From the Wikipedia article on geodesics: In Riemannian geometry geodesics are not the same as “shortest curves” between two points, though the two concepts are closely related. The ...
0
votes
0answers
96 views

Geodesics (1): Spaces with more than two geodesics between two points

From the Wikipedia article on geodesics: In Riemannian geometry geodesics are not the same as “shortest curves” between two points, though the two concepts are closely related. The ...
1
vote
2answers
93 views

Calculating coordinates

OK, I have a picture which will hopefully make my explanation a bit clearer. I have a line (a, b) to (x, y) and all I know is the end points of the line. I am trying to draw a line from the end ...
1
vote
2answers
158 views

Calculating 2D positions from a curve?

I am with quite a dylema here, as I need this for a game (so I am going to transform the given answers into programming code) to make a polygon around a curved line. from each line segment I have the ...
2
votes
5answers
123 views

A shorter way to prove the identity on vectors

I am trying to prove that $\vec{A}=(\vec{A}\cdot \vec{n})\vec{n}+(\vec{n}\times\vec{A})\times\vec{n} $ where $\vec{n}$ is a unit vector and $\times$ indicates the cross product. I am dealing with ...
1
vote
2answers
60 views

Geometry proof: Prove that A - B - C iff C - B - A

$A-B-C$ means that there is a function $f$ on the line containing $A,B,C$ such that $f(A)<f(B)<f(C)$. I think this is called the betweenness axiom in some geometry books. My professor said that ...
2
votes
2answers
118 views

Why do geometric sets such as $(\infty, x]$ never have infinity included?

I have a question about the use of infinity and geometric sets. Say I am trying to graph an equation, and the result is all values greater than or equal to, say, $3$. From what I've seen, the proper ...
7
votes
2answers
184 views

Prove that $EF$ perpendicular to $OI_a$

Let $O, I$ and $I_a,$ denote the circumcenter,incenter and excenter in the angle $A$ of a triangle $ABC$. $BI$ meets $AC$ at $E$. $CI$ meets $AB$ at $F$. Prove that $EF$ perpendicular to $OI_a$ It ...
11
votes
2answers
240 views

What is the smallest circle such that an arbitrary set of circles can be placed on the circumference without overlapping?

I have a set of circles of arbitrary radii: $r_1, r_2, r_3, ... r_n$. I wish to arrange them around an inner circle so that they are all touching the perimeter of the inner circle, and do not ...
1
vote
2answers
260 views

Is my proof that the medians of a triangle are concurrent valid?

Consider any triangle ABC. Connect the midpoints of each of the three sides. The inscribed triangle is equal to the other three triangles and they are all congruent. It turns out that the medians of ...
3
votes
3answers
362 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 ...
3
votes
1answer
110 views

Is there a geometric concept describing this?

$$ \frac{\int_{t+h}^{\infty} \lambda e^{-\lambda x} dx}{\int_{t}^{\infty} \lambda e^{-\lambda x} dx} = \frac{\int_{h}^{\infty} \lambda e^{-\lambda x} dx}{\int_{0}^{\infty} \lambda e^{-\lambda x} dx} ...
1
vote
1answer
131 views

Construction of given quadrilateral

There is given convex quadrilateral ABCD. And internal bisectors of angle $<A$ and $<C$ intersect in point X. And internal bisectors of angle $<B$ and $<D$ intersect in point Y. And ...
2
votes
1answer
70 views

Dense path in square, constructed from grid

Working on this stack overflow question led me to ask another, related question (here) . This first attempt was shown unsuccessful in the answer to it, so I try a rather different approach here. Let ...
2
votes
1answer
341 views

Equators and meridians on a discrete torus

Consider the 4 × 4 grid graph: Now torify it, i.e. connect its opposing vertices: How can one tell the difference between a “meridian” and an “equator”? The ...
2
votes
1answer
42 views

Converting vertex normals to face normals

I have a triangulated 3D polyhedron (not necessarily convex) and the following information: A list of the position of each vertex. A list of the vertex triples that define each face. A list of the ...
1
vote
1answer
502 views

The cosine of the angle between two vectors proof

Vector $\mathbf{v}$ and $\mathbf{v'}$ make angles $\alpha , \beta, \gamma$ and $\alpha ', \beta', \gamma$' with the coordinate axes respectively. $\phi$ is the angle between $\mathbf{v}$ and ...
13
votes
1answer
1k views

Floret Tessellation of a Sphere

I'm a programmer looking to create a 3D model of a Floret Tessellation of a sphere, like the one in this picture Class III 8,11 floret planar net (source) If anyone could point me in the right ...
3
votes
1answer
150 views

The unit square stays path-connected when you delete a cycle-free countable family of open segments?

This question was inspired to me by Lukas Geyer’s recent question. A positive answer to this question would easily entail a positive answer to Lukas’ question also, and a negative answer would ...
2
votes
1answer
292 views

Finding the minimum value of a function in an ellipse

so I have this problem for my homework: Consider the elipse: $\dfrac {x^2}{a^2} + \dfrac{y^2}{b^2}=1$ where $0<b<a$. For every point (x,y) on the ellipse find the the perpendicular line to the ...
0
votes
1answer
224 views

Collision detection of two circular sectors in mixed polar and Cartesian coordinate

I am trying to solve the following collision detection problem. Suppose we have two circular sectors, each described in their own polar coordinate system with four values $r_1$, $r_2$, $d_1$ and ...
0
votes
1answer
58 views

Geometry question on $S_3$

Consider the equilateral triangle above with the vertices laid out So I take two motions $R_1$ and $R_2$ both are reflections about their subscripted vertices and I take their composition $R_1 ...
3
votes
1answer
431 views

How to get the diameter of multiple circles?

How can I get the length of the red line, if I got the diameters of all black circles? I'd prefer to get the lengths of the right example but I think it's much more difficult.
3
votes
0answers
93 views

The geometric intuition behind the fact that $y-x^3=0$ in $\mathbb{P}^2(\mathbb{R})$ has a singularity at infinity

I apologize if this question is sophomoric as my knowledge of projective geometry is rather elementary. But I'm curious if there exists a good intuitive geometric explanation for why the curve ...
5
votes
1answer
1k views

Star-Shaped polygons

We call a polygon star-shaped if there exists at least one point for which the entire polygon is "visible" from that point. The set of such points we call the kernel of the polygon. The art-gallery ...
2
votes
2answers
65 views

Percentages, Points, and a Square

I know this is probably an easy problem, but I can't for the life of me figure it out. My problem is slightly hard to explain with words alone, so a picture can be found here. Point $a = (0, 0)$, ...
1
vote
1answer
183 views

Triangle inequalities, with angle bisector

I came across this question while I was taking one of the pratice Mu Alpha Theta tests for my school and I wasn't sure how to solve it. It reads: In $\Delta USA $, $\angle S$ is bisected by ...
2
votes
0answers
108 views

Blowup of a line and a point

I need to construct a morphism $f:X\to \mathbb{A}^3$ which is surjective, $X$ needs to be irreducible as does each fiber, and the dimension of $f^{-1}(0,0,0)$ must be $2$ while the dimension of the ...
1
vote
1answer
76 views

Finding the angle of a by-pass road given the lengths of two roads forming a right angle

The north-east section of a municipality is a rectangle whose 17 km west boundary is Highway 7 and whose 20 km south boundary is Highway 15. A by-pass road is planned to alleviate traffic congestion. ...
2
votes
1answer
56 views

What form does a function take which desribes a linear coordinate system as seen through a perspective projection?

Let's assume we have a linear coordinate system on a plane. If we make a 2d perspective projection of that plane from 3d, we get a "skewed" coordinate system. For example, one axis would look like ...
1
vote
1answer
62 views

formula for derivative of differentiable mapping

Let $M$ be a differentiable manifold and $f$ a differentiable mapping $f : (-\epsilon, \epsilon) \times M \to M$, $f(t,x) = f_t(x)$ Furthermore let $\gamma: (-\epsilon,\epsilon) \to M$ be a ...
0
votes
2answers
62 views

Angle bisector confusion

I fooled around with the concept of an angle bisector, and I (thought I) found out (and some websites confirmed this, but now I'm in doubt) that the angle bisector of a vertex is the collection of ...
0
votes
1answer
261 views

De Rham cohomology question

I'm trying to compute a certain DeRham cohomology. Consider $M = S^n-C$, where $C$ is the disjoint union of closed disks $C = \cup_{i=1}^m D_i$, and $m,n \geq 1$. How can we compute the cohomology ...
1
vote
1answer
91 views

Is there a simple way to turn a coordinate system “inside out”?

I have the following function, depicted on this image: The strange thing is, that I only know the coordinate system from the "inside out", that means I know the lengths ...
0
votes
1answer
253 views

Squares in a triangle?

I've got some trouble... IJKL is a square and B, I, J, C are aligned (alternatively, |IJ| is confounded with |BC|. h is the height of acute $\triangle$ ABC from A to side BC. C1 is the red ...
-1
votes
1answer
241 views

Finding Angle Between Lines represented by Homogenous Equations

I am trying to find angle between two lines represented by a homogeneous equation The equation is : $ 7x^2 + 4xy + 4y^2 = 0 $ When i use the standard formula $ \theta = \arctan \frac {2 \sqrt {h^2 ...
0
votes
1answer
120 views

How to move a one 3D line from three 3d parallel lines

I have 3 parallel line segments (say AB, CD, and EF are line segments and they are nearly horizontal) lay on 2 slanted planes which have been intersected through the CD. If I projected all the line ...
1
vote
2answers
59 views

What role does the last element of a 3D homogenous vector have?

If I have the following vector { x, y, z, w }, what role does w have in calculations such as dot product, cross product, vector ...
2
votes
1answer
650 views

Shortest distance between clothoid spline and point

Is there any way to analytically decide the shortest distance between a spline of clothoids and a point? Both lies in XY-plane. The clothoid spline has G2 continuity. The result should be used in ...
2
votes
1answer
289 views

Equation of Sheared/Skewed Cone

Let $C$ be an open ended cylinder shell running along the $z$-axis, with radius $1$ - i.e, $x^2+y^2=1$. Let $S$ be a flat plane defined by $z=x+3$. The cylinder $C$ and flat plane $S$ intersect to ...