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
votes
0answers
78 views

$n$ points on plane with sum of squares of L2 norm = 1

Let $p_1,p_2$ be two points on the 2 dimensional plane. $|p_1|$ denotes the $L^2$ norm of $p_1$ and $\delta(p_1,p_2)$ denotes the euclidean distance between $p_1$ and $p_2$. Let $f(p_1,\ldots, p_n) = ...
2
votes
1answer
62 views

angle inside a chordal quadrilateral

I am trying to solve this problem concerning this chordal quadrilateral. I'm supposed to find out $\beta$. Help is really needed since I study for an exam. $\beta$ should be in dependency of the angle ...
3
votes
2answers
144 views

Minimal length of non-contractible loops

Not self-intersecting loops on a connected closed orientable smooth surface $S$ must have a minimal length not to disconnect it, e.g. the equators of a torus. "Not to disconnect" is - on such surfaces ...
1
vote
1answer
151 views

References request: Introduction to K3 surface.

Are there some good books or survey papers about K3 surface? Thank you very much.
2
votes
0answers
65 views

volumes of balls under an affine transformation

Denote by $B_t(O,\rho) \subset \mathbb{R}^t$, the sphere centered at the origin with radius $\rho$, and $B_n(O,\delta) \subset \mathbb{R}^n$, the sphere centered at the origin with radius $\delta$. ...
1
vote
1answer
33 views

halfspaces question

How do I find the supporting halfspace inequality to epigraph of $$f(x) = \frac{x^2}{|x|+1}$$ at point $(1,0.5)$
0
votes
1answer
539 views

How to translate a slanted cylinder? ( iso-surface geometry)

A cylinder iso-surface formula is: $ x^2 + y^2 = 1 $ If you want to move the cylinder 1 higher on the Y axis it would be: $( x^2 + (y-1)^2 = 1 $ It gets a bit weird with any cylinder which ...
5
votes
3answers
271 views

How to show that all points are inside of unit circle?

There are $n$ points on the plane. Any $3$ of them are inside of a unit circle. How to show that all points are inside of unit circle? It is needed to prove that if there is a unit circle for each ...
6
votes
2answers
157 views

Concurrency of A'L, B'M, C'N.

Need some help with the following problem. Problem: In $\triangle ABC$ the midpoints of $BC$, $AC$, $AB$ are $L, M,$ and $N$ respectively, and the points on the circumcircle opposite to $A, B,$ and ...
1
vote
0answers
100 views

How to construct an orthogonal coordinate system from a smooth planar curve?

Given a planar curve $$\gamma:\mathbb R\to\mathbb R^2, t\mapsto \gamma(t) \text{, normalized to } |\gamma'(t)|\equiv1,$$ the tangential vector $$T(t) = \gamma'(t)$$ and the normal vector $$N(t) = ...
3
votes
2answers
320 views

Problems using idea of tangential quadrilaterals

I'm writing a ~60-page paper on cyclic, tangential and bicentric quadrilaterals. I need to give some problems (with solutions) where usage of those is "hidden". There are lots of problems that use ...
4
votes
0answers
329 views

Geometric intuition for Jordan normal forms (invariant subspaces, shearing, scaling, etc.)

I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form. For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear ...
6
votes
3answers
247 views

Prove or disprove inequality: $2a^2 + 2b^2 + 3c^2 \ge 16P$

Let $a,b,c$ be the lengths of the sides of a triangle with area $P$ prove or disprove inequality: $2a^2 + 2b^2 + 3c^2 \ge 16P$
4
votes
1answer
359 views

Rotation in 4 dimensions around an arbitrary plane

Rotations in 4 dimensions are performed around a fixed plane, they can be described by $SO(4)$, which is a group of orthogonal matrices with determinant equal to 1. It is easy to derive rotation ...
3
votes
1answer
189 views

Relation between Hadamard product and scalar product

Is there a known relation/formula for $$(A\circ B, C)$$ where $\circ$ is the Hadamard product and $(\cdot, \cdot)$ is the scalar (euclidean) product? In particular, I have a vector $y$ and a two ...
3
votes
1answer
11k views

About vector form of a line passing through 2 points.

According to my book: Equation of line passing through 2 points with position vectors $a$ and $b$ is $$r = a + K(b - a)$$ My question: If we are given 2 points how do we determine which point is ...
1
vote
1answer
181 views

Vector / geometry question

I've spent the last 2 hours trying to solve this question, but it's just too hard. Could someone please explain to me in a step by step manner on how I would go about this question. Help would be ...
0
votes
1answer
83 views

Finding the orthogonal projection

The angle between a line and a plane is thirty degrees. Segment $MN$ on the line has length $10$. What is the length of $MN's$ orthogonal projection on the plane? I got 5 as an answer. Is that ...
3
votes
4answers
7k views

Find the angle between the main diagonal of a cube and a skew diagonal of a face of the cube

I was told it was $90$ degrees, but then others say it is about $35.26$ degrees. Now I am unsure which one it is.
1
vote
2answers
509 views

Sum of all deflection angles.

If a polygon has 42 sides, what should the sum of all the deflection angles be? I know what a deflection angle is, but I have no clue how to answer this question with the information I've been ...
1
vote
1answer
668 views

What is “degenerate” about degenerate quadratic surfaces?

In Wikipedia the table of quadratic surfaces is divided into 2 parts, the second being "degenerate quadrics". Why is this distinction made? and what does the word degenerate means in this case?
0
votes
1answer
149 views

How can I calculate the size of a square block and the number of rows and columns needed to fit a known number of blocks on a page of known size?

I am a web developer and have a problem to solve that I thought might be suitable on here... I am developing an app that will be used on a range of phones and tablets all with different screen sizes. ...
5
votes
1answer
297 views

Maximize area of intersection between two rectangles

It is so simple but yet I am unable to solve it. Given two rectangles with sides x,y and a,b respectively. Determine the maximum possible common area of the two.
0
votes
2answers
1k views

proving F is the midpoint of a triangle segment

In triangle ABC, the points D, E and F are on respective segments BC, CA and AB. Also assume AD, BE and CF are concurrent (intersecting at a point P) and line DE is parallel to line AB. I have to ...
8
votes
1answer
435 views

Probem proposed for IMO 26

I have come across this problem and I really don't know how to construct this. Any ideas would be very much apreciated. Given 3 concentric circles, construct an equilateral triangle with a vertex on ...
9
votes
1answer
408 views

Bodyguards and Laser Beams

Suppose you are a point in a square room. The walls of the room are mirrors, and there is a man with a laser gun standing somewhere else in the room. The man is also a point, and both of your ...
1
vote
3answers
846 views

Classification of Euclidean plane isometries

I suppose this question has already been asked here, but I cannot find it. Is there any simple way to prove that there are 5 possibilities for isometries in the Euclidean plane? Namely: Identity, ...
1
vote
1answer
2k views

Area of a Quadrilateral proof

Prove that the area of a quadrilateral is one half the product of the lengths of its diagonals and the sine of the angle between the diagonals.
1
vote
2answers
165 views

Finding the angle of a sector

I have a circle, with an object lying at the edge. In the diagram the object is represented by the blue circle. I need to form a sector in the same way that is drawn in the diagram, given the ...
2
votes
3answers
17k views

Area of a parallelogram, vertices $(-1,-1), (4,1), (5,3), (10,5)$.

I need to find the area of a parallelogram with vertices $(-1,-1), (4,1), (5,3), (10,5)$. If I denote $A=(-1,-1)$, $B=(4,1)$, $C=(5,3)$, $D=(10,5)$, then I see that ...
1
vote
0answers
42 views

How do I calculate the distance between 2 unmeasured points in a cloud?

I know that different kinds of clouds exist at different altitudes. If I am looking at two points in a given cloud at a given altitude, how do I determine the distance between these two visual ...
2
votes
1answer
105 views

Is the intersection of a bunch of cylinders a sphere?

Suppose we have a 3-D shape $S$ with a center $C$, so that a point $p$ is in $S$ if and only if for any direction $\vec d$, $p$ is contained within a cylinder of radius $1$, extending infinitely both ...
0
votes
1answer
41 views

Closure of a set's cone

working in $\mathbb{R}^3$ , say I'm looking at the set : $\{1\} \times S^1$ denote E for the cone of the set above, is E a closed set? (I think it is) if not, what is it's closure? thanks.
3
votes
2answers
2k views

Does the orthocenter have any special properties?

Each of the commonly known triangle centers I know has some sort of special property. For example: The incenter is the center of the inscribed circle. The circumcenter is the center of the circle ...
2
votes
1answer
333 views

Closed Convex sets of $\mathbb{R^2}$

Can some one please list the closed convex sets in $\mathbb{R^2}$ up to homeomorphism. How many of them are compact
2
votes
2answers
130 views

Duplicate quadratic Bézier curve with new start point?

I have Bézier curve as shown by the wikipedia gif here: I would like to create a new curve that is a segment of the old one. For example, in this gif (from the same article): .. if I wanted B to ...
2
votes
3answers
896 views

geometry - triangle

ABC is a triangle in which $ \angle B = 2 \angle C$ D is a point on BC such that AD bisects $\angle BAC$ and AB = CD. Prove that $\angle BAC =72^{\circ}$ Here $\angle BAD = \angle CAD$ AB = DC Can ...
1
vote
0answers
119 views

Rotating and overlaying two equivalent equilateral triangles joined at one edge

I have two equilateral triangles with the same edge lengths, each defined by a set of three points: $(p_1, p_2, p_3)$ and $(q_1, q_2, q_3)$, respectively. Point $p_1$ overlaps with point $q_1$ and ...
3
votes
2answers
216 views

Computing the decay factor for a full rank wide matrix, or finding a unit vector farthest away from a set of spanning unit vectors

Let $A$ be a tall matrix that is not rank-deficient and has normalized columns. That is $A$ is $m\times n$, $m<n$ and rank$(A)=m$, and $||a_i||_2=1$ for all columns $a_i$. Define ...
6
votes
2answers
136 views

Find the most vertical line in a point set in $O(n \log n)$ time

Input: a set of $n$ points in general position in $\mathbb{R}^2$. Output: the pair of points whose slope has the largest magnitude. Time constraint: $O(n \log n)$ or better. Please don't spoil the ...
1
vote
1answer
255 views

Limiting Degrees of Freedom in 3D Point Registration

I'm search for some assistance in my application of Arun's algorithm for registration (fitting) of two 3D point sets using the Singular Value Decomposition: ...
0
votes
3answers
437 views

How to translate a point by $90$ degree counter clockwise direction?

I would like to find a point which is rotated $90$ degree counter clockwise direction about the origin. For example, the point $(2,0)$ is taken to $(0,2)$. Given $(x,y)$, how do i find the new ...
4
votes
5answers
223 views

A question about an equilateral triangle

Suppose that $\triangle ABC$ is an equilateral triangle. Let $D$ be a point inside the triangle so that $\overline{DA}=13$, $\overline{DB}=12$, and $\overline{DC}=5$. Find the length of ...
2
votes
1answer
339 views

Obtaining Least square adjusted single line by intersecting many 3D planes

I am working with many 3D planes and looking for a Least square solution for below case. IF I am having many number of 3D planes knowing only one point and the normal vector (for eg. O1 and N1), ...
4
votes
2answers
4k views

The distance from a point to a line segment

I'm pretty sure this may be a duplicate post somewhere, but I've searched all through the internet looking for a definite formula to calculate the distance between a point and a line segment. There ...
3
votes
0answers
238 views

Difference between crossing and touching curves

Maybe my first attempt to ask the following question was a bit confusing, so let me try it again, without ado: (How) can – in a 2-dimensional topological space – the concept of two ...
2
votes
1answer
2k views

Approximate arc length of cubic bezier curve?

I want to divide a cubic bezier curve, with 4 points, start, end and 2 control points, into segments that are not bigger then a certain distance. So, am looking for a computationally quick way to ...
2
votes
0answers
42 views

Bisecting a line segment with compass only (no ruler) [duplicate]

We all know the construction for bisecting a line segment with compass and ruler However, I wondered if it's still possible to find the midpoint of the segment without use of the ruler, by just ...
3
votes
2answers
369 views

3D geometry proof help (high school)

My textbook is very different from regular high school textbooks because I go to a Christian academy. No one can help me though I was told that there are real math experts here. I need someone to ...
1
vote
1answer
384 views

Geometry questions (high school)

I did my work but I need someone to check my answers and help me with another. A ship leaves port at noon and has a bearing of $\text{S} 29^{\circ} \text{W}$. The ship sails at $20 \text{ knots}$. ...