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Questions tagged [geometric-transformation]

A geometric transformation is any bijection of a set having some geometric structure to itself or another such set. (from Wikipedia)

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isometries as products of reflections [closed]

I am taking a course on geometric transformations in the 2-dimensional Euclidean plane. I have been told that all isometries (length-preserving transformations) are products of reflections about ...
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Rotate and Scale Around Different Origins using Local Coordinates

How can I chain rotate and scale operations (each with different origins) while keeping each operation origin in relation to the original local component space? Using matrix denotation (i.e. T for ...
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Is there a rotation formula for the gamma function: $\Gamma(e^{i\theta} z) = \Gamma(z)\cdot F_z(\theta)$, $0\le \theta <2\pi$?

The gamma function is essentially given by its functional equation $$\Gamma(z+1)=z\Gamma(z),$$ together with $\Gamma(1)=1$. This is a translation formula. Likewise, there is a reflection formula: $$\...
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systematic method to compare monic quadratics in order to find their roots

but I am wondering if anyone has worked on this idea: Notice that among all monic quadratics, the non-trivial equation $x^2+x-2$ has it's coefficient equal to it's roots, ie. $x=1,-2$. Let's consider ...
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If $T$ maps the lines to the lines then $T$ is additive

Let $T:\, \mathbb R^n\to\mathbb R^n\ \, (n\geq 2)$ be a bijective map that takes straight lines to straight lines, and $T(0)=0$. I want to show that $T$ is linear. So far, I have proved that $T$ maps ...
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Geometry inquiry on inversion.

Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ which is tangent to the sides $AB, BC, CD, DA$ at points $P, Q, R, S$ respectively. We want to show that $AC, BD, PR, QS$ meet at one ...
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Rotation of angle $2k\pi/p$ generates the group of all rotation

In the book Geometric Transformation of Razvan Gelca, there is an argument as follows: I could understand most of the proof there, however is there any easier explanation for the yellow painted part, ...
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projective transformation that maps lines through a point to parallel lines

A construction is as follows: Given a fixed line $L$ and two fixed points $S,S'$. For any point $P$ on the plane, let the line $PS'$ intersect $L$ at $Z$. Draw the line through $S'$ parallel to $SP$, ...
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Conjecture: Every shear transformation of the plane is congruent to a dilation

I conjecture that every shear transformation of the plane is geometrically congruent to an orthogonal dilation of the plane. That is, in Euclidean geometry of the plane, if I shear figure $F$, I can ...
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Transformation between space partitions

We have a space S, being partitioned into a set of polygons P containing $n$ polygons $P_1, P_2,..., P_n$. Given $n$ constants $k_1,k_2,...,k_n $. Apply a transformation $T$ from partition $P$ to ...
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Showing that a cylinder transformation maps rays to horizontal lines and circles to circles

I am reading through a text on time scale calculus and I came upon a certain transformation whose follow up question I cannot solve. Its defined as follows For $h >0$, let $\mathbb{Z}_h$ be the ...
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Find the image of spherical surface x^2+y^2+z^2=1 under the composite transformation

\begin{array}{l} ( 5-1) \ From\ the\ question,\ the\ normal\ vector\ of\ plane\ x+y+z=0,\ \vec{u} =( 1,1,1)^{T}\\ According\ to\ Householder\ Transformation,\ g=I-2\vec{u}\vec{u} =\begin{pmatrix} 1 &...
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What theory describe such images?

I make naive, visual exploration of 2D images, which can be described by simple functions, like: $ y = f(x)$ $ y = f(x,y)$ $f(z) = \begin{cases} k(x,y) \\ h(x,y) \end{cases}$ I have read about ...
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If I have two congruent circles that reflect each other, how would I find its reflection line?

I have these two circles, $\left(-6,\ 0.8\right)$ for the blue circle, and $\left(-3.988,\ -8.159\right)$ for the black circle. Using the formula to find the mid-point of these circles, I got $\left(-...
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Spherical transformation from world coordinates

I have a camera pose (position and orientation) in a 'world coordinate' frame described by a (4, 4) transformation matrix. I want to transform this in a spherical coordinate frame, assuming the ...
aktabit's user avatar
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Diffeomorphism between star-space and sphere-space

I am a robotics student who has very poor knowledge of topology, thus I hope my question is not ill-posed. Studying the classical textbook [1], I found an interesting diffeomorphism from stars* to ...
Antonio Bono's user avatar
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Rotate a body to align z axis in a particular direction

I have the orientation of a body in world frame, $_wP_b$. Let us say, a bottle with the z axis representing the height, lying on its sides on the table. Now, I have a direction vector $\textbf{v}$, ...
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How to transform global coordinates to local coordinates?

For example, I have 4 points with the following global coordinates $(4,2),(5,3),(6,4),(8,5)$. Graph How to transform these global coordinates into local, such that the first point is $(0,0)$ in the ...
vspredator's user avatar
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Geometry transformation in function translation

Let $f(x)=x+3$. let's say we want a new function that is a translation of $f(x)$ by $5$ units to the right. If we will denote by $x'$ the new coordinate, Then For all $x$ , $x'=x+5\,\Rightarrow\, x=x'...
Frime196's user avatar
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Can you map $\mathbb{R}^2$ onto half of $\mathbb{R}^2$ as a bijection?

Basically, can you map all the points on a coordinate plane to unique points on the space above the x-axis? The way I'm imagining it is that the vertical lines infinitely close to the right and left ...
Awesome_fire's user avatar
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Locus of point with constant angle

It's been a few years since I dealt with Euclidean Geometry and I now fell onto this problem: We are given the square $ABCD$, and point $F$ on $AB$. We construct a right angle $FEH$ for which $EH = EF$...
Juan Manuel Prada's user avatar
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Concurrency of three lines in a multiple tangent circles configuration

Let $ABC$ be a triangle, $\mathcal{C}$ its circumscribed circle and $\mathcal{I}$ its inscribed circle. We construct a circle that is tangent on the interior to $\mathcal{C}$ in the point $A$ and is ...
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Doing a complex 2D transformation

Given the following graphical representation of shape that has been transformed: What is the final transformation matrix for it? I am doing the following: Translate center to origin: $ T_1=\begin{...
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Transform point in the Poincaré disc to point in tile

I have a set of image files that I want to use to use to texture a set of tiles with (one image per tile), in order to render a textured version of the Poincaré disc with a specific tiling by using ...
HelloGoodbye's user avatar
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What's the value of the area of the triangle $ABC$ below?

For reference: Calculate the area of ​​triangle $ABC$; if $ED = 16$; $AB = 10$ and $D = \angle15^o$(Answer:$20$) My progress: I didn't get much. $\triangle ECD - (15^o, 75^o) \implies EC = 4(\sqrt6-\...
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Ultrasonic anemometer: Transformation of space diagonal components to Cartesian components

We have built an ultrasonic anemometer measuring 4 components of air velocity along the 4 space diagonals of a cube. The space diagonals can be characterized by vectors (1,1,1), (1,-1,1), (-1,1,1) and ...
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Find $\angle CAD$ if $\triangle ABC$ is right angled at $B$, $\angle BAD = 30^\circ, \angle ADB = \angle ADC = 15^\circ$

Find angle $\theta$ in the below diagram. This is a question that was brought to me by a high school student. While I came up with a trigonometric solution and a synthetic solution, I am posting here ...
Math Lover's user avatar
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Angle chasing in a square [closed]

Attempt:- I also tried some constructions but couldn't solve it.
Navdeep Singh's user avatar
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What is the value of the measure of the segment $MN$?

In an ABC triangle. plot the height AH, then $ HM \perp AB$ and $HN \perp AC$. Calculate $MN$. if the perimeter of the pedal triangle (DEH) of the triangle ABC is 26 (Answer:13) My progress: I made ...
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How could a "fan-shaped" transformation of the angles be mathematically described?

I'm no real mathematician (OK, I'm a statistician, but here it doesn't help ...), and this is my first question here, so excuse me if my question doesn't make much mathematical sense. I need to get my ...
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How can I extract the equations from this 3D projection graph?

I'm trying to transform these plots to functions, but I'm having a hard time figuring out a formulaic 2D->3D transformation on this type of mapping. They collapsed an axis flat on the abscissa, and ...
Esteban's user avatar
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non-rigid transformation for data correction

I am seeking a mathematical method to perform non-rigid transformations of quadrilaterals as demonstrated below. Here you can imagine the four coordinates of a rectangular figure where the data ...
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Name for the relation between two surfaces when one is equal to the other after an affine transform

I'm trying to wrap my head around what I think is 50% a definitions problem and 50% me not understanding affine/vector spaces and subspaces well enough. I have an operation that I can apply to any set ...
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1 answer
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Show that it is a point reflection at $A$.

We have the points $Z=(-1,1)$, $A=(-1,6)$ and $B=(3,4)$. Let $\delta$ be the rotation with center $Z$ and $\delta (A)=B$. Let $C$ be the point on the circumcircle of the triangle $ABZ$ such that the ...
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Transformation properties of surface measures under flow maps - Alternativ to Nanson's formula

I am aware of the so-called Nanson's formula which relates area elements in a reference configuration to those in a deformed configuration through the adjugate of the jacobian matrix. In Chadwick's ...
Not a chance's user avatar
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Show that projective lines pass through a fixed point

In the projective plane $P(\mathbb{R}^3)=\mathbb{R}P^2$ a triangle $\Delta ABC$ and a point $D$, not on either side of the triangle are given. Denote $P_1=AD\cap BC$ and let $l$ be a changeable line ...
TheHunter's user avatar
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Question about a series of distance preserving transformations on points

I have a problem that asks me to Find all length preserving transformations of the plane that send point A to point A’ and point B to point B’ where: $A=(0,1), B=(1,1), A’=(3,2), B’=(3- \frac{\sqrt3}...
alb.j789's user avatar
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1 answer
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How to figure out the transformation matrix for rotation and then sheer?

I was watching this video. (Actually, I watched it 3 times because I couldn't understand it.) And right then, he showed that the trasformation matrix for rotation and then sheer is I understood how ...
Shubham Gupta - TCH's user avatar
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On the proof about the dimension of the conformal group of a manifold

I have been reading the book "Transformation Groups in Differential Geometry" by S. Kobayashi. More concretely, I am trying to understand the proof of the Theorem 6.1 of Chapter IV. Theorem ...
I. Roperval's user avatar
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3 answers
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Geometry problem proving that all lines $DE$ passes through one point

Let $I$ is the incenter of $\triangle ABC$. Let $K$ be the circumcircle of $ABC$. Let $D$ be a variable point on arc $AB$ on $K$ not containing $C$. Let $E$ be a point on line segment $BC$ such that $\...
Maths explorer's user avatar
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Although the transformation $t: x^2 \rightarrow x^2 + \frac{1}{x^2+1} - 1$ is not a linear transformation, is it another sort of transformation?

Although it clearly isn't a linear transformation, is there any sort of other definite transformation that could carry us from $x^2$ to $x^2 + \frac{1}{x^2 + 1}$, such that we could test whether $t$ ...
shintuku's user avatar
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4 answers
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Calculate the measure in degrees of the acute angle formed by $ NMA $?

Consider the $PAT$ triangle such that the angle $ \angle P = 36 $ degrees, $ \angle A = 56 $ degrees and $ PA = 10 $. Knowing that the points $ U $ and $ G $ belong, respectively, to the sides $ TP $ ...
Lambert macuse's user avatar
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1 answer
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Geometric transformations in Olympiad problems - Putnam 2001, A-4

I have been reading Pasumarty's article A Fine Use of Transformations in Mathematical Reflections 4 (2016). The first part of the article deals with question A-4 on Putnam 2001: Triangle $ABC$ has an ...
Paolo's user avatar
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Midpoints of diagonals

Let $ABCD$ be convex quadrilateral such that $AB=CD$. And $E\neq F$ where $E, F$ is midpoint of $AC, BD$ respectively. Then prove that angle between$(AB, EF)$ and $(CD, EF)$ are equal. I'll prove $\...
Mutse's user avatar
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6 answers
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Point within the interior of a given angle

The point $M$ is within the interior of given angle $\alpha$. Find the distance between $M$ and the vertex of the angle ($OM=?$) if $a$ and $b$ are the distances from $M$ to the sides of the angle. ...
Math Student's user avatar
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Midpoint of a line segment passing through two points.

Let $C_1$ be a circle defined with $X(-2,7)$ and $Y(2,-5)$ as the endpoints of the diameter of the circle. Let $C_2$ be a circle defined with $Y(2,-5)$ and $Z(4,-11)$ as the endpoints of the diameter ...
Sid's user avatar
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The image of a line through an inversion

I am going to start by saying that geometry is not my strong suit, but I am taking a course on analytic geometry where I learnt about inversions and there is this question that bugs me. The following ...
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Conformal map $\mathbb{C} \setminus (\{ z \mid \mathrm{Im}(z) \leq 0\} \cup\{z = x+ iy \mid y \geq \sqrt{x^2 + 1}, x \leq 0\})$ onto unit disk

For figure $\mathbb{C} \setminus (\{ z \mid\mathrm{Im}(z) \leq 0\} \cup\{z = x+ iy \mid y \geq \sqrt{x^2 + 1}, x \leq 0\})$ find a conformal mapping to closed unit disk. What I have done so far: I ...
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How to remap points on 2d plane to another plane with regression kind of approach?

The title can not be clear enough as I really don't know how to call this problem. Let me describe it briefly. I have some points in coordinate system as (x,y). All points have to move to a new ...
Horizon1710's user avatar
4 votes
4 answers
291 views

A geometry question...

In the given figure, $ABCD$ is a square of side $3$cm. If $BEMN$ is another square of side $5$cm & $BCE$ is a triangle right angled at $C$. Then the length of $CN$ will be:- I plotted this on ...
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