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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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 preserves angle at a point
I try to find all projective transformations on $\mathbb RP^2$ that fix the point $[0,0,1]$, and preserve the angle between any two lines through $[0,0,1]$.
Using SageMath to calculate Jacobian I ...
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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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Calculating a homography matrix between a conic and a unit circle
Given that I have a conic in the form of $ x^T A_Q x=0$ where $x$ is the homogeneous coordinate vector in three variables. $A_Q$ is the matrix:
$${\displaystyle A_{Q}={\begin{pmatrix}A&B/2&D/2\...
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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 ...
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Describe all the situations in $\mathbb{R}^3$ when $R_{u,\phi} \circ R_{w,\psi} = \tau_v \circ R_{v,\xi}$
Describe all the situations in $\mathbb{R}^3$ when $R_{u,\phi} \circ R_{w,\psi} = \tau_v \circ R_{v,\xi}$
where $R_{w,\psi}$ - rotation around a line with direction vector $w$, counterclockwise, at ...
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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 ...
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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 ...
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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'...
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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 ...
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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 $EH = EF$...
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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 ...
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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 ...
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Angle chasing in a square [closed]
Attempt:-
I also tried some constructions but couldn't solve it.
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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 ...
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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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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 ...
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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 ...
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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}...
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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 ...
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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 ...
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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 $\...
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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 $t$ ...
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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 $ ...
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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 ...
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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 $\...
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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.
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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 ...
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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 ...
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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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To prove $EA = FB$ or that $CQ'$ is radical axis
Given disjoint circles $c_1 = \odot(P,PA), c_2 = \odot(O,OB)$ such that $B$ and $A$ are in the same half-plane wrt $OP$ and that $PA \parallel OB \perp OP$.
Line $CDQ$ is the perpendicular bisector ...
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Problem with right angle triangle, circumscribed circle, tangent and the half of its height
(An interesting problem inspired by this one but still different. And, no, I'm not looking for your help to solve a detail here in order to provide a full solution elsewhere. I'll stop here).
A right-...
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Prove that IK, AC, BD are concurrent.
Given a circle (O, R) with diameter AB. Point M on (O), A, B are not coincident. Two lines through O and perpendicular to AM, BM intersects the tangent of (O) through M at C, D, respectively. OC ...
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A Transformation of a cross-shaped grid filled with 1s (Proof of impossibility?)
Consider a cross-shaped grid of size 7 as it shows on the figure (compared to one of size 3). Each cell contains a 1. Le'ts define a transformation $\pi$ of the grid as follows: take any 3 sized sub-...
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