# 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)

228 questions
Filter by
Sorted by
Tagged with
37 views

### 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 ...
• 29
1 vote
49 views

### 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 ...
1 vote
87 views

### 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: \...
• 8,369
68 views

### 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 ...
64 views

### 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 ...
• 579
60 views

### 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 ...
• 1,898
82 views

### 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, ...
• 499
49 views

### 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 $SP$, ...
• 3,047
137 views

### 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 ...
• 4,450
1 vote
134 views

### 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 ...
• 13
29 views

### 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 ...
• 266
53 views

### 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 &...
• 125
84 views

### 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 ...
• 1,746
26 views

• 21
1 vote
52 views

### 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 ...
209 views

### 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 $EH = EF$...
• 1,031
202 views

### 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 ...
• 1,250
73 views

• 7,031
84 views

### 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 ...
362 views

### 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 ...
• 51.9k
2k views

### Angle chasing in a square [closed]

Attempt:- I also tried some constructions but couldn't solve it.
142 views

### 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 ...
• 7,031
107 views

### 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 ...
• 101
1 vote
40 views

### 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 ...
• 165
194 views

### 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 ...
1 vote
60 views

### 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 ...
• 111
1 vote
57 views

### 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 ...
• 14k
55 views

### 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 ...
164 views

### 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 ...
• 305
100 views

58 views

### 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 $t$ ...
• 59
125 views

### 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 $TP$ ...
• 4,266
316 views

### 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 ...
• 935