Dgdrxrt

Is there such a thing as a proper verb, like a proper noun?

Traveling with my 5 year old daughter (as the father) without the mother from Germany to Mexico

How to set page number in right side in chapter title page?

Prepend last line of stdin to entire stdin

Expectation in a stochastic differential equation

Which one is the true statement?

How to use ReplaceAll on an expression that contains a rule

IC has pull-down resistors on SMBus lines?

In the "Harry Potter and the Order of the Phoenix" video game, what potion is used to sabotage Umbridge's speakers?

Calculate the Mean mean of two numbers

Does higher Oxidation/ reduction potential translate to higher energy storage in battery?

"Eavesdropping" vs "Listen in on"

Does destroying a Lich's phylactery destroy the soul within it?

Why is the US ranked as #45 in Press Freedom ratings, despite its extremely permissive free speech laws?

Is it okay to majorly distort historical facts while writing a fiction story?

Lucky Feat: How can "more than one creature spend a luck point to influence the outcome of a roll"?

Why the last AS PATH item always is `I` or `?`?

Yu-Gi-Oh cards in Python 3

(How) Could a medieval fantasy world survive a magic-induced "nuclear winter"?

How did Beeri the Hittite come up with naming his daughter Yehudit?

Film where the government was corrupt with aliens, people sent to kill aliens are given rigged visors not showing the right aliens

Defamation due to breach of confidentiality

Help understanding this unsettling image of Titan, Epimetheus, and Saturn's rings?

Can you teleport closer to a creature you are Frightened of?

Translating points in hyperbolic geometry

1

$begingroup$

The hyperboloid given by:

$$
x^2 + y^2 - z^2 = -1
$$

can be parameterized as:

$$
beginalign
x &= sinh(r) cos(theta)\
y &= sinh(r) sin(theta)\
z &= cosh(r)\
endalign
$$

Conversely, given $(x, y, z)$ we can find $(r, theta)$:

$$
beginalign
r &= cosh^-1(z)\
theta &= tan(y/x)\
endalign
$$

Translating an $(x, y, z)$ point $a$ units along the $x$ axis is accomplished by a hyperbolic rotation.

$$
beginbmatrix
cosh(a) & 0 & sinh(a) \
0 & 1 & 0 \
sinh(a) & 0 & cosh(a) \
endbmatrixbeginbmatrix
x \ y \ z
endbmatrix
$$

Questions

---

1. How is an $(r, theta)$ point translated?


2. Suppose I have the three points $A$ at $(0, 0°)$, $B$ at $(1, 0°)$, and $C$ at $(1, 45°)$. If these three points are translated 1 unit to the right, will $C$ still be at $45°$ relative to $A$?


3. Will the alternate angle to $45°$ be $180°-45° = 135°$?


4. Will $C$ still be $1$ unit away from $A$?

I've drawn a little picture to help illustrate the problem.

$$
beginarrayc
point & before & after \
hline
A & (0, 0°) & (1, 0°) \
B & (1, 0°) & (2, 0°) \
C & (1, 45°) & (?, ?) \
endarray
$$

hyperbolic-geometry

share|cite|improve this question

edited Mar 28 at 2:11

thndrwrks

asked Mar 28 at 1:23

thndrwrksthndrwrks

1264

$endgroup$

add a comment | 

1

$begingroup$

The hyperboloid given by:

$$
x^2 + y^2 - z^2 = -1
$$

can be parameterized as:

$$
beginalign
x &= sinh(r) cos(theta)\
y &= sinh(r) sin(theta)\
z &= cosh(r)\
endalign
$$

Conversely, given $(x, y, z)$ we can find $(r, theta)$:

$$
beginalign
r &= cosh^-1(z)\
theta &= tan(y/x)\
endalign
$$

Translating an $(x, y, z)$ point $a$ units along the $x$ axis is accomplished by a hyperbolic rotation.

$$
beginbmatrix
cosh(a) & 0 & sinh(a) \
0 & 1 & 0 \
sinh(a) & 0 & cosh(a) \
endbmatrixbeginbmatrix
x \ y \ z
endbmatrix
$$

Questions

---

1. How is an $(r, theta)$ point translated?


2. Suppose I have the three points $A$ at $(0, 0°)$, $B$ at $(1, 0°)$, and $C$ at $(1, 45°)$. If these three points are translated 1 unit to the right, will $C$ still be at $45°$ relative to $A$?


3. Will the alternate angle to $45°$ be $180°-45° = 135°$?


4. Will $C$ still be $1$ unit away from $A$?

I've drawn a little picture to help illustrate the problem.

$$
beginarrayc
point & before & after \
hline
A & (0, 0°) & (1, 0°) \
B & (1, 0°) & (2, 0°) \
C & (1, 45°) & (?, ?) \
endarray
$$

hyperbolic-geometry

share|cite|improve this question

edited Mar 28 at 2:11

thndrwrks

asked Mar 28 at 1:23

thndrwrksthndrwrks

1264

$endgroup$

add a comment | 

$begingroup$

The hyperboloid given by:

$$
x^2 + y^2 - z^2 = -1
$$

can be parameterized as:

$$
beginalign
x &= sinh(r) cos(theta)\
y &= sinh(r) sin(theta)\
z &= cosh(r)\
endalign
$$

Conversely, given $(x, y, z)$ we can find $(r, theta)$:

$$
beginalign
r &= cosh^-1(z)\
theta &= tan(y/x)\
endalign
$$

Translating an $(x, y, z)$ point $a$ units along the $x$ axis is accomplished by a hyperbolic rotation.

$$
beginbmatrix
cosh(a) & 0 & sinh(a) \
0 & 1 & 0 \
sinh(a) & 0 & cosh(a) \
endbmatrixbeginbmatrix
x \ y \ z
endbmatrix
$$

Questions

---

1. How is an $(r, theta)$ point translated?


2. Suppose I have the three points $A$ at $(0, 0°)$, $B$ at $(1, 0°)$, and $C$ at $(1, 45°)$. If these three points are translated 1 unit to the right, will $C$ still be at $45°$ relative to $A$?


3. Will the alternate angle to $45°$ be $180°-45° = 135°$?


4. Will $C$ still be $1$ unit away from $A$?

I've drawn a little picture to help illustrate the problem.

$$
beginarrayc
point & before & after \
hline
A & (0, 0°) & (1, 0°) \
B & (1, 0°) & (2, 0°) \
C & (1, 45°) & (?, ?) \
endarray
$$

hyperbolic-geometry

share|cite|improve this question

edited Mar 28 at 2:11

thndrwrks

asked Mar 28 at 1:23

thndrwrksthndrwrks

1264

$endgroup$

share|cite|improve this question

edited Mar 28 at 2:11

thndrwrks

asked Mar 28 at 1:23

thndrwrksthndrwrks

1264

share|cite|improve this question

edited Mar 28 at 2:11

thndrwrks

asked Mar 28 at 1:23

thndrwrksthndrwrks

1264

asked Mar 28 at 1:23

thndrwrksthndrwrks

1264

asked Mar 28 at 1:23

thndrwrksthndrwrks

1264