How to plot point in polar coordinate?

I’m trying to make a figure like this below:

enter image description here

I already have 3d polar coordinate where 3rd dimension is amplitude (Q in above figure)
ex) {{110,10,0.15},{30,170,0.02},…}

Can anyone help me to make such a figure please?
Thank you!

differential geometry – coordinate transformation of the Laplace Beltrami Operator

My Laplace-Beltrami operator isn’t transforming correctly under a change of coordinates. What am I doing wrong?


The Laplace-Betrami operator has the following expressing in local coordinates of a Riemannian manifold $(M, g)$,
begin{align}
Delta f(x) = g^{ij}(x) frac{partial^2 f(x)}{partial x_i partial x_j} + g^{jk}Gamma^i_{jk} frac{partial f(x)}{partial x_i},
end{align}

where $Gamma^i_{jk}$ are Christoffel symbols. Written in this way, it is clear that the Laplace-Beltrami operator is a second-order elliptic operator on the manifold.

Many books such as (Markov Processes by Dynkin)(1) (page 151) or (Functional Analysis by Yosida)(2) (page 426) state that the coefficient functions of elliptic operators must transform in a prescribed way in order to give a consistent result on the manifold. Namely, consider an elliptic operator of the form (written in local coordinates $x$)
begin{align}
Af(x) = b^i(x) frac{partial f(x)}{partial x_i} + a^{ij}(x) frac{partial^2 f(x)}{partial x_ipartial x_j}.
end{align}

Then in another coordinate system $tilde{x}$ the coefficient functions transform as
begin{align}
tilde{b}^i(x) &= b^k(x) frac{partial tilde{x}_i}{partial x_k} + a^{kl}(x)frac{partial^2 tilde{x}_i}{partial x_kpartial x_l} \\
tilde{a}^{ij}(x) &= a^{kl}(x) frac{partial tilde{x}_i}{partial x_k}frac{partial tilde{x}_j}{partial x_l}.
end{align}

The Laplace-Beltrami operator should then obey this transformation rule with $a^{ij} =g^{ij}$ and $b^i = g^{jk}Gamma^i_{ij}$. The fact that $g^{ij}$ obeys the correct transformation rule is apparent; however, I am having a hard time seeing that $g^{jk}Gamma^i_{ij}$ obeys the correct transformation.


What I want to show is that
begin{align}
tilde{g}^{jk}tilde{Gamma}^i_{jk} = g^{pq}Gamma^k_{pq} frac{partial tilde{x}_i}{partial x_k} + g^{pq}frac{partial^2 tilde{x}_p}{partial x_kpartial x_q}
end{align}

I know that the transformation rule for the Christoffel symbols is as follows:
begin{align}
tilde{Gamma}^i_{jk} &= frac{partial x_p}{partialtilde{x_j}}frac{partial x_q}{partialtilde{x}_k} Gamma_{pq}^r frac{tilde{x}_i}{partial x_r} + frac{partial^2 x_r}{partial tilde{x}_jpartial tilde{x}_k} frac{partial tilde{x}_i}{partial x_r} \
&= frac{partial x_p}{partialtilde{x_j}}frac{partial x_q}{partialtilde{x}_k} Gamma_{pq}^r frac{tilde{x}_i}{partial x_r} – frac{partial x_p}{partial tilde{x}_j} frac{partial^2 tilde{x}_i}{partial x_ppartial x_q} frac{partial x_q}{partial tilde{x}_k}
end{align}

If I multiply both sides by $tilde{g}^{jk}$ and use the transformation law of the inverse metric in coordinates, I obtain,
begin{align}
tilde{g}^{jk} tilde{Gamma}^i_{jk} &= tilde{g}^{jk} frac{partial x_p}{partialtilde{x_j}}frac{partial x_q}{partialtilde{x}_k} Gamma_{pq}^r frac{tilde{x}_i}{partial x_r} – tilde{g}^{jk}frac{partial x_p}{partial tilde{x}_j} frac{partial^2 tilde{x}_i}{partial x_ppartial x_q} frac{partial x_q}{partial tilde{x}_k} \
&= g^{pq} Gamma_{pq}^r frac{tilde{x}_i}{partial x_r} – g^{pq} frac{partial^2 tilde{x}_i}{partial x_ppartial x_q}.
end{align}

This is very nearly what I wanted to show, but differs from the expected result by a negative sign in the second term. What have I done wrong?

(1) https://www.springer.com/gp/book/9783662000335
(2) https://www.springer.com/gp/book/9783540586548

geometry – Geometrically finding reflected coordinate of $(x_{1},y_{1})$ about the line $y=xtantheta$

Let us say we require to find the reflection of $(x_{1},y_{1})$ about the line $y=xtantheta$. So it can be done by using the following formula easily.

$$frac{h-x_{1}}{tantheta}=frac{k-y_{1}}{-1}=frac{-2(x_{1}tantheta-y_{1})}{1+tan^2theta}$$

We also have, $k=htan(theta-alpha)$ and $y_{1}=x_{1}tan(theta+alpha)$. I wish to prove this result geometrically, is there a way to do so. Any hints are appreciated. Thanks.

c++ – How do game engines enforce global, engine-specific coordinate systems?

In Unreal, the 3D coordinate system is defined as a right-handed, X-forward, Y-left, Z-up frame of reference; but, the default 3D coordinate system for, say, DirectX is a left-handed, X-right, Y-up, Z-forward frame of reference.

How does an engine enforce this behavior? Or, how do engines define a global coordinate system that “just works”? When an object is asked to get its “forward direction” it doesn’t do any matrix or quaternion conversions that I can tell, it just returns the I-basis (or X-component) of its transform. Its obvious something is being done to translate local transform matrices’ coordinate systems into the engine-specific transform matrices’ required coordinate systems. I just don’t know where or when these modifications are being applied.

EDIT:

A concrete example:

My confusion is compounded by how 3D cameras work; they ultimately determine how the scene is viewed. In my own engine I don’t change the default behavior that DirectX provides, so my coordinate system is a left-handed, X-right, Y-up, Z-forward. I don’t like this. I want it to be X-forward, Y-right, Z-up globally throughout the engine. Where would I do this? During the view calculation for every camera? Would that affect intuitive translations where if I translate “right” with a vector (+10,0,0) the camera erroneously appears to move forward?

Calculate the double integral in polar coordinate and domains D

Calculate the double integral in polar coordinate and domains D (imgur)

graphics3d – Are there commands other than ListPlot3D for plotting 3D points, where one coordinate ls either 0 or 1, and the other two lie in [0,1]?

I have a set of 3D points, the first coordinate of which, say, is confined to the values 0 and 1, while the other two can assume values within the unit interval [0,1]. ListPlot3D shows a continuous variation of the first value–which perhaps is somewhat misleading.

Can I represent the discreteness of the first coordinate, while simultaneously showing the continuous nature of the other two?

Placing an object as near to a specific coordinate without overlapping another object

I am working on a 2D game in Java. My goal is to place a ball as close as possible to the center without overlapping the balls that are already there. How can this be done?

plotting – Labeling coordinate curves in ParametricPlot

I want to display a coordinate grid in parabolic coordinates:

ParametricPlot({-parabolic(r, phi), parabolic(r, phi)}, {r,0,1}, {phi,-1, 1},
  PlotStyle -> {Gray, Gray}, 
  BoundaryStyle -> Dashed, Mesh -> 9, Frame -> False, Ticks -> None)

Is there a relatively simply way to include labels for a selection of coordinate curves within the plot (essentially a ‘non-dynamic’ Tooltip)?

coordinate transformation – How to troubleshoot TransformedField functionality in Mathematica?

I list up this method to transform a complex function to Cartestian form, which can be used on virtually any function:

such as:

u0(r_, phi_) := Sum(I^(-n) BesselJ(n, r) Exp(I n phi), {n, -5, 5});

TransformedField("Polar" -> "Cartesian",  u0(r, phi), {r, phi} -> {x, y})

which yields:

  u0(x_, y_) := 
 BesselJ(0, Sqrt(x^2 + y^2)) + 
  1/(x^2 + y^2)^(5/2)
    2 (-I x (x^2 + y^2)^2 BesselJ(1, Sqrt(x^2 + y^2)) + 
      Sqrt(x^2 + y^2) (-x^4 + y^4) BesselJ(2, Sqrt(x^2 + y^2)) + 
      I x^5 BesselJ(3, Sqrt(x^2 + y^2)) - 
      2 I x^3 y^2 BesselJ(3, Sqrt(x^2 + y^2)) - 
      3 I x y^4 BesselJ(3, Sqrt(x^2 + y^2)) + 
      x^4 Sqrt(x^2 + y^2) BesselJ(4, Sqrt(x^2 + y^2)) - 
      6 x^2 y^2 Sqrt(x^2 + y^2) BesselJ(4, Sqrt(x^2 + y^2)) + 
      y^4 Sqrt(x^2 + y^2) BesselJ(4, Sqrt(x^2 + y^2)) - 
      I x^5 BesselJ(5, Sqrt(x^2 + y^2)) + 
      10 I x^3 y^2 BesselJ(5, Sqrt(x^2 + y^2)) - 
      5 I x y^4 BesselJ(5, Sqrt(x^2 + y^2)))

or

a Hankel and Bessel function together:

u(r_, phi_) :=  Piecewise({{BesselJ(1.5 r, 5)*Exp(I 5 phi), 
0 < r < 1/2}, {(BesselJ(3 r, 5) + BesselY(3 r, 5))*Exp(I 5 phi), 
1/2 < r < 1}, {HankelH1(r, 5)*Exp(I 5 phi), r > 1}})

which yields:

Hankel+Bessel

which yields the respective given plots, when plotted:

Plot_Bessel_Cartesian

and

Plot_Hankel+Bessel_Cartesian

However, my supervisor thinks these plots look “strange”.

How can I verify that the TransformedField command did the right job – for such extended functions?

In other words, how to trobleshoot TransformedField?

orthogonality – How do I find the basis of an orthogonal coordinate system?

Given two arbitrary orthogonal coordinates x1 and x2,
with orthonormal basis i1 and i2,
where a is a constant and coordinates F1 and F2 are defined by:

x1 = a frac{sinh(F^{1})}{coshF^{1}+cosF^{2}}

x2 = a frac{sin(F^{2})}{coshF^{1}+cosF^{2}}

How would I find the basis vectors, e1 and e2?
I know that for e1 and e2 to form a basis then i1 and i2 have to be in the span of e1 and e2 but I’m having a hard time trying to find something online that’s written in simple enough terms to understand