## Slice in volumetric data, via Plotly¶

A volume included in a parallelepiped is described by the values of a scalar field, $f(x,y,z)$, with $x\in[a,b]$, $y\in [c,d]$, $z\in[e,f]$. A slice in this volume is visualized by coloring the surface of the slice, according to the values of the function f, restricted to that surface.

In order to plot a planar or a nonlinear slice of equation z=s(x,y) one proceeds as follows:

• define a meshgrid in x,y;
• evaluate z=s(x,y)
• define an instance of the Plotly Surface class, that represents the surface z=s(x,y)
• this surface is colored according to the values, f(x,y,z), at its points. More precisely, the normalized values of the function f are mapped to a colormap/colorscale.

With obvious modications we get slices of equation $x=s(y,z), y=s(z,x)$.

In [ ]:
import numpy as np
import plotly.graph_objects as go
from IPython


Define a function that returns a slice as a Plotly Surface:

In [ ]:
def get_the_slice(x,y,z, surfacecolor):
return go.Surface(x=x,
y=y,
z=z,
surfacecolor=surfacecolor,
coloraxis='coloraxis')

In [ ]:
def get_lims_colors(surfacecolor):# color limits for a slice
return np.min(surfacecolor), np.max(surfacecolor)


Let us plot the slices z=0 and y=-0.5 in the volume defined by:

In [ ]:
scalar_f = lambda x,y,z: x*np.exp(-x**2-y**2-z**2)

In [ ]:
x = np.linspace(-2,2, 50)
y = np.linspace(-2,2, 50)
x, y = np.meshgrid(x,y)
z = np.zeros(x.shape)
surfcolor_z = scalar_f(x,y,z)
sminz, smaxz = get_lims_colors(surfcolor_z)

slice_z = get_the_slice(x, y, z, surfcolor_z)

In [ ]:
x = np.linspace(-2,2, 50)
z = np.linspace(-2,2, 50)
x, z = np.meshgrid(x,y)
y = -0.5 * np.ones(x.shape)
surfcolor_y = scalar_f(x,y,z)
sminy, smaxy = get_lims_colors(surfcolor_y)
vmin = min([sminz, sminy])
vmax = max([smaxz, smaxy])
slice_y = get_the_slice(x, y, z, surfcolor_y)


In order to be able to compare the two slices, we choose a unique interval of values to be mapped to the colorscale:

In [ ]:
def colorax(vmin, vmax):
return dict(cmin=vmin,
cmax=vmax)

In [ ]:
fig1 = go.Figure(data=[slice_z, slice_y])
fig1.update_layout(
title_text='Slices in volumetric data',
title_x=0.5,
width=700,
height=700,
scene_zaxis_range=[-2,2],
coloraxis=dict(colorscale='BrBG',
colorbar_thickness=25,
colorbar_len=0.75,
**colorax(vmin, vmax)))

#fig1.show()

In :
from IPython.display import IFrame
IFrame('https://chart-studio.plotly.com/~empet/13862', width=700, height=700)

Out:

### Oblique slice in volumetric data¶

As an example we plot comparatively two slices: a slice through $z=0$ and an oblique planar slice, that is defined by rotating the plane z=0 by $\alpha=\pi/4$, about Oy.

Rotating the plane $z=c$ about Oy (from Oz towards Ox) with $\alpha$ radians we get the plane of equation $z=c/\cos(\alpha)-x\tan(\alpha)$

In [ ]:
alpha = np.pi/4
x = np.linspace(-2, 2, 50)
y = np.linspace(-2, 2, 50)
x, y = np.meshgrid(x,y)
z = -x * np.tan(alpha)

surfcolor_obl = scalar_f(x,y,z)

In [ ]:
smino, smaxo = get_lims_colors(surfcolor_obl)
vmin = min([sminz, smino])
vmax = max([smaxz, smaxo])

In [ ]:
slice_obl = get_the_slice(x,y,z, surfcolor_obl)

In [ ]:
fig2 = go.Figure(data=[slice_z, slice_obl], layout=fig1.layout)
fig2.update_layout( coloraxis=colorax(vmin, vmax))

#fig2.show()

In :
IFrame('https://chart-studio.plotly.com/~empet/13864', width=700, height=700)

Out:
In [ ]:
from IPython.core.display import HTML
def  css_styling():
styles = open("./custom.css", "r").read()
return HTML(styles)
css_styling()