How to render ( y - x + 1 - ( (y-x+1)^2 -4y )^(1/2) ) / (2) in Blender?

Question

I need to render:

$$z = \left( y - x + 1 - \left[\left(y-x+1\right)^2 -4y\,\right]^\left(\frac{1}{2}\right) \right) \div 2$$

as a 3d function in Blender to make a visual using domain $$0<=x<=1$$ $$0<=y<=1$$

The software reports an error when I type **$(1/2)$ to add the square root part.

Answer 1

This is a Geometry nodes solution:

Answer 2

You can easily plot the points using a python script. This script iterates through a list of $x$ & $y$ values from $-70$ to $70$ where as @Robin Betts has pointed out $4y > (y-x+1)^2$ are imaginary values and cannot be plotted. So this script ignores negative values where the term $4y$ is greater than the term $(y-x+1)^2$ and only plots the real points.

import bpy

def get_object(name):
    objects = bpy.context.scene.objects
    if name in objects:
        return objects[name]
    m = bpy.data.meshes.new(name + "-mesh")
    o = bpy.data.objects.new(name, m)
    #o.modifiers.new(name, 'SKIN')
    bpy.context.collection.objects.link(o)
    return o 
 
# ==================================================================================================
# Equation:
# Descritpion: plot the graph ( y - x + 1 - ( (y-x+1)^2 -4y )^(1/2) ) / (2)
# ==================================================================================================

def get_range(start, end, step = 2):
    return [x * 0.1 for x in range(start * 10, end * 10, step)]
 
def get_graph_z_real(x, y):
    return (y - x + 1)**2 - 4*y

def get_graph_z(x, y, real):
    return ( y - x + 1 - ( real )**(1/2) ) / (2)

def draw_graph():
    verts = []

    for py in range(-70, 70):
        for px in range(-70, 70):
            real = get_graph_z_real(px, py)
            if real < 0:
                continue
            pz = get_graph_z(px, py, real)
            verts.append([px, py, pz])

    o = get_object("graph")
    m = o.data
    m.clear_geometry()
    m.from_pydata(verts, (), ())


draw_graph()

Or you can use the Z Math Surface under object menu Add > Math Function > Z Math Surface

But since you have that imaginary part you cannot directly use the equation $(y-x+1-((y-x+1)^2-4y)^{1/2} )/(2)$ but instead need to use a condition to filter out the imaginary part. The best you can do is probably set the term $(y-x+1)^2-4y$ to zero if $4y$ is greater than $(y-x+1)^2$ like so (or experiment with other non-imaginary values):

(y-x+1-((y-x+1)**2 -4*y if 4*y < (y-x+1)**2 else 0 )**(1/2) ) / (2)

Another sample to restrict the domain to $0 <= x <= 1$ & $0 <= y <= 1$

import bpy

def get_object(name):
    objects = bpy.context.scene.objects
    if name in objects:
        return objects[name]
    m = bpy.data.meshes.new(name + "-mesh")
    o = bpy.data.objects.new(name, m)
    #o.modifiers.new(name, 'SKIN')
    bpy.context.collection.objects.link(o)
    return o 
 
# ==================================================================================================
# Equation:
# Descritpion: plot the graph ( y - x + 1 - ( (y-x+1)^2 -4y )^(1/2) ) / (2)
# ==================================================================================================

def get_range(start, end, step = 2):
    return [x * 0.001 for x in range(start * 1000, end * 1000, step)]
 
def get_graph_z_real(x, y):
    return (y - x + 1)**2 - 4*y

def get_graph_z(x, y, real):
    return ( y - x + 1 - ( real )**(1/2) ) / (2)

def draw_graph():
    verts = []

    for py in get_range(0, 1):
        for px in get_range(0, 1):
            real = get_graph_z_real(px, py)
            if real < 0:
                continue
            pz = get_graph_z(px, py, real)
            verts.append([px, py, pz])

    o = get_object("graph")
    m = o.data
    m.clear_geometry()
    m.from_pydata(verts, (), ())

draw_graph()

Answer 3

Here's a Geometry Nodes solution to cater to math domain $0<=x<=1$ & $0<=y<=1$

Note: In case you are wondering why it has jagged edges, it actually is an accurate representation of the boundary between real and complex numbers. See comments in GN: How to smooth out jagged edges after deleting geometry?

This one confirms the output of my python solution in the other answer.

Answer 4

Here's a Geometry Nodes solution using Volume Cube node. Take note that values for $4y > (y-x+1)^2$ are imaginary and cannot be plotted because there is no square root for negative values hence the imaginary check before the Density socket input.