using a linear equation to create a radially interpolated circle

Here is one way to solve it, the idea is to create a mesh, calculate the colors with a function then use imshow to visualize the mesh.

from matplotlib import pyplot as plt 
import numpy as np


def create_mesh(slope,center,radius,t_x,t_y,ax,xlim,ylim):
    """
    slope: the slope of the linear function
    center: the center of the circle
    raadius: the radius of the circle 
    t_x: the number of grids in x direction 
    t_y: the number of grids in y direction 
    ax: the canvas 
    xlim,ylim: the lims of the ax
    """
    def cart2pol(x,y):
        rho = np.sqrt(x**2 + y**2)
        phi = np.arctan2(y,x)
        return rho,phi
    
    def linear_func(slope):
        # initialize a patch and grids  
        patch = np.empty((t_x,t_y))
        patch[:,:] = np.nan
        x = np.linspace(xlim[0],xlim[1],t_x)
        y = np.linspace(ylim[0],ylim[1],t_y)
        x_grid,y_grid = np.meshgrid(x, y)
        
        # centered grid
        xc = np.linspace(xlim[0]-center[0],xlim[1]-center[0],t_x)
        yc = np.linspace(ylim[0]-center[1],ylim[1]-center[1],t_y)
        xc_grid,yc_grid = np.meshgrid(xc, yc)
        rho,phi = cart2pol(xc_grid,yc_grid)
        linear_values = slope * rho
        
        # threshold controls the size of the gaussian 
        circle_mask = (x_grid-center[0])**2 + (y_grid-center[1])**2 < radius
        patch[circle_mask] = linear_values[circle_mask]
    
        return patch

    # modify the patch
    patch = linear_func(slope)
    extent = xlim[0],xlim[1],ylim[0],ylim[1]
    ax.imshow(patch,alpha=.6,interpolation='bilinear',extent=extent,
              cmap=plt.cm.YlGn,vmin=v_min,vmax=v_max)
        
        
fig,ax = plt.subplots(nrows=1,ncols=2,figsize=(12,6))
slopes = [40,30]
centroids = [[2,2],[4,3]]
radii = [1,4]

for item in ax:item.set_xlim(0,8);item.set_ylim(0,8)
v_max,v_min = max(slopes),0

create_mesh(slopes[0],centroids[0],radii[0],t_x=300,t_y=300,ax=ax[0],xlim=(0,8),ylim=(0,8))
create_mesh(slopes[1],centroids[1],radii[1],t_x=300,t_y=300,ax=ax[1],xlim=(0,8),ylim=(0,8))

plt.show()

The output of this code is

As you can see, the color gradient of the figure on the left is not as sharp as the figure on the right because of the different slopes ([40,30]).

Also note that, these two lines of code

v_max,v_min = max(slopes),0

ax.imshow(patch,alpha=.6,interpolation='bilinear',extent=extent,
          cmap=plt.cm.YlGn,vmin=v_min,vmax=v_max)

are added in order to let the two subplots share the same colormap.