Shap3d50.dat, 50th 'shape' file for CPO3D
5 holes in disc, details, by users equation.
For information on the option for users equations, see users equations.
2 symmetry planes and 6 electrodes are used in shap3d49 to give a complete disc.
Here the symmetry planes have been removed and also the last set of 3 electrodes has been removed so that only one octant appears, to enable an explanation to be given of the users equations.
As with shap3d49, the radius of the inner hole is r1 = 2, the radii of the 4 surrounding holes is r2 = 1 and the outer radius is R = 6. The disc lies in the plane z = 0.
The octant is divided into 3 electrodes, each of which has 4 distinct parts for the boundary.
As explained elsewhere, suppose that the equations g1(x,y,z) define the 'inner' part of the boundary, for example x = r*cos(phi), y = r*sin(phi), z = 0, phi = 0 to pi/2, which defines an arc of a circle.
Then suppose that the equations g2(x,y,z) define the opposite 'outer' part of the boundary, for example an arc of a circle of radius R.
In the usual fashion we can combine these by using the general equation
g(x,y,z) = (1 - f)*g1 + f*g2, where f goes from 0 to 1.
This reproduces g1 when f = 0, and g2 when f = 1.
The other 2 edges will go linearly between the ends of the 'inner' and 'outer' edges.
Now we can examine the equations for the 3 electrodes.
In each electrode one or both of the 'inner' and 'outer' edges is curved while the other 2 edges are straight. We therefore satisfy the conditions for using the general method outlined above.
In electrode 1, g1 defines the 'inner' edge as an arc of radius r1. The equations g2 for the 'outer' edge are linear in the variable phi and therefore define a straight line. As phi goes from 0 to pi/4, we can see that (x,y) goes from (0,(s-r2)) to (0.7071*(s-r2),0.7071*(s-r2)), so giving the straight line that can be seen in the simulation.
In electrode 2, g1 defines the 'inner' edge as an arc of radius r2, centred at (x,y) = (0,s), giving the first part of the outer hole. The equations g2 for the 'outer' edge are again linear, with (x,y) going from (0.7071*(s-r2),0.7071*(s-r2)) to (0.7071*R,0.7071*R), giving the second part of the diagonal edge of the complete simulation.
In electrode 3, g1 again defines the 'inner' edge as an arc of radius r2, thus completing the outer hole. The equations g2 for the 'outer' edge also define an arc, this time of radius R.
In shap3d49 the electrodes 1 to 3 above are copied as electrodes 4 to 6 and are transformed by simply interchanging the equations for x and y (which effectively reflects about the x=y plane). Also the symmetry planes x=0 and y-0 are added to give a complete disc.