Shap3d33.dat, 33rd 'shape' file for CPO3D

A toroidal surface , defined by users equations, replaces shap3d16.

This toroidal surface is the same as that generated in shap3d16, but here the option for 'users equations' is used.

See users equations for information on the option.

This toroidal surface is generated by drawing an arc of a circle of radius 0.5 and then rotating it around an axis in the plane of the arc with the centre of the arc at a distance of 1 from the axis.

The equations are

b*cos(theta)

(a+b*sin(theta))*cos(phi)

(a+b*sin(theta))*sin(phi)

where theta = 0.785 to 2.356 (that is pi/4 to 3*pi/4), phi = -0.785 to 0.785, a = 0.5, b = 0.5.

(The program recognises that 0.785 represents pi/4 and so changes this value to the exact value, as with 1.57.)

Two planes of reflection symmetry have been used, xz (y = 0), yz (x = 0).

The parts of the equations for x and y that read

x=b*cos(theta), y=(a+b*sin(theta))

define an arc of a circle drawn in the xy plane, with radius 0.5 and centre at (x,y,z) = (0,0.5,0).

Rotation by phi around the x axis transforms a point (x,y,z) to

(x,y*cos(phi) - z*sin(phi),z*cos(phi) + y*sin(phi)).

In the present case the initial z is zero, so we therefore obtain the equations for y and z shown in the data file.

We have used values of a and b different from those used in shap3d16, to make the toroidal shape more obvious.

The equations, as they appear in the data file, are:

theta 0.785 1.57 name of variable number 1 and its limits

phi -0.785 0.785 name of variable number 2 and its limits

a 0.5 name of parameter number 3 and its fixed value

b 0.5 name of parameter number 4 and its fixed value

0 0 name of parameter number 5 and its fixed value

0 0 name of parameter number 6 and its fixed value

b*cos(theta)

(a+b*sin(theta))*cos(phi)

(a+b*sin(theta))*sin(phi)

1 1 numbers of 2 applied voltages (can be same)

10 10 numbers of subdivisions of variables 1 and 2

(The program has corrected the less accurate values that were entered for pi/4 etc.)