Enter the parametric equation of the surface here. Note that the surface must be defined and
continuous for all u, v in or near the range (minimum distance less than 0.0001*(amax-amin)/acnt,
a is u or v), and must be smooth for all u and v in the range. (Singular
points will be plotted inaccurately, sometimes the calculation even fails)

x(u,v)=
y(u,v)=
z(u,v)=

Enter the range for u and v:

umin= , umax=
vmin= , vmax=

In order to draw the image, we must subdivide [umin, umax] and [vmin, vmax] into ucnt
and vcnt equal subdivisions, and calculate the coordinates and normals (for smoothening)
of the resulting ucnt * vcnt lattice points. Of course, the larger ucnt
and vcnt is, the more accurate the resulting surface is. Generally 12x12 is already quite
satisfactory, and can usually be calculated in several seconds. However, if you need high
quality, 24x24 should be OK, but you will need to wait for 10 or more seconds. Note that
transforming the equations to a good form is vital to the quality of the image. Enter
ucnt and vcnt here:

ucnt= , vcnt=

Now enter the ranges for x, y, and z. Portions of the surface outside of this range is clipped.
This is useful for examing the interior of the image. However, if you want to see the full image,
specify a wide range that will fit in all the points on the surface. Note that this does not
affect your camera view, thus a moderately large range and a extremely large range acts exactly
the same for finite surfaces.

xmin= , xmax=
ymin= , ymax=
zmin= , zmax=

Now enter the location of the camera and the light. For the beginners, you can specify a point
having a moderate distance from the surface as the location of the camera, and putting the light
near (or in the same place as) the camera generally is OK. Only beware not to put the light or
the camera inside the surface if you are not intended to. Note that the camera is always pointing
to (0,0,0). For best results, it is often needed to tune these paraeters several times.

Camera: x= , y= ,
z=
Light: x= , y= ,
z=

The bottom note: The surfaces are plotted in a left-handed
coordinate system, so they are really the mirror images of the usual surfaces in a right-handed
coordinate system. However, for many symmetric surfaces, this makes no difference. If you
do want to see the surfaces in a right-handed coordinate system, transform them by yourself.
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Description: generates raytraced smooth 3D surfaces from parametric equations. interactive exercises, online calculators and plotters, mathematical recreation and games

Keywords: interactive mathematics, interactive math, server side interactivity, analysis, geometry, parametric, surfaces, plot