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Examples of parameterised surfaces

For details of the basic operation of the program see the Main Help page. Some special options specific to the calculation of parameterised surfaces is below. Example definitions plus some explanation are also below.

X steps and Y steps

These parameters control number of steps used to define the grid representing the surface. Higher values will give finer meshes.


A very simple algorithm is used to ensure that the surface does not run off to infinity. If any value is greater than the clipping parameter, then the value is set to that parameter.


Monge Form Surface
(x,y,(A x^2 + B y^2)/2 + (a x^3 + 3 b x^2 y + 3 c x y^2 + d y^3)/6);
Any smooth surface, can be rotated and translated until it is this form (with the possible addition of higher order terms). This can be useful for examining the local behaviour of surfaces.
(x,x y,y^2);
Unlike the algebraic form this does not include a handle.
A sphere
r (l,m,n)+(x0,y0,z0);
(l,m,n);   # This equation specifies the normal allows us to get a correct normal at poles
l = cos(Pi th) COs(Pi phi);
m = COs(Pi th) sin(Pi phi);
n = sin(Pi th);
psurf_normals = 3; # switches on eqn for normal calculation
r = 1; th = x; phi = y;
Rather more complicated than it need to be. The first line gives the equation for the surface and the second gives a definition for the normals. This is needed as the standard way of calculating the normal does not work at the poles.
Kline Bottle
X = t COs(2v); Y = t sin(2v);
Z = sin(u)sin(v)+sin(2u) COs(v);
t = 4 + sin(u) COs(v) - sin(2 u) sin(v);
u = x; v = y;
This surface is not orientable, i.e. it can not be given a global direction for the normals. The equation for this form comes from Alfred Gray.
Swallowtail surface
(3 x^4 + y x^2,-4 x^3 - 2 y x,y);
See the Algebraic surfaces page for a discussion of this surface.
A torus
(r sin(th),r COs(th),r1 sin(phi));
r = r0 + r1 COs(phi);r0=2;r1=1; th = x; phi = y;
(v COs(u),v sin(u),b u);
b = 1;
u = Pi x; v = y;
Tangent developable of a curve
v + y * diff(v,x);
v = (x,x^2,x^3);
This surface is constructed by taking a curve v = (x,x^2,x^3); and calculating the tangent at each point. A surface is then constructed by taking points along the tangent. This example illustrates the use of the symbolic differential operator diff.
Gauss map or spherical image of a surface
n / sqrt( n . n ); # a unit length normal
n = dx ^ dy;       # find normal direction
dx = diff(f,x);    # the two tangents to the surface
dy = diff(f,y);
f = (x,y,(A x^2 + B y^2)/2 + (a x^3 + 3 b x^2 y + 3 c x y^2 + d y^3)/6);
A = 1; B = 0; a = 0.5; b = -0.5; c = 0.3; d = 2;
A surface can be constructed by taking the unit length normal at each point of a surface and translating the normal to the origin so the end of the normal give a point on a sphere. The Gauss map will have a fold whenever the original surface has a parabolic line.
Bohemian Dome
X = a*COs(u);
Y = b*COs(v) + a*sin(u);
Z = c*sin(v);
a = 0.5; b = 0.5; c = 0.5;
u = Pi x; v = Pi y;
This surface is generated by taking a set of circles in planes parallel to the x-z plane whose centres lie on a circle in the y-z plane.

Web page, applet and Algebraic Surface program by Richard Morris
home page Email Copyright Jan 2001.