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Torus

The Torus, or donut, is an example of rational surface. The torus' surface is formed by sweeping a circle 360 degrees around the z-axis.

3 key values define the structure of the Torus:

R = Radius of the torus
r = Radius of the circles forming the torus' body
n = the number of points on the torus, where the circles will be located, and the number of points on the circles themselves.

The Torus surface is described by the parametric representation:

X = (r + R cos v) cos u
Y = (r + R cos v) sin u
Z = R sin v

Where u (angle of circles) and v (angle of torus body) = [0, 2PI), the angles u and v are set according to the angle the point being calculated lies on. As we have seen the torus surface is formed of n equidistant circles thus the angle between each circle is 2PI/n. Equally the angle between each point on the circles is 2PI/n.

Thus to calculate the points on the surface involves visiting each of the n points on the torus (v = [0,n) * 2PI/n) and at each point calculating the points of the circle to be placed there (u = [0,n) * 2PI/n).

Pluckers Conoid

The Pluckers Conoid, or Cylindroid, is a surface of revolution formed by rotating a line about the connecting axis.

The surface is known for the presence of 2 of more folds formed by the application of a cylindrical equation to the line during this rotation. This equation defines the path of the line along the axis of rotation.

The Pluckers Conoid is represented by the polar parameterisation:

X = Rcos(u)
Y = Rsin(u)
Z = 2cos(u)sin(u)

And the generalised parameterisation for Plucker's Conoid with f folds:

X = Rcos(u)
Y = Rsin(u)
Z = Rsin(fu)

The difference between these two forms is the function in the z-axis. The polar form is a specialised function that outputs only one type of curvature with two undulations while the generalised form is more flexible with the number of undulations of the outputted curvature being determined by the value of f.

Ruled Hyperboloid

The ruled hyperboloid is a surface of revolution formed by rotating a line about the perpendicular axis.

The surface is defined by three key values:

r = maximum radius of hyperboloid, at its center
h = height of hyperboloid
n = number of lines on surface.

The ruled hyperboloid is represented by the parametric equation:

X = r(cos(u) + v*sin(u))
Y = r(sin(u) - v*cos(u))
Z = hv

Where u = [0,2PI) and represents the angle of rotation and is set according to the line being calculated. There are n lines on the surface and therefore the angle between them is 2PI/n. V is the vector that defines the number of levels on the hyperboloid surface and the number of points on each line. The vector value ranges from [-v, v].

Ellipsoid

The ellipsoid is a quadratic surface that is composed of a number of elliptic curves swept along a given axis, usually z. The ellipsoid surface possesses two umbilics present at the start and end points of the surface.

The surface is defined by four key values:

rX = radius of ellipse in the x-axis
rY = radius of ellipse in the y-axis
rZ = radius of ellipse in the z-axis
n = n/2 is the number of ellipses located at equidistant points on the Ellipsoid, n is the number of points on the ellipses themselves.

The ellipsoid is represented parametrically as:

X = rX * cos(v)sin(u)
Y = rY * sin(v)sin(u)
Z = rZ * cos(u)

Where v = [0,2PI) and represents the angle of the points on elliptic curves in the xy plane, u = [0,PI] and represents the angle of the points on the elliptic curve in the xz the ellipses are located. The angles u and v are set according to the angle the point being calculated lies on. As we have seen the ellipsoid surface is formed of n/2 equidistant ellipses thus the angle between each ellipsoid is PI/n. Equally the angle between each point on the ellipses is 2PI/n.