The lathe is an object generated from rotating a twodimensional curve about an axis. This curve is
defined by a set of points which are connected by linear, quadratic, cubic or bezier spline curves. The syntax is:
LATHE:
lathe
{
[SPLINE_TYPE] Number_Of_Points, <Point_1>
<Point_2>... <Point_n>
[LATHE_MODIFIER...]
}
SPLINE_TYPE:
linear_spline  quadratic_spline  cubic_spline  bezier_spline
LATHE_MODIFIER:
sturm  OBJECT_MODIFIER
Lathe default values:
SPLINE_TYPE : linear_spline
sturm : off
The first item is a keyword specifying the type of spline. The default if none is specified is linear_spline .
The required integer value Number_Of_Points specifies how many twodimensional points are used
to define the curve. The points follow and are specified by 2D vectors. The curve is not automatically closed, i.e.
the first and last points are not automatically connected. You will have to do this yourself if you want a closed
curve. The curve thus defined is rotated about the yaxis to form the lathe object, centered at the origin.
The following examples creates a simple lathe object that looks like a thick cylinder, i.e. a cylinder with a thick
wall:
lathe {
linear_spline
5,
<2, 0>, <3, 0>, <3, 5>, <2, 5>, <2, 0>
pigment {Red}
}
The cylinder has an inner radius of 2 and an outer radius of 3, giving a wall width of 1. It's height is 5 and it's
located at the origin pointing up, i.e. the rotation axis is the yaxis.
Note: the first and last point are equal to get a closed curve.
The splines that are used by the lathe and prism objects are a little bit difficult to understand. The basic
concept of splines is to draw a curve through a given set of points in a determined way. The default
linear_spline is the simplest spline because it's nothing more than connecting consecutive points with a line.
This means the curve that is drawn between two points only depends on those two points. No additional information is
taken into account. The other splines are different in that they do take other points into account when connecting two
points. This creates a smooth curve and, in the case of the cubic spline, produces smoother transitions at each point.
The quadratic_spline keyword creates
splines that are made of quadratic curves. Each of them connects two consecutive points. Since those two points (call
them second and third point) are not sufficient to describe a quadratic curve, the predecessor of the second point is
taken into account when the curve is drawn. Mathematically, the relationship (their relative locations on the 2D
plane) between the first and second point determines the slope of the curve at the second point. The slope of the
curve at the third point is out of control. Thus quadratic splines look much smoother than linear splines but the
transitions at each point are generally not smooth because the slopes on both sides of the point are different.
The cubic_spline keyword creates splines which
overcome the transition problem of quadratic splines because they also take a fourth point into account when drawing
the curve between the second and third point. The slope at the fourth point is under control now and allows a smooth
transition at each point. Thus cubic splines produce the most flexible and smooth curves.
The bezier_spline is an alternate kind of
cubic spline. Points 1 and 4 specify the end points of a segment and points 2 and 3 are control points which specify
the slope at the endpoints. Points 2 and 3 do not actually lie on the spline. They adjust the slope of the spline. If
you draw an imaginary line between point 1 and 2, it represents the slope at point 1. It is a line tangent to the
curve at point 1. The greater the distance between 1 and 2, the flatter the curve. With a short tangent the spline can
bend more. The same holds true for control point 3 and endpoint 4. If you want the spline to be smooth between
segments, points 3 and 4 on one segment and points 1 and 2 on the next segment must form a straight line and point 4
of one segment must be the same as point 1 on the next segment.
You should note that the number of spline segments, i. e. curves between two points, depends on the spline type
used. For linear splines you get n1 segments connecting the points P[i], i=1,...,n. A quadratic spline gives you n2
segments because the last point is only used for determining the slope, as explained above (thus you will need at
least three points to define a quadratic spline). The same holds for cubic splines where you get n3 segments with the
first and last point used only for slope calculations (thus needing at least four points). The bezier spline requires
4 points per segment, creating n/4 segments.
If you want to get a closed quadratic and cubic spline with smooth transitions at the end points you have to make
sure that in the cubic case P[n1] = P[2] (to get a closed curve), P[n] = P[3] and P[n2] = P[1] (to smooth the
transition). In the quadratic case P[n1] = P[1] (to close the curve) and P[n] = P[2].
The sturm keyword can be used to specify that the slower, but more
accurate, Sturmian root solver should be used. Use it, if the shape does not render properly. Since a quadratic
polynomial has to be solved for the linear spline lathe, the Sturmian root solver is not needed.
