Reference:Lathe - Revision history

Jr: add link to (splines) tutorial section. (thanks to Bald Eagle)

2025-08-29T13:34:48Z

add link to (splines) tutorial section. (thanks to Bald Eagle)

Jholsenback: caveat

2017-10-01T14:31:02Z

caveat

Jholsenback: corrected link issue

2016-07-17T13:59:50Z

corrected link issue

Jholsenback: user contributed update

2016-06-06T10:01:29Z

user contributed update

Jholsenback: consistency

2016-05-31T17:02:20Z

consistency

Jholsenback: clarification

2016-05-28T13:59:48Z

clarification

Jholsenback: consistency edits

2016-05-26T23:54:28Z

consistency edits

Jholsenback: clean up and user contributed additions

2016-05-25T22:38:01Z

clean up and user contributed additions

Jholsenback: 1 revision: Reference Migration Initial Load

2012-03-15T19:07:20Z

1 revision: Reference Migration Initial Load

Jholsenback: 1 revision: Initial Load (TF)

2012-03-11T22:26:59Z

1 revision: Initial Load (TF)

New page

[[Category:Objects]]
[[Category:Finite Solid Primitives]]
{{#indexentry:lathe, keyword}}
{{#indexentry:keyword, lathe}}
The <code>lathe</code> is an object generated from rotating a
two-dimensional 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:
<pre>
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
</pre>

{{#indexentry:default values, lathe}}
Lathe default values:
<pre>
SPLINE_TYPE : linear_spline
sturm : off
</pre>

{{#indexentry:linear_spline, lathe}}
{{#indexentry:keyword, linear_spline}}
The first item is a keyword specifying the type of spline. The default if
none is specified is <code>linear_spline</code>. The required integer value
<code>Number_Of_Points</code> specifies how many two-dimensional
points are used to define the curve. The points follow and are specified by
2-D 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
y-axis 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:
<pre>
lathe {
linear_spline
5,
<2, 0>, <3, 0>, <3, 5>, <2, 5>, <2, 0>
pigment {Red}
}
</pre>

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 y-axis.
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 <code>
linear_spline</code> 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.

{{#indexentry:quadratic_spline, lathe}}
{{#indexentry:keyword, quadratic_spline}}
The <code>quadratic_spline</code> 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 2-D 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.

{{#indexentry:cubic_spline, lathe}}
{{#indexentry:keyword, cubic_spline}}
The <code>cubic_spline</code> 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.

{{#indexentry:bezier_spline, lathe}}
{{#indexentry:keyword, bezier_spline}}
The <code>bezier_spline</code> 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 n-1
segments connecting the points P[i], i=1,...,n. A quadratic spline gives you
n-2 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 n-3
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[n-1] = P[2] (to get a closed curve), P[n] = P[3] and P[n-2] = P[1] (to
smooth the transition). In the quadratic case P[n-1] = P[1] (to close the
curve) and P[n] = P[2].

{{#indexentry:sturm, lathe}}
{{#indexentry:keyword, sturm}}
The <code>sturm</code> 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.

@@ Line 41: / Line 41: @@
 <p class="Note"><strong>Note:</strong> The first and last point are equal to get a closed curve.</p>
 <p>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 <code>linear_spline</code> 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.</p>
 {{#indexentry:quadratic_spline, lathe}}

@@ Line 59: / Line 59: @@
 quadratic spline). The same holds for cubic splines where you get n-3 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.</p>
 <p>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[n-1] = P[2] (to get a closed curve), P[n] = P[3] and P[n-2] = P[1] (to smooth the transition). In the quadratic case P[n-1] = P[1] (to close the curve) and P[n] = P[2].</p>
-<p>The surface normal determination for lathes depends upon the order in which the splines points are specified.  The following code will render with the color Red on the outside and the color Blue on the inside.</p>
+<p>The surface normal determination for lathes depends upon the order in which the splines points are specified. For <code>interior_texture</code> to work as expected, the outline of the underlying 2D shape must be specified in counter-clockwise order, except for holes which must be specified in clockwise order, and they must not self-intersect.</p>
 <pre>
 #declare Lathe_InitialOrder = lathe {

@@ Line 92: / Line 92: @@
 {{#indexentry:sturm, lathe}}
 {{#indexentry:keyword, sturm}}
-<p>If additional accuracy is required you can add the <code>[[Reference:Sturm_Object_Modifier|sturm]]</code> object modifier.</p>
+<p>If additional accuracy is required you can add the <code>[[Reference:Sturm Object Modifier|sturm]]</code> object modifier.</p>

@@ Line 2: / Line 2: @@
 [[Category:Finite Solid Primitives]]
 {{#indexentry:lathe, keyword}}
 {{#indexentry:keyword, lathe}}
 <p>The <code>lathe</code> is an object generated from rotating a two-dimensional 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:</p>
 <pre>
@@ Line 59: / Line 59: @@
 quadratic spline). The same holds for cubic splines where you get n-3 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.</p>
 <p>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[n-1] = P[2] (to get a closed curve), P[n] = P[3] and P[n-2] = P[1] (to smooth the transition). In the quadratic case P[n-1] = P[1] (to close the curve) and P[n] = P[2].</p>
-<p class="Note"><strong>Note:</strong> The use of <em>clockwise</em> and <em>counter-clockwise</em> in the following examples should not be taken imply any notion of [[Reference:Camera#Handedness|handedness]].</p>
+<p>The surface normal determination for lathes depends upon the order in which the splines points are specified.  The following code will render with the color Red on the outside and the color Blue on the inside.</p>
 <pre>
-#declare Lathe_CounterClockwise = lathe {
+#declare Lathe_InitialOrder = lathe {
      bezier_spline
 ,
@@ Line 74: / Line 73: @@
+}
 </pre>
-<p>While the following example, where the point order for the bezier spine is <em>clockwise</em> in the x-y plane, will render inside-out, with the color Blue on the outside and the color Red on the inside.</p>
+<p>While the following example will render inside-out, with the color Blue on the outside and the color Red on the inside.</p>
 <pre>
-#declare Lathe_Clockwise = lathe {
+#declare Lathe_ReverseOrder = lathe {
      bezier_spline
 ,
@@ Line 88: / Line 87: @@
+}
 </pre>
-<p>The actual normal determination is a bit more complicated than just <em>clockwise</em> vs <em>counter-clockwise</em> orientation for complex splines. If the surface normal is important to the visual result, it is best to check how the lathe is being rendered by testing with substantially different inside and outside textures.</p>
+<p>The actual normal determination is more complicated for complex splines. If the surface normal is important to the visual result, it is best to check how the lathe is being rendered by testing with substantially different inside and outside textures.</p>
 <p class="Note"><strong>Note:</strong> With the bezier spline, unlike all the other spline types used with the lathe, it is possible to create independent closed curves within a single lathe definition.</p>

@@ Line 93: / Line 93: @@
 {{#indexentry:sturm, lathe}}
 {{#indexentry:keyword, sturm}}
-<p>The <code>sturm</code> 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.</p>
+<p>If additional accuracy is required you can add the <code>[[Reference:Sturm_Object_Modifier|sturm]]</code> object modifier.</p>

@@ Line 60: / Line 60: @@
 <p>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[n-1] = P[2] (to get a closed curve), P[n] = P[3] and P[n-2] = P[1] (to smooth the transition). In the quadratic case P[n-1] = P[1] (to close the curve) and P[n] = P[2].</p>
-<p>The surface normal determination for lathes depends upon the order in which the splines points are specified. The following code, where the bezier spline point order is <em>counter-clockwise</em> in the xy plane, will render with the color Red on the outside and the color Blue on the inside.</p>
+<p>The surface normal determination for lathes depends upon the order in which the splines points are specified. The following code, where the bezier spline point order is <em>counter-clockwise</em> in the x-y plane, will render with the color Red on the outside and the color Blue on the inside.</p>
 <pre>
 #declare Lathe_CounterClockwise = lathe {
@@ Line 74: / Line 74: @@
+}
 </pre>
-<p>While the following example, where the point order for the bezier spine is <em>clockwise</em> in the xy plane, will render inside-out, with the color Blue on the outside and the color Red on the inside.</p>
+<p>While the following example, where the point order for the bezier spine is <em>clockwise</em> in the x-y plane, will render inside-out, with the color Blue on the outside and the color Red on the inside.</p>
 <pre>
 #declare Lathe_Clockwise = lathe {

@@ Line 3: / Line 3: @@
 {{#indexentry:lathe, keyword}}
 {{#indexentry:keyword, lathe}}
-<p>The <code>lathe</code> is an object generated from rotating a
+<p>The <code>lathe</code> is an object generated from rotating a two-dimensional 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:</p>
 <pre>
 LATHE:
@@ Line 29: / Line 26: @@
 {{#indexentry:linear_spline, lathe}}
 {{#indexentry:keyword, linear_spline}}
-<p>The first item is a keyword specifying the type of spline. The default if
+<p>The first item is a keyword specifying the type of spline. The default if none is specified is <code>linear_spline</code>. The required integer value <em><code>Number_Of_Points</code></em> specifies how many two-dimensional points are used to define the curve. The points follow and are specified by 2-D vectors. The curve is not automatically closed, i.e. the first and last
-none is specified is <code>linear_spline</code>. The required integer value
+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 y-axis to form the lathe object, centered at the origin.</p>
-<em><code>Number_Of_Points</code></em> specifies how many two-dimensional
+<p>The following example creates a simple lathe object that looks like a thick cylinder, i.e. a cylinder with a thick wall:</p>
 <pre>
 lathe {
@@ Line 49: / Line 38: @@
 </pre>
-<p>The cylinder has an inner radius of 2 and an outer radius of 3, giving a
+<p>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 y-axis.</p>
 <p class="Note"><strong>Note:</strong> The first and last point are equal to get a closed curve.</p>
-<p>
+<p>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 <code>linear_spline</code> 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.</p>
 {{#indexentry:quadratic_spline, lathe}}
 {{#indexentry:keyword, quadratic_spline}}
-<p>The <code>quadratic_spline</code> keyword creates splines that are made of
+<p>The <code>quadratic_spline</code> 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 2-D 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
-quadratic curves. Each of them connects two consecutive points. Since those
+the transitions at each point are generally not smooth because the slopes on both sides of the point are different.</p>
 {{#indexentry:cubic_spline, lathe}}
 {{#indexentry:keyword, cubic_spline}}
-<p>The <code>cubic_spline</code> keyword creates splines which overcome the
+<p>The <code>cubic_spline</code> 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.</p>
 {{#indexentry:bezier_spline, lathe}}
 {{#indexentry:keyword, bezier_spline}}
-<p>The <code>bezier_spline</code> is an alternate kind of cubic spline. Points
+<p>The <code>bezier_spline</code> 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.
-and 4 specify the end points of a segment and points 2 and 3 are control
+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.</p>
-points which specify the slope at the endpoints. Points 2 and 3 do not
+<p>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 n-1 segments connecting the points P[i], i=1,...,n. A quadratic spline gives you n-2 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
-actually lie on the spline. They adjust the slope of the spline. If you draw
+quadratic spline). The same holds for cubic splines where you get n-3 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.</p>
-an imaginary line between point 1 and 2, it represents the slope at point 1.
+<p>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[n-1] = P[2] (to get a closed curve), P[n] = P[3] and P[n-2] = P[1] (to smooth the transition). In the quadratic case P[n-1] = P[1] (to close the curve) and P[n] = P[2].</p>
 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
+<p>The surface normal determination for lathes depends upon the order in which the splines points are specified. The following code, where the bezier spline point order is <em>counter-clockwise</em> in the xy plane, will render with the color Red on the outside and the color Blue on the inside.</p>
-bend more. The same holds true for control point 3 and endpoint 4. If you
+<pre>
-want the spline to be smooth between segments, points 3 and 4 on one segment
+#declare Lathe_CounterClockwise = lathe {
-and points 1 and 2 on the next segment must form a straight line and point 4
+    bezier_spline
-of one segment must be the same as point 1 on the next segment.</p>
+,
-<p>
+    <0.45,0>,<0.45,0.0828427>,<0.382843,0.15>,<0.3,0.15>
-You should note that the number of spline segments, i. e. curves between two
+    <0.3,0.15>,<0.217157,0.15>,<0.15,0.0828427>,<0.15,0>
-points, depends on the spline type used. For linear splines you get n-1
+    <0.15,0>,<0.15,-0.0828427>,<0.217157,-0.15>,<0.3,-0.15>
-segments connecting the points P[i], i=1,...,n. A quadratic spline gives you
+    <0.3,-0.15>,<0.382843,-0.15>,<0.45,-0.0828427>,<0.45,0>
-n-2 segments because the last point is only used for determining the slope, as
+    sturm
-explained above (thus you will need at least three points to define a
+    texture          { pigment { Red } }
-quadratic spline). The same holds for cubic splines where you get n-3
+    interior_texture { pigment { Blue } }
 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
+</pre>
-segment, creating n/4 segments.</p>
+<p>While the following example, where the point order for the bezier spine is <em>clockwise</em> in the xy plane, will render inside-out, with the color Blue on the outside and the color Red on the inside.</p>
-<p>
+<pre>
-If you want to get a closed quadratic and cubic spline with smooth
+#declare Lathe_Clockwise = lathe {
-transitions at the end points you have to make sure that in the cubic case
+    bezier_spline
-P[n-1] = P[2] (to get a closed curve), P[n] = P[3] and P[n-2] = P[1] (to
+,
-smooth the transition). In the quadratic case P[n-1] = P[1] (to close the
+    <0.45,0>,<0.45,-0.0828427>,<0.382843,-0.15>,<0.3,-0.15>
-curve) and P[n] = P[2].</p>
+    <0.3,-0.15>,<0.217157,-0.15>,<0.15,-0.0828427>,<0.15,0>
 {{#indexentry:sturm, lathe}}
 {{#indexentry:keyword, sturm}}
-<p>The <code>sturm</code> keyword can be used to specify that the slower, but
+<p>The <code>sturm</code> 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.</p>

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