User:Le Forgeron/splines - Revision history

Le Forgeron at 10:31, 26 June 2018

2018-06-26T10:31:03Z

Le Forgeron: /* basic x spline */

2018-06-25T19:41:37Z

basic x spline

Le Forgeron: /* general x spline */

2018-06-24T12:00:29Z

general x spline

Le Forgeron: /* extended x spline */

2018-06-24T11:59:33Z

extended x spline

Le Forgeron: /* basic x spline */

2018-06-24T11:59:09Z

basic x spline

Le Forgeron: /* Splines */

2018-06-24T11:55:49Z

Splines

Le Forgeron: /* extended x spline */

2018-06-24T11:54:16Z

extended x spline

Le Forgeron at 11:50, 24 June 2018

2018-06-24T11:50:02Z

Le Forgeron: Created page with "{{User:Le_Forgeron/HgPovray/soon}} ==Splines== Image:LeForgeronSpline03.png Image:LeForgeronSpline09.png Image:LeForgeronSpline10.png [[Image:LeForgeronSpline11.p..."

2018-06-24T11:49:08Z

Created page with "{{User:Le_Forgeron/HgPovray/soon}} ==Splines== Image:LeForgeronSpline03.png Image:LeForgeronSpline09.png Image:LeForgeronSpline10.png [[Image:LeForgeronSpline11.p..."

New page

{{User:Le_Forgeron/HgPovray/soon}}
==Splines==

[[Image:LeForgeronSpline03.png]]
[[Image:LeForgeronSpline09.png]]
[[Image:LeForgeronSpline10.png]]
[[Image:LeForgeronSpline11.png]]

===Accessing splines data===
Not only the value of a spline can be evaluated with the traditional <code>SPLINE_IDENTIFIER ( FLOAT [, SPLINE_TYPE] )</code>, it is also possible to get back the actual pieces of information from a spline.
<code>dimension_size( SPLINE_IDENTIFIER )</code> provides the number of entries in the spline.
For each entry, the spline can be accessed like an array, the first element is the float of the path list and the second element the associated vector.
* <code>SPLINE_IDENTIFIER[ INDEX ][0]</code> is a float
* <code>SPLINE_IDENTIFIER[ INDEX ][1]</code> is a vector
* INDEX should evolve as an integer from 0 to <code>dimension_size(SPLINE_IDENTIFIER)-1</code>

===Additional types of splines===
====Sor spline====
[[Image:LeForgeronSpline12.png]]

the curve followed by a <code>sor</code> can be evaluted by a <code>spline</code> of <code>sor_spline</code> type with a bit of manipulation: the y value of the sor must be used as the float of the path, and the x value become one of the component of the vector. The component of the vector never get negative with a sor spline.
<source lang="pov">
spline{
sor_spline
-1.000000,0.000000*x
0.000000,0.118143*x
0.540084,0.620253*x
0.827004,0.210970*x
0.962025,0.194093*x
1.000000,0.286920*x
1.033755,0.468354*x
}

sor{
7
<0.000000, -1.000000>
<0.118143, 0.000000>
<0.620253, 0.540084>
<0.210970, 0.827004>
<0.194093, 0.962025>
<0.286920, 1.000000>
<0.468354, 1.033755>
}
</source>

====akima spline====
[[Image:LeForgeronSpline00.png]]
That spline will go through all its points, smoothly.
<pre>
spline {
akima_spline
time_Val_1, <Vector_1> [,]
time_Val_2, <Vector_2> [,]
...
time_Val_n, <Vector_n>
}
</pre>
====tcb spline====
[[Image:LeForgeronSpline13.png]]
[[Image:LeForgeronSpline14.png]]
[[Image:LeForgeronSpline15.png]]
[[Image:LeForgeronSpline16.png]]
[[Image:LeForgeronSpline17.png]]

Also know as Kochanek-Bartels spline, tcb stand for tension, continuity and bias.
The first and last point of such spline are not reached.
<pre>
spline {
tcb_spline [TCB_PARAMETERS]
time_Val_1 [TCB_PARAMETERS], <Vector_1> [TCB_PARAMETERS][,]
time_Val_2 [TCB_PARAMETERS], <Vector_2> [TCB_PARAMETERS][,]
...
time_Val_n [TCB_PARAMETERS], <Vector_n> [TCB_PARAMETERS]
}

TCB_PARAMETERS:
[tension FLOAT] [continuity FLOAT] [bias FLOAT]
</pre>
The <code>tension</code>, <code>continuity</code> and <code>bias</code> are fully optional. Depending on the place where they appear, they control the spline in different ways:
* Placed right after the tcb_spline keyword, they set the default values for all ends of the spline segments. This placement is ignored in case of copying spline without adding new controls because previous defaults were already propagated to each side of control points.
* Placed between the time_value and the corresponding vector, the tcb parameters determine the properties of the spline segment ending in the vector that follows these parameters (as well as for the spline segment beginning at this vector if not overriden with the third position).
* For tcb parameters following a vector, the properties of the spline segment beginning after this vector are set.
What is controlled by these parameters?
* <code>tension</code> controls how sharply the curve bends.
* <code>continuity</code> controls how rapid speed and direction change.
* <code>bias</code> controls the direction of the curve as it passes through the control point.

====x splines====
X-Splines are an alternative to traditional splines that was introduced by Carole Blanc and Christophe Schlick in 1995.

XSplines have the very nice property that they can interpolate (go through) a control point as well as just approximate it. Sharp edges are also possible, but the curve is always C2 continuous, hence the sharp edges are only possible when the first derivative drops to zero.

An X-Spline is completely defined by a set of control point (vertices) and a set of parameters. One parameter is associated with each control point.

There is 3 variations of x splines:
* basic
* extended
* general
=====basic x spline=====
The first and last point of such spline are not reached.
[[Image:LeForgeronSpline01.png]]
[[Image:LeForgeronSpline02.png]]
<pre>
spline {
basic_x_spline [freedom_degree FLOAT]
time_Val_1, <Vector_1> [freedom_degree FLOAT] [,]
time_Val_2, <Vector_2> [freedom_degree FLOAT] [,]
...
time_Val_n, <Vector_n> [freedom_degree FLOAT]
}
</pre>

The freedom_degree before the first point is the default value (0.0 by default)

======extended x spline======
[[Image:LeForgeronSpline04.png]]
[[Image:LeForgeronSpline05.png]]
Extended spline are always C², but allows sharp points (where both the first and second derivatives drop to zero). All the points are visited.

<pre>
spline {
extended_x_spline [freedom_degree FLOAT]
time_Val_1, <Vector_1> [freedom_degree FLOAT] [,]
time_Val_2, <Vector_2> [freedom_degree FLOAT] [,]
...
time_Val_n, <Vector_n> [freedom_degree FLOAT]
}
</pre>

The freedom_degree before the first point is the default value (0.0 by default)

=====general x spline=====
[[Image:LeForgeronSpline06.png]]
[[Image:LeForgeronSpline07.png]]
[[Image:LeForgeronSpline08.png]]

All points are visited.

<pre>
spline {
general_x_spline [freedom_degree FLOAT]
time_Val_1, <Vector_1> [freedom_degree FLOAT] [,]
time_Val_2, <Vector_2> [freedom_degree FLOAT] [,]
...
time_Val_n, <Vector_n> [freedom_degree FLOAT]
}
</pre>

The freedom_degree before the first point is the default value (0.0 by default)

← Older revision		Revision as of 10:31, 26 June 2018
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−	{{User:Le_Forgeron/HgPovray38/~~soon~~}}	+	{{User:Le_Forgeron/HgPovray38/available}}
	==Splines==		==Splines==

← Older revision		Revision as of 11:50, 24 June 2018
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−	{{User:Le_Forgeron/~~HgPovray~~/soon}}	+	{{User:Le_Forgeron/HgPovray38/soon}}
	==Splines==		==Splines==

@@ Line 113: / Line 113: @@
 spline {
    basic_x_spline [freedom_degree FLOAT]
-   time_Val_1, <Vector_1> [freedom_degree FLOAT] [,]
+   time_Val_1, <Vector_1> [,]
-   time_Val_2, <Vector_2> [freedom_degree FLOAT] [,]
+   time_Val_2, <Vector_2> [,]
    ...
-   time_Val_n, <Vector_n> [freedom_degree FLOAT]
+   time_Val_n, <Vector_n>
+}
 </pre>
-<p>The <span class="code">freedom_degree</span> before the first point is the default value (0.0 by default)</p>
+<p>The <span class="code">freedom_degree</span> before the first point is the value for all points (0.0 by default)</p>
 =====extended x spline=====

@@ Line 142: / Line 142: @@
 =====general x spline=====
 [[Image:LeForgeronSpline06.png]]
 [[Image:LeForgeronSpline07.png]]
 [[Image:LeForgeronSpline08.png]]

@@ Line 124: / Line 124: @@
 =====extended x spline=====
 [[Image:LeForgeronSpline04.png]]
 [[Image:LeForgeronSpline05.png]]
 <p>Extended spline are always C², but allows sharp points (where both the first and second derivatives drop to zero). All the points are visited.</p>

@@ Line 108: / Line 108: @@
 <p>The first and last point of such spline are not reached.</p>
 [[Image:LeForgeronSpline01.png]]
 [[Image:LeForgeronSpline02.png]]
 <pre>

@@ Line 3: / Line 3: @@
 [[Image:LeForgeronSpline03.png]]
 [[Image:LeForgeronSpline09.png]]
 [[Image:LeForgeronSpline10.png]]
 [[Image:LeForgeronSpline11.png]]
@@ Line 57: / Line 60: @@
 </pre>
 ====tcb spline====
 [[Image:LeForgeronSpline13.png]]
 [[Image:LeForgeronSpline14.png]]
 [[Image:LeForgeronSpline15.png]]
 [[Image:LeForgeronSpline16.png]]
 [[Image:LeForgeronSpline17.png]]