POV-Wiki - User contributions [en]

HowTo:Use Splines and Bezier Curves

2007-12-09T10:04:26Z

Bathurstfreak: /* About Bicubic Patches */

== Introduction ==

All Curves and Patches utilised by POVRay are influenced by a number of control points. All curves pass through the start and the end control point, while most of the curves pass through the intermediate control points.

== Splines ==

Splines may be of Linear, Quadratic or Cubic type of Spline and are accessed with the command #spline as shown below.

<source lang="pov">
#declare MYSPLINE = spline {
cubic_spline
-1, <-1,0,0>
0, <2,-1,0>
1, <3,-1,2>
2, <5,0,5>
3, <3,3,3>
4, <2,2,2>
5, <2,-1,0>
6, <3,-1,2>
7, <5,0,5>
}

#declare count = 0;
#while (count <= 5)
#sphere { MYSPLINE (count), 0.25 pigment {Blue} }
#declare count = count + 0.01;
#end

</source>

All "spline" type splines pass through the intermediate control points, these are:
#A linear spline will move linearly from one control point to the next.
#A Quadratic Spline will pass from one point to the next with reference only to where it is going.
#A Cubic Spline will pass from one control point to the next with reference to where it cam from and where it will go to next.

The important thing to remember about using splines is that as the control number exceeds a control point, the next increment of control points is taken to control the path of the spline. Sufficient control points need to be defined otherwise POVRay will yield an error or misrepresent the scene.

== Bezier Curves ==

Bezier Curves differ from other types of spline in that they only pass through the first and last control point and are simply influenced by the other intermediate control points.
The equation for a bezier curve as used by POVRay is cubic and is of the form.

<math>B\left( t\right) = A \left( 1 - t\right)^3 + 3 B t \left( 1 - t\right)^2 + 3 C t^2\left( 1-t\right) + D t^3</math>

where t is from 0 to 1 and A, B, C and D represent a variable that may be a scalar or vector.

Consecutive bezier curves may be placed one after the other however the direction of the tangents at the start and end control point of joining curves must be identical.

== About Bicubic Patches ==

If a Bezier curve is a 1 dimensional line on three dimensional space, then a bicubic patch is a 2 dimensional sheet on a three dimensional plane.

The bezier patch has only four fixed points that is controlled by 12 reference points.

<source lang="pov">
#bicubic_patch {
type 1 flatness 1 u_steps 3 v_steps 3
<0, 0, 2> <1, 0, 0> <2, 0, 0> <3, 0, -2>
<0, 1, 0> <1, 1, 0> <2, 1, 0> <3, 1, 0>
<0, 2, 0> <1, 2, 0> <2, 2, 0> <3, 2, 0>
<0, 3, 2> <1, 3, 0> <2, 3, 0> <3, 3, -2>
}
</source>

HowTo:Use Splines and Bezier Curves

2007-12-09T10:03:16Z

Bathurstfreak: /* About Bicubic Patches */

== Introduction ==

All Curves and Patches utilised by POVRay are influenced by a number of control points. All curves pass through the start and the end control point, while most of the curves pass through the intermediate control points.

== Splines ==

Splines may be of Linear, Quadratic or Cubic type of Spline and are accessed with the command #spline as shown below.

<source lang="pov">
#declare MYSPLINE = spline {
cubic_spline
-1, <-1,0,0>
0, <2,-1,0>
1, <3,-1,2>
2, <5,0,5>
3, <3,3,3>
4, <2,2,2>
5, <2,-1,0>
6, <3,-1,2>
7, <5,0,5>
}

#declare count = 0;
#while (count <= 5)
#sphere { MYSPLINE (count), 0.25 pigment {Blue} }
#declare count = count + 0.01;
#end

</source>

All "spline" type splines pass through the intermediate control points, these are:
#A linear spline will move linearly from one control point to the next.
#A Quadratic Spline will pass from one point to the next with reference only to where it is going.
#A Cubic Spline will pass from one control point to the next with reference to where it cam from and where it will go to next.

The important thing to remember about using splines is that as the control number exceeds a control point, the next increment of control points is taken to control the path of the spline. Sufficient control points need to be defined otherwise POVRay will yield an error or misrepresent the scene.

== Bezier Curves ==

Bezier Curves differ from other types of spline in that they only pass through the first and last control point and are simply influenced by the other intermediate control points.
The equation for a bezier curve as used by POVRay is cubic and is of the form.

<math>B\left( t\right) = A \left( 1 - t\right)^3 + 3 B t \left( 1 - t\right)^2 + 3 C t^2\left( 1-t\right) + D t^3</math>

where t is from 0 to 1 and A, B, C and D represent a variable that may be a scalar or vector.

Consecutive bezier curves may be placed one after the other however the direction of the tangents at the start and end control point of joining curves must be identical.

== About Bicubic Patches ==

If a Bezier curve is a 1 dimensional line on three dimensional space, then a bicubic patch is a 2 dimensional sheet on a three dimensional plane.

The bezier patch has only four fixed points that is controlled by 12 reference points.

#bicubic_patch {
type 1 flatness 1 u_steps 3 v_steps 3
<0, 0, 2> <1, 0, 0> <2, 0, 0> <3, 0, -2>
<0, 1, 0> <1, 1, 0> <2, 1, 0> <3, 1, 0>
<0, 2, 0> <1, 2, 0> <2, 2, 0> <3, 2, 0>
<0, 3, 2> <1, 3, 0> <2, 3, 0> <3, 3, -2>
}

HowTo:Use Splines and Bezier Curves

2007-12-09T09:20:35Z

Bathurstfreak: /* Bezier Curves */

HowTo:Use Splines and Bezier Curves

2007-12-09T08:55:39Z

Bathurstfreak: /* Splines */

HowTo:Use Splines and Bezier Curves

2007-12-09T08:52:11Z

Bathurstfreak: /* Splines */

HowTo:Use Splines and Bezier Curves

2007-12-09T07:42:18Z

Bathurstfreak: /* Splines */

HowTo:Use Splines and Bezier Curves

2007-12-09T07:39:24Z

Bathurstfreak: /* Introduction */

HowTo:Contents

2007-12-08T22:08:17Z

Bathurstfreak: /* Work-in-Progress Articles */

The following is a list of the currently-available 'How To' articles. If you add a new article, please list it here.

Please also read our '''[[Help:Editing_Guidelines|Editing Guidelines]]''' prior to creating any articles.

==Suggestions==

It would be handy if some users co-operated in working out a 'wishlist' of articles and a hierarchy for them, and then created the links to the empty pages from this page.

Discussions should be held on the [[HowTo_Talk:Contents|talk]] page, not here.

==Completed Articles==

''(None so far)''

==Work-in-Progress Articles==

*[[HowTo:Create animations]]
*[[HowTo:Create anaglyph images]]
*[[HowTo:Create multi-phase web buttons]]
*[[HowTo:Plan your scenes]]
*[[HowTo:Use the blob object]]
*[[HowTo:Use the lathe object]]
*[[HowTo:Use the plane object]]
*[[HowTo:Use constructive solid geometry]]
*[[HowTo:Use macros and loops]]
*[[HowTo:Use photons]]
*[[HowTo:Use radiosity]]
*[[HowTo:Use Splines and Bezier Curves]]
*[[HowTo:Use UV-mapping]]

HowTo:Use Splines and Bezier Curves

2007-12-08T22:06:53Z

Bathurstfreak: New page: == Introduction == == Splines == == Bezier Curves == == About Bicubic Patches ==

== Introduction ==

== Splines ==

== Bezier Curves ==

== About Bicubic Patches ==

HowTo:Use macros and loops

2007-12-08T22:00:54Z

Bathurstfreak:

Loops and Macros are usually created to do one set of code thousands of times. The beauty of the loop is that the code can reference an array or a spline to write a rollercoaster track or a roadway such that very complicated scenes can be created with the use of simple declared variables.

== Loops ==

Loops may be constructed through the use of the #while statement. A typical loop may be constructed as follows.

<source lang="pov">
#declare step = 0;
#declare steps = 100;
#while (step <= steps)
#sphere { step*x, 0.4 pigment {Blue} }
#declare step = step + 1;
#end
</source>

Nested loops may be created to generate a two or three dimensional aspect to the scene. Beware that creating extra nested loops will dramatically increase the parse time for the scene.

The following nested loop will draw a series of Blue balls on the XZ plain

<source lang="pov">
#declare xpos = 0;
#declare zpos = 0;
#declare xfinal = 10;
#declare zfinal = 10;
#while (xpos <= xfinal)
#while (zpos <= zfinal)
#sphere { <xpos,0,zpos> 0.4 pigment {Blue} }
#declare zpos = zpos + 1;
#end
#declare xpos = xpos + 1;
#declare zpos = 0;
#end
</source>

== Macros ==

HowTo:Use macros and loops

2007-12-07T23:46:56Z

Bathurstfreak: New page: == Loops == Loops are usually created to do one set of code thousands of times. The beauty of the loop is that the code can reference an array or a spline to write a rollercoaster track or...

== Loops ==
Loops are usually created to do one set of code thousands of times. The beauty of the loop is that the code can reference an array or a spline to write a rollercoaster track or a roadway such that very complicated scenes can be created with the use of simple structures.

Loops may be constructed through the use of the #while statement. A typical loop may be constructed as follows.

#declare step = 0;
#declare steps = 100;
#while (step <= steps)
#sphere { step*x, 0.4 pigment {Blue} }
#declare step = step + 1;
#end

Nested loops may be created to generate a two or three dimensional aspect to the scene. Beware that creating extra nested loops will dramatically increase the parse time for the scene.

The following nested loop will draw a series of Blue balls on the XZ plain

#declare xpos = 0;
#declare zpos = 0;
#declare xfinal = 10;
#declare zfinal = 10;
#while (xpos <= xfinal)
#while (zpos <= zfinal)
#sphere { <xpos,0,zpos> 0.4 pigment {Blue} }
#declare zpos = zpos + 1;
#end
#declare xpos = xpos + 1;
#declare zpos = 0;
#end