2 advanced topics in modelling with JOGL2 tủ tài liệu bách khoa

Dr. Manuel Carcenac - European University of Lefke

Advanced topics
in modeling
with Java OpenGL - JOGL 2
Spatial orientation: Euler angles, gimbal lock, a practical solution
Textures
NURBS surfaces
Fractal landscape generated with diamond-square algorithm

References:
/>
(for JOGL 2.0)

/> /> />(for practical introduction to JOGL 2.0)

Advanced Animation and Rendering Techniques – Theory and Practice
Alan Watt, Mark Watt; Addison-Wesley, ACM press
/> /> />
(for NURBS)

(for beautiful free planet textures)
(for beautiful free planet textures)

1


Dr. Manuel Carcenac - European University of Lefke

Spatial orientation: Euler angles, gimbal lock,
a practical solution

Euler angles:
any spatial orientation of an object can be described by 3 angles = 3 degrees of freedom

how to get from the absolute coordinate system
to the local coordinate system of the object ?
 global rotation = succession of 3 rotations around coordinate system axes

several possibilities: in which order do we rotate around axes x , y , z ?
 one possible solution:

absolute
(x,y,z)

rot. of angle a
around axis z

rot. of angle b
around axis y'

(x',y',z)

(x'',y',z')

local
(x'',y'',z'')

rot. of angle c
around axis x''

z

z'
z''

c

b
y''
c
a
a

y'
y

x''

b
x'

x

2


Dr. Manuel Carcenac - European University of Lefke

gimbal lock:
decoupled degrees of freedom:
axes of rotation remain more or less orthogonal
 angles remain decoupled = they have independent actions

big problem: loss of a degree of freedom:
in certain configurations, 2 of the 3 axes of rotation become aligned!
 their 2 associated angles are equivalent = they produce the same rotation in space
 only 2 effective degrees of freedom left...
example: b = -  / 2  a and c both rotate around axis z
 once b is fixed, only possible to rotate around z axis with a and c
z
x''

z'

y''
c

z''

y'

c
a

y

a
x'

x

annoying problem: degrees of freedom partially coupled:

b tends toward  / 2



a and c become increasingly coupled

 varying a has an increasing effect on the rotation that should be controled only by c
= very disturbing for a human operator (especially to control camera orientation...)

3


Dr. Manuel Carcenac - European University of Lefke

a practical solution:
instead of Euler angles,
define directly the local coordinate system of the object
with 3 orthogonal unit vectors u , v , w
assume u , v , w given initially  how to adjust the space orientation of u , v , w ?

incremental approach:
start with an inital u , v , w
then rotate gradually u , v , w in space
 new u , v , w after each incremental rotation δ

u'

u
w'
w

v'
v
at each time step, apply small rotations around u , v or w

= applied relatively to the object
 intuitive for operator
 axes of rotation remain practically orthogonal over a time step
 remain far away from any gimbal lock configuration

4


Dr. Manuel Carcenac - European University of Lefke

3D rotation around a vector:
rotation of angle δ around unit vector n : v  v'
v'

n×v
v'
δ
v
v=

v

n

v= = (n . v) n

v = v - (n . v) n


v' = v= + cos δ v + sin δ n×v



v' = cos δ v + (1 - cos δ) (n . v) n + sin δ n×v

5


Dr. Manuel Carcenac - European University of Lefke

application to attitude control of virtual camera:
initial orientation of camera:
w aiming direction
uup up vector = up direction of screen

 u orthogonalized relative to w : u = uup - (u.w) w

uup
u = uup - (uup.w) w
w

v=w×u

real-time control of evolution of camera orientation:
we adjust camera orientation in real time
= we are in a spacecraft and we control its orientation (attitude control system,

attitude rockets)
 small rotations over roll , pitch , yaw at each time step
u
w
yaw
roll

pitch

v

6


Dr. Manuel Carcenac - European University of Lefke

controlling orientation with some keyboard action keys:

fast rotation:
rotation 10 δ

Insert

Page
Up

roll incline left:
rot(-δ,w) over u,v

Delete

Page
Down

roll incline right:
rot(δ,w) over u,v

pitch up:
rot(δ,v) over u,w


yaw left:
rot(δ,u) over v,w







yaw right:
rot(-δ,u) over v,w

pitch down:
rot(-δ,v) over u,w

Insert key pressed while other keys used  fast rotation by 10 δ per time step

7

Dr. Manuel Carcenac - European University of Lefke

extra: position control of virtual camera:
how to move the camera?
 incremental approach:
initial position p
small translation over p at each time step

translation expressed relatively to camera orientation:
move backward - forward:

 intuitive for operator

translation along w

move left - right:

translation along v

move down - up:

translation along u

fast move:
translation 10 δ

Insert

Page

Up

move forward:
pp+δw

moving mode

Delete

Page
Down

move backward:
pp-δw


move left:
pp-δv





move up:
pp+δu


move right:
pp+δv

move down:
pp-δu

Delete key pressed while other keys used  moving mode, else rotation mode
Insert key pressed while other keys used  fast move by 10 δ per time step
8


Dr. Manuel Carcenac - European University of Lefke

class V3d: operations on 3D vectors
class V3d
{
double x , y , z;
V3d() {}
V3d(double x , double y , double z) { this.x = x;
void V3d_move(V3d n , double d)
{ this.x += d * n.x;
this.y += d * n.y;
this.z += d * n.z; }

this.y = y; this.z = z; }

// this <-- this + d * n

void V3d_rotate(V3d n , double theta) // this <-- rotation (theta , n) of this; n unit vector; theta in deg.
{
double c = Math.cos(theta * Math.PI / 180.0) , s = Math.sin(theta * Math.PI / 180.0);
double k = (1 - c) * (this.x * n.x + this.y * n.y + this.z * n.z);
double x = c * this.x + k * n.x + s * (n.y * this.z - n.z * this.y)

, y = c * this.y + k * n.y + s * (n.z * this.x - n.x * this.z)
, z = c * this.z + k * n.z + s * (n.x * this.y - n.y * this.x);
this.x = x; this.y = y; this.z = z;
}
void V3d_orthogonalize(V3d b) // this <-- this - (this . b) b
{ double d = this.x * b.x + this.y * b.y + this.z * b.z;
this.x -= d * b.x; this.y -= d * b.y; this.z -= d * b.z; }
void V3d_normalize() // this <-- unit vector (this)
{ double d = Math.sqrt(this.x * this.x + this.y * this.y + this.z * this.z);
if (d < 1.0e-10) { System.out.println("in V3d_normalize: d < 1.0e-10");
this.x /= d; this.y /= d; this.z /= d; }
void V3d_vector_product(V3d a , V3d b) // this <-- a x b
{ this.x = a.y * b.z - a.z * b.y; this.y = a.z * b.x - a.x * b.z;

System.exit(0); }

this.z = a.x * b.y - a.y * b.x; }

void normal_3_points(V3d p0 , V3d p1 , V3d p2)
{ x = (p1.y - p0.y) * (p2.z - p0.z) - (p1.z - p0.z) * (p2.y - p0.y);
y = (p1.z - p0.z) * (p2.x - p0.x) - (p1.x - p0.x) * (p2.z - p0.z);
z = (p1.x - p0.x) * (p2.y - p0.y) - (p1.y - p0.y) * (p2.x - p0.x);
double d = Math.sqrt( x * x + y * y + z * z);
if (d < 1.0e-10) { System.out.println("d < 1.0e-10 in normal_3_points");
x /= d; y /= d; z /= d; }

System.exit(0); }

}

9


Dr. Manuel Carcenac - European University of Lefke

class Craft: virtual craft (orientation and position controlled in real time with keyboard)
class Craft
{
V3d p , u , v , w;
double dmove , drot;

boolean move , fast;

Craft( double posx , double posy , double posz
, double aimx , double aimy , double aimz
, double upx , double upy , double upz
, final JFrame f
, double dm , double dr)
{
p = new V3d(posx , posy , posz);
w = new V3d(aimx , aimy , aimz); w.V3d_normalize();
u = new V3d(upx , upy , upz); u.V3d_orthogonalize(w); u.V3d_normalize();
v = new V3d(); v.V3d_vector_product(w , u);
this.dmove = dm; this.drot = dr; move = false; fast = false;
f.addKeyListener(new KeyAdapter()
{ public void keyPressed(KeyEvent e)
{ int key = e.getKeyCode();
double d; if (move) d = (fast ? 10 * dmove : dmove); else d = (fast ? 10 * drot : drot);
switch (key)
{ case KeyEvent.VK_LEFT:

if (move) p.V3d_move(v , - d); else { v.V3d_rotate(u , d); w.V3d_rotate(u , d); }; break;
case KeyEvent.VK_RIGHT:
if (move) p.V3d_move(v , d); else { v.V3d_rotate(u , - d); w.V3d_rotate(u , - d); }; break;
case KeyEvent.VK_DOWN:
if (move) p.V3d_move(u , - d); else { u.V3d_rotate(v , - d); w.V3d_rotate(v , - d); }; break;
case KeyEvent.VK_UP:
if (move) p.V3d_move(u , d); else { u.V3d_rotate(v , d); w.V3d_rotate(v , d); }; break;
case KeyEvent.VK_PAGE_DOWN:
if (move) p.V3d_move(w , - d); else { u.V3d_rotate(w , d); v.V3d_rotate(w , d); }; break;
case KeyEvent.VK_PAGE_UP:
if (move) p.V3d_move(w , d); else { u.V3d_rotate(w , - d); v.V3d_rotate(w , - d); }; break;
case KeyEvent.VK_INSERT:
fast = true;
break;
case KeyEvent.VK_DELETE:
move = true; break; }
f.repaint(); } } );
f.addKeyListener(new KeyAdapter()
{ public void keyReleased(KeyEvent e)
{ int key = e.getKeyCode();
switch (key)
{ case KeyEvent.VK_INSERT: fast = false;
break;
case KeyEvent.VK_DELETE: move = false; break; } } } );
}
}

what if we need to use keyPressed and keyReleased for other keys as well ?
10

Dr. Manuel Carcenac - European University of Lefke

class Gui: create a virtual craft and use its orientation and position to input gluLookAt
class Gui
{
JFrame f; DrawingPanel p;
Craft craft;
class DrawingPanel extends GLJPanel
{
............
DrawingPanel()
{
super(new GLCapabilities(GLProfile.getDefault()));
this.addGLEventListener(new GLEventListener()
{
............
public void display(GLAutoDrawable drawable) //**** DISPLAY
{
GL2 gl = drawable.getGL().getGL2();
............
gl.glMatrixMode(GLMatrixFunc.GL_MODELVIEW);
gl.glLoadIdentity();
glu.gluLookAt(
(float)craft.p.x , (float)craft.p.y , (float)craft.p.z
, (float)(craft.p.x + craft.w.x) , (float)(craft.p.y + craft.w.y) , (float)(craft.p.z + craft.w.z)
, (float)craft.u.x , (float)craft.u.y , (float)craft.u.z );
............
}
............

} );
}
}

Gui()
{
f = new JFrame(); f.setFocusable(true); f.setVisible(true);
p = new DrawingPanel(); f.getContentPane().add(p , BorderLayout.CENTER);
craft = new Craft(100.0 , 0.0 , 0.0

,

-1.0 , 0.0 , 0.0 ,

0.0 , 0.0 , 1.0 , f , 0.5 , 0.5);

f.setSize(new Dimension(900 + 16 , 600 + 38));
}
}

11


Dr. Manuel Carcenac - European University of Lefke

Textures
create a texture object from a jpeg file
map the texture over a surface:
 precise at each vertex the texture coordinates



one or several polygons



GLU quadric (sphere, cylinder, disk, partial disk)

 automatic mapping

assign white color to the surface to preserve the colors of the texture
(else: red surface  texture becomes reddish when applied to surface)

creating a texture object:
import texture package:
import com.jogamp.opengl.util.texture.*;

declare texture object:
Texture texture;
true if antialiasing (mipmap)
false if aliasing tolerated
create texture object from image file:
try { texture = TextureIO.newTexture(new File("texture.jpg") , true); }
catch(Exception e) {}

texture == null



error reading the image file

12


Dr. Manuel Carcenac - European University of Lefke

texture coordinate system:
ut parameter: left to right
vt parameter: top to down

example:

 flip image vertically and save image file

earth map (Earth_flipV.jpg)
ut

vt

do not forget to flip vertically your jpeg image, else confusion over coordinates

13


Dr. Manuel Carcenac - European University of Lefke

mapping the texture over polygons:
enable texturing:

texture.enable();

bind texture to GL context:

texture.bind();

define polygon with texture coordinates at each vertex:
between glBegin and glEnd , for each vertex:
gl.glNormal3f(nx , ny , nz);

normal at vertex

gl.glTexCoord2f(ut , vt);

texture coordinates at vertex

gl.glVertex3f(x , y, z);

position of vertex

disable texturing with texture:

texture.disable();

 texture will not be applied over following surface

14


Dr. Manuel Carcenac - European University of Lefke

template for textured polygon(s):
import com.jogamp.opengl.util.texture.*;
..........
class Gui
{ ..........
class DrawingPanel extends GLJPanel
{ ..........
Texture texture;
DrawingPanel()
{ ..........
this.addGLEventListener(new GLEventListener()
{
public void init(GLAutoDrawable drawable)
{ ..........
try { texture = TextureIO.newTexture(new File("texture.jpg") , true); }
catch(Exception e) {}
}
..........
public void display(GLAutoDrawable drawable)
{ ..........
texture.enable(); texture.bind();
gl.glBegin(GL2.GL_);
..........
gl.glNormal3f(nx , ny , nz);
gl.glTexCoord2f(ut , vt);
gl.glVertex3f(x , y, z);
..........
gl.glEnd();
texture.disable();
}

..........
} );
}
}
}

15


Dr. Manuel Carcenac - European University of Lefke

example: Earth_flipV.jpg mapped over a flat quadrangle
.........
Texture tEv;
.........
try { tEv = TextureIO.newTexture(new File("Earth_flipV.jpg") , true); } catch(Exception e) {}
if (tEv == null) { System.out.println("tEv is null"); System.exit(0); }
.........
////------- QUADRANGLE
gl.glMaterialfv(GL.GL_FRONT_AND_BACK , GLLightingFunc.GL_AMBIENT_AND_DIFFUSE
, new float[] { 1.0f , 1.0f , 1.0f , 1.0f } , 0);
tEv.enable();

tEv.bind();

gl.glBegin(GL2.GL_QUADS);
gl.glNormal3f(1.0f , 0.0f , 0.0f); gl.glTexCoord2f(0.0f , 0.0f);
gl.glNormal3f(1.0f , 0.0f , 0.0f); gl.glTexCoord2f(1.0f , 0.0f);
gl.glNormal3f(1.0f , 0.0f , 0.0f); gl.glTexCoord2f(1.0f , 1.0f);
gl.glNormal3f(1.0f , 0.0f , 0.0f); gl.glTexCoord2f(0.0f , 1.0f);

gl.glEnd();

gl.glVertex3f(0.0f , 0.0f , 0.0f);
gl.glVertex3f(0.0f , 100.0f , 0.0f);
gl.glVertex3f(0.0f , 100.0f , 50.0f);
gl.glVertex3f(0.0f , 0.0f , 50.0f);

tEv.disable();
.........

4th vertex
(ut=0; vt=1)

3rd vertex
(ut=1; vt=1)

1st vertex
(ut=0; vt=0)
2nd vertex
(ut=1; vt=0)

16


Dr. Manuel Carcenac - European University of Lefke

example: Earth_flipV.jpg mapped over a single triangle
.........
Texture tEv;
.........

try { tEv = TextureIO.newTexture(new File("Earth_flipV.jpg") , true); } catch(Exception e) {}
if (tEv == null) { System.out.println("tEv is null"); System.exit(0); }
.........
////------- TRIANGLE
gl.glMaterialfv(GL.GL_FRONT_AND_BACK , GLLightingFunc.GL_AMBIENT_AND_DIFFUSE
, new float[] { 1.0f , 1.0f , 1.0f , 1.0f } , 0);
tEv.enable();

tEv.bind();

gl.glBegin(GL2.GL_TRIANGLES);
gl.glNormal3f(1.0f , 0.0f , 0.0f); gl.glTexCoord2f(0.0f , 0.0f);
gl.glNormal3f(1.0f , 0.0f , 0.0f); gl.glTexCoord2f(1.0f , 0.0f);
gl.glNormal3f(1.0f , 0.0f , 0.0f); gl.glTexCoord2f(0.5f , 1.0f);
gl.glEnd();
tEv.disable();
.........

gl.glVertex3f(0.0f , 0.0f , 0.0f);
gl.glVertex3f(0.0f , 100.0f , 0.0f);
gl.glVertex3f(0.0f , 50.0f , 50.0f);

3rd vertex
(ut=0.5; vt=1)

1st vertex
(ut=0; vt=0)
2nd vertex
(ut=1; vt=0)

17


Dr. Manuel Carcenac - European University of Lefke

example: Earth_flipV.jpg mapped over a triangle strip with 4 triangles
.........
Texture tEv;
.........
try { tEv = TextureIO.newTexture(new File("Earth_flipV.jpg") , true); } catch(Exception e) {}
if (tEv == null) { System.out.println("tEv is null"); System.exit(0); }
.........
////------- TRIANGLE STRIP
gl.glMaterialfv(GL.GL_FRONT_AND_BACK , GLLightingFunc.GL_AMBIENT_AND_DIFFUSE
, new float[] { 1.0f , 1.0f , 1.0f , 1.0f } , 0);
tEv.enable();

tEv.bind();

gl.glBegin(GL2.GL_TRIANGLES);
gl.glNormal3f(0.0f , 1.0f , 0.0f); gl.glTexCoord2f(0.0f , 0.0f);
gl.glNormal3f(0.0f , 1.0f , 0.0f); gl.glTexCoord2f(0.5f , 1.0f);
gl.glNormal3f(0.0f , 1.0f , 0.0f); gl.glTexCoord2f(0.0f , 1.0f);

gl.glVertex3f( 0.0f , 0.0f , 0.0f);
gl.glVertex3f( 0.0f , 50.0f , 50.0f);
gl.glVertex3f(43.0f , 25.0f , 50.0f);

gl.glNormal3f(1.0f , 0.0f , 0.0f);
gl.glNormal3f(1.0f , 0.0f , 0.0f);

gl.glNormal3f(1.0f , 0.0f , 0.0f);

gl.glVertex3f(0.0f , 0.0f , 0.0f);
gl.glVertex3f(0.0f , 100.0f , 0.0f);
gl.glVertex3f(0.0f , 50.0f , 50.0f);

gl.glTexCoord2f(0.0f , 0.0f);
gl.glTexCoord2f(1.0f , 0.0f);
gl.glTexCoord2f(0.5f , 1.0f);

gl.glNormal3f(0.0f , 1.0f , 0.0f); gl.glTexCoord2f(1.0f , 0.0f);
gl.glNormal3f(0.0f , 1.0f , 0.0f); gl.glTexCoord2f(1.0f , 1.0f);
gl.glNormal3f(0.0f , 1.0f , 0.0f); gl.glTexCoord2f(0.5f , 1.0f);
gl.glEnd();

gl.glVertex3f( 0.0f , 100.0f , 0.0f);
gl.glVertex3f(-50.0f , 50.0f , 50.0f);
gl.glVertex3f( 0.0f , 50.0f , 50.0f);

tEv.disable();
.........

3rd triangle

1st triangle
2nd triangle

18

Dr. Manuel Carcenac - European University of Lefke

mapping the texture over GLU quadrics:
enable texturing:

texture.enable();

bind texture to GL context:

texture.bind();

enable texturing for quadric object quad:

define quadric:

automatic mapping
according to rules below

glu.gluQuadricTexture(quad , true);

glu.gluSphere(quad , ......);
glu.gluCylinder(quad , ......);
glu.gluDisk(quad , ......);
glu.gluPartialDisk(quad , ......);

disable texturing with texture:

texture.disable();

 texture will not be applied over following surface

automatic mapping over sphere:
z
ut

x

vt

y

flip vertically your jpeg image

19


Dr. Manuel Carcenac - European University of Lefke

automatic mapping over cylinder:

z

ut

x

vt

y

flip vertically and horizontally your jpeg image

automatic mapping over disk / partial disk:
z
ut

x

vt

y

flip vertically your jpeg image

20


Dr. Manuel Carcenac - European University of Lefke

template for textured quadric(s):
import com.jogamp.opengl.util.texture.*;
..........
class Gui
{ ..........
class DrawingPanel extends GLJPanel
{ ..........
Texture texture;
DrawingPanel()
{ ..........
this.addGLEventListener(new GLEventListener()

{
public void init(GLAutoDrawable drawable)
{ ..........
try { texture = TextureIO.newTexture(new File("texture.jpg") , true); }
catch(Exception e) {}
}
..........
public void display(GLAutoDrawable drawable)
{ ..........
texture.enable(); texture.bind(); glu.gluQuadricTexture(quad , true);
glu.gluSphere(quad , ......);
glu.gluCylinder(quad , ......);
glu.gluDisk(quad , ......);
glu.gluPartialDisk(quad , ......);
texture.disable();
}
..........
} );
}
}
}

21


Dr. Manuel Carcenac - European University of Lefke

example: Earth_flipV.jpg mapped over a sphere, a disk, a partial disk
Earth_flipVH.jpg mapped over a cylinder
.........

try { tEv = TextureIO.newTexture(new File("Earth_flipV.jpg")
, true); } catch(Exception e) {}
try { tEvh = TextureIO.newTexture(new File("Earth_flipVH.jpg") , true); } catch(Exception e) {}
.........
tEv.enable(); tEv.bind(); glu.gluQuadricTexture(quad , true);
glu.gluSphere(quad , 30.0f , 50 , 50);
tEv.disable();
.........
tEvh.enable(); tEvh.bind(); glu.gluQuadricTexture(quad , true);
gl.glPushMatrix(); gl.glTranslatef(0.0f , 50.0f , 0.0f);
glu.gluCylinder(quad , 15.0f , 20.0f , 30.0f , 50 , 50);
gl.glPopMatrix();
tEvh.disable();
.........
tEv.enable(); tEv.bind(); glu.gluQuadricTexture(quad , true);
gl.glPushMatrix(); gl.glTranslatef(50.0f , 0.0f , 0.0f);
glu.gluDisk(quad , 5.0f , 15.0f , 20 , 20);
gl.glPopMatrix();
tEv.disable();
.........
tEv.enable(); tEv.bind(); glu.gluQuadricTexture(quad , true);
gl.glPushMatrix(); gl.glTranslatef(50.0f , 40.0f , 0.0f);
glu.gluPartialDisk(quad , 5.0f , 15.0f , 20 , 20 , 0.0f , +135.0f);
gl.glPopMatrix();
tEv.disable();
.........

22

Dr. Manuel Carcenac - European University of Lefke

example: Earth_flipV.jpg , Moon_flipV.jpg , Sun_flipV.jpg , Heaven_flipV.jpg
mapped on spheres ; virtual spacecraft
.........
try { tEv = TextureIO.newTexture(new File("Earth_flipV.jpg") , true);
try { tMv = TextureIO.newTexture(new File("Moon_flipV.jpg") , true);
try { tSv = TextureIO.newTexture(new File("Sun_flipV.jpg")
, true);
try { tHv = TextureIO.newTexture(new File("Heaven_flipV.jpg") , true);
.........
tEv.enable(); tEv.bind(); glu.gluQuadricTexture(quad , true);
glu.gluSphere(quad , 30.0f , 50 , 50);
tEv.disable();
.........
tMv.enable(); tMv.bind(); glu.gluQuadricTexture(quad , true);
gl.glPushMatrix(); gl.glTranslatef(0.0f , 80.0f , 0.0f);
glu.gluSphere(quad , 15.0f , 50 , 50);
gl.glPopMatrix();
tMv.disable();
.........
tSv.enable(); tSv.bind(); glu.gluQuadricTexture(quad , true);
gl.glPushMatrix(); gl.glTranslatef(20000.0f , 0.0f , 0.0f);
glu.gluSphere(quad , 1000.0f , 50 , 50);
gl.glPopMatrix();
tSv.disable();
.........
tHv.enable(); tHv.bind(); glu.gluQuadricTexture(quad , true);
gl.glPushMatrix();
glu.gluSphere(quad , 100000.0f , 50 , 50);

gl.glPopMatrix();
tHv.disable();
.........

}
}
}
}

catch(Exception e) {}
catch(Exception e) {}
catch(Exception e) {}
catch(Exception e) {}

23


Dr. Manuel Carcenac - European University of Lefke

24


Dr. Manuel Carcenac - European University of Lefke

NURBS surfaces
grid of control points along surface parameters u and v :
n_ctrlpts_u control points along u
v

n_ctrlpts_v control points along v

u

"Non Uniform":
knots = help us define intervals for parameters u , v between control points

define non uniform intervals (stretched, squeezed intervals)
 smoothen, sharpen locally the surface

"Rational":
eventually, specific weight for each control point

25