2IV60 Computer graphics
set 7: Basic Geometric Modeling
Jack van Wijk
TU/e
Geometric modeling 1
The world is full of all kind of objects:
Trees, people, cars, housed, clouds, rocks, waves,
pencil sharpeners, fire, mountains, plants, …
How can we describe these, such that they are
- easy to enter;
- easy to process;
- easy to display?
Complex problem, HUGE topic!
Geometric modeling 2
Input:
• by user
- preferably on a high level, customized to
application
• by scanning objects
- laser scanners: points
- medical scanners: voxels
Geometric modeling 3
User
Instructions,
specifications
conversion
triangles
Scanner
3D points,
voxels
conversion
graphics
pipeline
image
Smooth Curves (2D)
Two standard methods:
• Parametric: X(t) = (x(t), y(t))
• Implicit: f(x, y) = 0
H&B A-8,9:772-774
Parametric description line 1
Given point P and vector V:
X(t) = P + Vt
y
Given two points P en Q:
X(t) = P + (Q  P)t , or
= P(1t) + Qt
Segment: tmin  t  tmax
t
Q
P
V
x
H&B A-8,9:772-774
Parametric description curve
• X(t) = (x(t), y(t))
• Drawing a curve in a simple way,
approximate with a polyline:
MoveTo(X(0));
for i := 1 to N do LineTo(X(iDt));
H&B A-8,9:772-774
Drawing a curve with OpenGL 1
X(float t):Point;
{ P[0]= …; P[1]=…; P[2]=…; return P; };
DrawX(); {
N = 12; // use symbolic name for the number of segments
int i;
float dt := (tmax – tmin)/N;
glBegin(GL_LINE_STRIP); // GL_LINE_LOOP if closed curve
for (i = 0; i <= N; i++) // bounds here: 0 - N
glVertex3fv(X(tmin + i*dt));
glEnd();
}
H&B 4-5:82-83
Drawing a curve with OpenGL 2
• Using too few points:
– jagged appearance;
• Using too many points:
– slow,
– fat appearance (many segments per pixel).
H&B 4-5:82-83
Drawing a curve with OpenGL 3
• Decide on #points:
– characteristics shape;
– size of shape on screen.
– beforehand or adjust automatically.
H&B 4-5:82-83
Implicit description line
• (X  P).N = 0
with N.V = 0
(N: normal vector)
• Also:
ax+by+c=0
y
N
V
P
x
H&B A-8,9:772-774
Implicit description curve
f=1
f=0
f >0
f =-1
f =-2
f <0
H&B A-8,9:772-774
Circle
Parametric :
( x, y )  (r cos 2t , r sin 2t ),
0  t 1
y
r
x
Implicit (or nonparametric) :
x  y r 0
2
2
2
H&B A-8,9:772-774
Curves (3D)
Two standard methods:
• Parametric: X(t) = (x(t), y(t), z(t))
• Implicit: f(x, y, z) = 0 and g(x, y, z) = 0
Intersection of two surfaces
H&B A-8,9:772-774
Circle in 3D
Parametric :
( x, y, z )  (r cos 2t , r sin 2t ,0),
z
y
0  t 1
r
Implicit :
x
x  y  r  0 and
z0
2
2
2
H&B A-8,9:772-774
Surfaces
• Polyhedra
• Parametric surfaces
• Implicit surfaces
H&B A-8,9:772-774
Polyhedra
• Set of polygons that describe the surface of an
object
• Often only triangles (hardware!)
• Many variations for storage (zie H&B 4-7)
• Often additional information per vertex
(color, normal, texture, …)
H&B 13-1:418
Curved surfaces
• Parametric: X(u,v) = (x(u,v), y(u,v), z(u,v))
• Implicit: f(x, y, z) = 0
H&B 13-3:421-422
Sphere 1
z
Parametric :
y
x(u , v)  r cos(2u ) sin( v)
y (u , v)  r sin( 2u ) sin( v)
z (u , v)  r cos(v)
0  u, v  1
x
u constant, v varies
v constant, u varies
H&B 13-4:422-424
Sphere 2
z
Display :
y
Make N  N facets with vertices :
X(ui ,v j ), X(ui 1,v j ),
X(ui ,v j 1 ), X(ui 1,v j 1 ),
x
with ui  i / N , v j  j / N ,
u constant, v varies
v constant, u varies
0  i, j  N  1
v
u
H&B A-8,9:807-808
Sphere 3
z
Implicit descriptio n :
y
f ( x, y , z )  x 2  y 2  z 2  r 2  0
Test if punt ( x, y, z ) is inside object :
x
f ( x, y, z )  0 : Yes
f ( x, y, z )  0 : No
Again, f divides space in two parts.
H&B 13-4:422-424
Partial Derivatives 1
Standard derivative for function of one variable :
df ( x)
f ( x  Dx)  f ( x)
f ' ( x) 
 lim
Dx 0
dx
Dx
Partial derivative s for function of two variables :
f ( x, y )
f ( x  Dx, y )  f ( x, y )
f x ( x, y ) 
 lim
Dx 0
x
Dx
f ( x, y )
f ( x, y  Dy )  f ( x, y )
f y ( x, y ) 
 lim
Dy 0
y
Dy
H&B A10:774
Partial Derivatives 2
Example : f ( x, y )  2 x  xy  3 y
For the partial derivative to x, consider y as constant, and apply
standard differenti ation to x :
f ( x, y )
 4x  y
x
For the partial derivative to y, consider x as constant, and apply
standard differenti ation to y :
f ( x, y )
 x  9 y2
x
H&B A10:774
2
3
Normal on surface 1
Given :
Parametric curve P(u )  ( x(u ), y (u ), z (u ))
u
Tangent vector :
 dx(u ) dy(u ) dz(u ) 
T(u )  
,
,

du
du 
 du
T(u)
P(u)
Note :if u is time, then T(u ) is a velocity vector.
H&B A-10:774-775
Normal on surface 2
Given :
Parametric surface.P (u , v)  ( x(u , v), y (u , v), z (u , v))
Tangent vector in u - direction :
 x(u , v) y (u , v) z (u , v) 
Tu (u , v)  
,
,

u
u 
 u
Tv(u,v)
u
Tu(u,v)
P(u,v)
Tangent vector in v - direction :
 x(u , v) y (u , v) z (u , v) 
Tv (u , v)  
,
,

v
v 
 v
v
H&B A-10:774-775
Normal on surface 3
Tangent vector in u - direction :
 x(u , v) y (u , v) z (u , v) 
Tu (u , v)  
,
,

N

u

u

u


Tangent vector in v - direction :
Tv(u,v)
u
Tu(u,v)
 x(u , v) y (u , v) z (u , v) 
Tv (u , v)  
,
,


v

v

v


P(u,v)
Tu (u , v) and Tv (u , v) are in tangent plane
of surface. Normal to surface :
N(u , v)  Tu (u , v)  Tv (u , v)
v
H&B A-10:774-775
Normal on surface 4
Example cylinder w ith radius r and height h, with 0  u , v  1 :
z
x(u , v)  r cos(2u )
y (u , v)  r sin( 2u )
z (u , v)  hv
Tv
Tu (u , v)  x(u , v) / u , y (u , v) / u , z (u , v) / u 
  2r sin( 2u ), 2r cos(2u ), 0
Tv (u , v)  x(u , v) / v, y (u , v) / v, z (u , v) / v 
 0, 0, h 
N(u , v)  Tu (u , v)  Tv (u , v)
 2rh cos(2u ), 2r sin( 2u ), 0
 2rhcos(2u ), sin( 2u ), 0
v
N
Tu
x
y
u
H&B A-10:774-775
Normal on surface 5
Example sphere :
x(u , v)  r cos(2u ) sin( v)
y (u , v)  r sin( 2u ) sin( v)
z (u , v)  r cos(v)
Tu (u , v)   2r sin( 2u ) sin( v), 2r cos(2u ) sin( v), 0
Tv (u , v)  r cos(2u ) cos(v), r sin( 2u ) cos(v),  r sin( v) 


 2 2 r 2 cos(2u ) sin 2 (v)


2 2
2

Tu (u , v)  Tv (u , v)  
 2 r sin( 2u ) sin (v)


2 2
2
2 2
2

2

r
sin
(
2

u
)
sin(

v
)
cos(

v
)

2

r
cos
(
2

u
)
sin(

v
)
cos(

v
)


 x(u , v) 


2
  2 r sin( v) y (u , v) 
 z (u , v) 


H&B A-10:774-775
Normal on surface 6
Given :
Implicit surface f ( x, y, z )  0
Normal vector :
N ( x , y , z )  f ( x , y , z )
( is grad)
 f f f 
  , , 
 x y z 
 direction of fastest increase f
N(x,y,z)
(x,y,z)
f(x,y,z)=0
H&B A-10:774-775
Normal on surface 7
f >0
f=0
f <0
H&B A-10:774-775
Normal on surface 8
Example cylinder,
radius r and height h,
with 0  u , v  1 :
z
f ( x, y , z )  x 2  y 2  r 2  0
N
x
 f f f 
N( x, y, z )   , , 
 x y z 
 (2 x,2 y,0)
y
H&B A-10:774-775
Quadratic surfaces 1
Quadratic surfaces:
Second degree equation f ( x, y, z )  0
z
ax2  by 2  cz 2  dxy  exz  fyz  gx  hy  iz  j  0
Example :
cylinder : f ( x, y, z )  x  y  r  0
2
2
2
x
y
sphere : f ( x, y, z )  x 2  y 2  z 2  r 2  0
cone :
f ( x, y , z )  x 2  y 2  z 2  0
Can also easily be described parametrically
H&B 13-4:422-424
Quadratic surfaces 2
Ellipsoid : scaled sphere (mentos, cigar)
sphere :
f ( x, y , z )  x 2  y 2  z 2  r 2  0
ellipsoid :
 x  y z
f ( x, y , z )           1  0
a b c
2
2
2
z
c
x
a
b
y
H&B 13-4:422-424
Quadratic surfaces 3
Torus : donut, car tyre (degree 4 surface)
In x  0 plane : f ( x, y, z )  ( y  ra ) 2  z 2  r 2  0
f ( x, y, z )  ( x 2  y 2  ra ) 2  z 2  r 2  0
Generic :
z as
y
(x, y, z)
r
ra
(0, y, z)
y as
y as
x as
H&B 13-4:422-424
Drawing surfaces with OpenGL 1
• Quadratic surfaces: GLU and GLUT offer
some basic functions:
–
–
–
–
glutSolidSphere(r, nLon, nLat);
glutSolidCone(rbase, rheight, nLon, nLat);
glutSolidTeapot(size);
gluSphere, gluCylinder,
• Teapot?
– Google: Utah teapot;
– Google: Stanford bunny.
H&B 13-6:425-431
Drawing surfaces with OpenGL 1
• Quadratic surfaces: GLU and GLUT offer
some basic functions:
–
–
–
–
glutSolidSphere(r, nLon, nLat);
glutSolidCone(rbase, rheight, nLon, nLat);
glutSolidTeapot(size);
gluSphere, gluCylinder,
• Alternative: Define your own, custom
versions, based on standard way to render
parametric surfaces.
H&B 13-6:425-431
Drawing surfaces with OpenGL 2
X(float u, v):Point;
{ P[0]= …; P[1]=…; P[2]=…; return P; };
DrawX(); {
NU, NV = 12; // use symbolic names
int i, j;
float du := (umax – umin)/NU;
float dv := (vmax – vmin)/NV;
for numbers of points again
for (j = 0; j < NV; j++) {
glBegin(GL_QUAD_STRIP);
for (i = 0; i <= NU; j++) {
glVertex3fv(X(umin + i*du, vmin + j*dv));
glVertex3fv(X(umin + i*du, vmin + (j+1)*dv));
}
glEnd();
}
Drawing surfaces with OpenGL 3
Many variations possible
– using triangle strips;
– inserting normals, shading info, texture, ...
Selecting the number of facets:
– too few: jagged appearance;
– too many: slow;
– decide beforehand or automatically, based
on size on screen and curvature surface.
H&B 4-5:82-83
Next…
• We now know how to define basic objects.
• Stay tuned for splines: arbitrary curves and
curved surfaces.
• But first: let’s consider illumination and
shading.