CG OpenGL 3D object representations-course 8

3D object
representations
Chen Jing-Fung (2006/12/5)
Assistant Research Fellow,
Digital Media Center,
National Taiwan Normal University

Video Processing Lab
臺灣師範大學數位媒體中心視訊處理研究室

Ch8: Computer Graphics with OpenGL 3th, Hearn Baker
Ch6: Interactive Computer Graphics 3th, Addison Wesley

3D objects
• Regular Polyhedrons

– Solid

– Mesh surface

Video Processing Lab 2

Graphics scenes
• Contain many different kinds of objects
and material surfaces
– Trees, flowers, clouds, rocks, water, bricks …
• Polygon and quadric surfaces provide
precise descriptions for simple Euclidean
objects
– Polyhedrons & ellipsoids
– Quadric surfaces (Ex: spline surfaces…)
• Mathematical method modify those objects


Useful design
• The constructive solid-geometry
techniques are useful design
– Aircraft wings, gear and other
engineering structures
The F-15 Eagle has a large relatively
Air France 747-200(Boeing)
lightly loaded wing (large and low..)

wing loading landing gear

Polyhedrons
• The most commonly used boundary to show
3D graphics object
– A set of surface polygons to enclose the object
interior
• Simply & speed up the surface rendering
and display of object
– Graphics systems store all object descriptions
• Standard graphics objects


OpenGL polyhedron
functions
• Polygon Fill-Area functions
– OpenGL primitive constants
• GL_POLYGON, GL_TRIANGLES,
GL_TRIANGLE_STRIP,
GL_TRIANGLE_FAN, GL_QUADS and
GL_QUAD_STRIP
– All faces compose by a group of parallelograms,
all faces compose by triangular surfaces or other
polygon from..


GLUT regular Polyhedron
functions (1)
• Five regular polyhedrons are pre-defined
by routines in GLUT library
– These polyhedrons are also called Platonic
solids
• All the faces are same regular polygons
– All edges are equal, All edge angles are equal and All
face angles are equal
– Polyhedron are named according to numberface
• Ex: regular tetrahedron (4 faces), regular
hexahedron ( or cube, 6 faces) ….


GLUT regular Polyhedron
functions (2)
– Each regular polyhedron is described in
modeling coordinates
• Each object’s center at world-coordinaate origin
– Wire-frame object functions
• glutWire*()
– Tetrahedron, Octahedron, Dodecahedron and
Icosahedron
– Fill areas polyhedron functions
• glutSolid*()
– Tetrahedron, Octahedron, Dodecahedron and
Icosahedron
edgeLength must be pre-defined
glutWireCube (edgeLength)
glutSolidCube (edgeLength)


Design notices
• Display function
– Set viewing transformation
• gluLookAt()
• Reshape function
– glMatrixMode(GL_PROJECTION)
• 3D projection
– glFrustum() or glOrtho() ….

demo


Regular 3D object’s mesh
• What is 3D object’s mesh?
– Like the object’s skin

• How to construct it?
– Curved surfaces coordinate
– What polygon can be used?


Curved surfaces
• General coordinates z

– Cartesian coordinate
y

• Quadric surfaces x

– Spherical surface
– Ellipsoid’s surface
– Torus’s surface


Spherical Surface
• In cartesian coordinates, a spherical
z
surface (x,y,z)

– radius r, any point (x,y,z) φ
y
• x2+y2+z2=r2 θ
x
– Describe the spherical surface
• x  r cos cos  / 2     / 2
y  r cos sin     
z  r sin


Think!!
• We change the angle φ of the point’s axis
– the angle range will be z
re-computed φ (x,y,z)

x  r cos cos 0     y
y  r cos sin 0    2 θ
x
z  r sin
– Want to re-range two angles in 0~1
• substituting    u,   2 v

Why?


Draw a sphere wire
• Plot longitude and latitude to
approximates the spherical surface
as a quadrilateral mesh
– Divide to three parts
• Top, floor
– The primary element is triangle
Top & floor means covered the surface
• Side
– Construct it with rectangles
– A surface


Top (+z) & floor (-z)

glBegin(GL_TRIANGLE_FAN);
glVertex3d(x,y,z);
c=3.14159/180.0; //interval Δφ, φ=80.0*c
x  r cos cos z=sin(c*80.0); // z=+..top, z=-.. floor Δθ
for(thet=-180.0; thet<=180.0;thet+=20.0){
y  r cos sin
x=cos(c*80.0)*cos(c*thet);
z  r sin y=cos(c*80.0)*sin(c*thet);
glVertex3d(x,y,z);
}
glEnd();


Side
for(phi=-80.0; phi<=80.0; phi+=20.0){
glBegin(GL_QUAD_STRIP); Δθ
for(thet=-180.0; thet<=180.0;thet+=20.0) {
x=cos(c*phi)*cos(c*thet);
y=cos(c*phi)*sin(c*thet);
x2+y2+z2=r2
z=sin(c*phi);
x  r cos cos glVertex3d(x,y,z);
x=cos(c*(phi+20.0))*cos(c*thet);
y  r cos sin
y=cos(c*(phi+20.0))*sin(c*thet);
z  r sin z=sin(c*(phi+20.0));
glVertex3d(x,y,z);
}
glEnd();
}
p5 glBegin(GL_QUAD_STRIP);
p1 p4 p8 glVertex3d(p1)….
glVertex3d(p8)
glEnd();

p2 p3 p6
p7

Advantage of this design
• Put each design component to list
function in initial function
– Ex: top, floor or side
• In display function call our design to
let them move
• Next page will show how to design


Display architecture

Display-processor architecture
Simple graphics architecture
server
Display
server processor
Host
Host (DPU)

Display file
Immediate mode (display list)
Ex : progressive
Retained mode
Ex : interlace


Definition & execution of
display lists
• Display lists have much in common
with ordinary files.
– Define (create)
• What contents of display list
– Manipulate (place information in)
• What functions of display lists
– Such as: flexible enough….


OpenGL display lists
• Display lists in OpenGL are defined
similarly to geometric primitive
Display function:
Object’s name in the list
*Each time that we wish to draw:
glNewList(BOX, flag); (Ex: do transformation or set up time
control…)
glBegin(GL_POLYGON)
GL_COMPILE: send list glCallList(BOX);
… to server, not to display Matrix and attribute stacks:
glEnd(); GL_COMPILE_AND_EXE glPushAttrib(GL_ALL_ATTRIB_BITS);
CUTE: immediate display glPushMatrix();
glEndList();
Display list function:
glPopAttrib();
glPopMatrix();

OpenGL GLUT Quadric-
Surface sphere functions
• GLUT sphere functions
– Approximate the spherical surface as a
quadrilateral mesh
– glutWireSphere (r,nLongitudes, nLatitudes);
glutSolidSphere (…)
• r (sphere radius): double or float point
• nLongitudes & nLatitudes: the integer number about
longitude lines & latitude lines
– The sphere is defined in modeling coordinates,
centered at the originworld-coordinate with its polar
axis along the z-axis


height
GLUT Cone functions
rBase

• GLUT cone is obtained with
– glutWireCone(rBase,height,nLongitudes,nLatitudes);
glutSolidCone(…);
• rbase & height: float or double
• nLongitudes & nLatitudes: integer Ex: 5, 6
– Each latitude line is parallel with theirseves
• the quadrilateral mesh approximation


Ellipsoid’s surface
• Describe it an extension of a
spherical surface z
– Three radii rz

– In Cartesian coordinates rx y
2 ry
x y z
2 2

      1
  x
 rx   ry   rz 

– Representation by two angle (φ&θ)
x  rx cos  cos   / 2     / 2
y  ry cos  sin     
z  rz sin  This mesh is similar the spherical mesh

Torus’s surface (1) raxial

• What is Torus?
– a doughnut-shaped object or called ring
– Describe the surface
• rotating a circle or an ellipse about a co-
planar axis line
z axis

(0,y,z)
r φ 0 y axis
(x,y,z)
0 y axis θ
raxial
x axis
Side view
Top view

Torus’s surface (2) raxial

• In Cartesian equation
( x 2  y 2  raxial )2  z2  r 2

• Using a circular cross section
x  (raxial  r cos  )cos     
y  (raxial  r cos  )sin     
z  r sin


• If generate a torus by rotating an
ellipse & round circle about the z axis
– In Cartesian equation
2
 x2  y 2  r   z 2
 axial
   1
 ry   rz 
 

– The torus with an elliptical cross section
x  (raxial  ry cos  )cos      
y  (raxial  ry cos  )sin     
z  rz sin 

GLUT torus rout

rin

functions
• Wire-frame or surface-shaded displays of a
torus with a circular cross section
– glutWireTorus(rCrossSection, rAxial,
nConcentrics,nRadialSlices);
glutSolidTorus(…);
• rcrossSection (=inner Radius)& rAxial (=outer Radius)
– float or double
• nConcentrics(sides) & nRadialSlices(rings)
– Integer
– sides is parallel together(nCon..: the number of sides),
nRadial..: the number of rings


GLU Quadric-Surface
functions
• Quadric surface can used GLU functions to
design
– Assign a name to the quadric
– Activate the GLU quadric render
– Designate values for surface parameters
• Set other parameter values to control the
appearance of a GLU quadric surface.


Some GLU functions
• Sphere:
– gluSphere(name,r,nLongitudes,nLatitudes);
• Cylinder:
– gluCylinder(name,rBase,rTop,height,
nLongitudes,nLatitudes);
• A section of ring:
– gluPartialDisk(name, rin,rout,nRadii,nRings, Anglestart,
Anglesweep);


How to use?
• Example: to draw the sphere
GLUquadricObj *sphere1;
sphere1 = gluNewQuadric ();
gluQuadricDrawStyle(sphere1, display mode);
gluSphere(sphere1,…);
gluQuadricOrientation(sphere1, normalvectordirection);//signal face surface
gluQuadricNormal(sphere1, generationMode);//total object
pluDeleteQuadric (sphere1);//delete the object
• Display mode:
– GLU_LINE, GLU_POINT, GLU_FULL
• Normal vector direction
– GLU_OUTSIDE or GLU_INSIDE (inside: back-face,
outside:fact-face)
• General mode
– default=GLU_NONE, flat surface=GLU_FLAT(object of
constant color)


Approximation of a sphere
by recursive subdivision

• Recursive subdivision is a powerful
technique for generating
approximations to curves and
surfaces to any desired level of
accuracy
– Start from four equilateral triangles (a
tetrahedron)
• Four vertices (points) in unit sphere:
(0,0,1),(0,2 2 / 3, 1/ 3),( 6 / 3,  2 / 3, 1/ 3),( 6 / 3,  2 / 3, 1/ 3)


Triangle -> tetrahedron

void triangle( point a, point b, point void tetrahedron() {
c)// display one triangle using a line triangle(v[0],v[1],v[2]);
loop for wire frame triangle(v[3],v[2],v[1]);
{ triangle(v[0],v[3],v[1]);
glBegin(GL_LINE_LOOP); triangle(v[0],v[2],v[3]);
}
glVertex3fv(a);
glVertex3fv(b); v0
glVertex3fv(c);
glEnd();
}

v1 v2 v3


void normal(point p){ void tetrahedron( int m){
/* normalize a vector */ /* Apply triangle subdivision to faces of
double sqrt(); tetrahedron */
float d =0.0; divide_triangle(v[0], v[1], v[2], m);
int i; divide_triangle(v[3], v[2], v[1], m);
for(i=0; i<3; i++) d+=p[i]*p[i]; divide_triangle(v[0], v[3], v[1], m);
d=sqrt(d); divide_triangle(v[0], v[2], v[3], m);}
if(d>0.0) for(i=0; i<3; i++) p[i]/=d;}
 void divide_triangle(point a, point b, point c, int m){
p
Vnor  /* triangle subdivision using vertex numbers
det(p) righthand rule applied to create outward pointing faces */
point v1, v2, v3;
v0 int j;
if(m>0) {
for(j=0; j<3; j++) v1[j]=a[j]+b[j];
normal(v1);
for(j=0; j<3; j++) v2[j]=a[j]+c[j];
v3 normal(v2);
v1 v2
for(j=0; j<3; j++) v3[j]=b[j]+c[j];
normal(v3);
divide_triangle(a, v1, v2, m-1);
divide_triangle(c, v2, v3, m-1);
divide_triangle(b, v3, v1, m-1);
divide_triangle(v1, v3, v2, m-1); }
else(triangle(a,b,c)); /* draw triangle at end of recursion */}

Approximate object
• Recursive subdivision
– We could design a object’s surface
numbers.
– Recursive from coarse to fine
• Mesh object related its surface
– Direction: Longitude and latitude


How to plot 3D object
• Take apart of object to several polygon
facets
– First, list the vertices to show one surface
• primitive:
– GL_POLYGON, GL_QUADS or GL_TRIANGLES…
– Function: glVertex3* (pointcoordinate)
» Key point: triangle: 3 points or triangle+quadrangle
– Second, list its surface direction
• Find function about the approximate point
– Ex: approximate sphere glNormal3* (normcoordinate)
• Surface function construct from those vertices


Homework 1
• Draw two object’s mesh (wire & solid)
name Implicit function (F=0) Parametric form v range u range
Hyperboloid (sec(v )cos(u ),sec(v )sin(u ), (  / 2, / 2)
x2+y2-z2-1 (  , )
tan(v ))
Elliptic cone x2+y2-z2 (v cos(u ),v sin(u ),v ) any real nums,( , )

Hyperbolic v  0,( , )
-x2+y2-z (v tan(u ),v sec(u ),v 2 )
superellipsoid x 2 / s2
y 
2 / s2 s2 / s1
z 2 / s1
1
(coss1 (v )coss2 (u ),coss1 (v )sins2 (u ), (  / 2, / 2),(  , )
sins1 (v )) s1, s2 : 0.5 ~ 3.0(# 6)
superhyperbol s2 / s1 (sec s (v )sec s (u ),sec s (v )tans (u ), v : (  / 2, / 2)
 x 2 / s2  y 2 / s2   z2 / s1  1
1 2 1 2

oid   tans1 (v )) u : sheet#1 or sheet#2

sheet #1: ( / 2, / 2)
sheet # 2 : ( / 2,3 / 2)
Superellipse demo


Homework 2
• Practice “Recursive subdivision”
– Draw a magic box
• Hint: Some things jump out the cube or this
box’s sub-part can be rotated


Appendix
name Implicit function (F=0) Parametric form v, u range
Ellipsoid (cos(v )cos(u ),cos(v )sin(u ), v : (  / 2, / 2)
x2+y2+z2-1 sin(v )) u : (  , )

Hyperboloid of v : (  / 2, / 2)
x2+y2-z2-1 (sec(v )cos(u ),sec(v )sin(u ),
one sheet tan(v )) u : (  , )
Hyperboloid of (sec(v )cos(u ),sec(v )tan(u ), v : ( / 2, / 2)
x2-y2-z2-1  
two sheet tan(v )) sheet# 1: u : ( , )
2 2
 3
sheet# 2 : u : ( , )
2 2

Ellipsoid Hyperboloid of one sheet Hyperboloid of two sheet
Ch9: Computer Graphics, Hill 臺灣師範大學數位媒體中心視訊處理研究室

name Implicit function (F=0) Parametric form v, u range
Elliptic cone v : any real
x2+y2-z2 (v cos(u ),v sin(u ),v )
u : (  , )
Elliptic x2+y2-z (v cos(u ),v sin(u ),v 2 )
v0
paraboloid u : (  , )
Hyperboloid v0
-x2+y2-z (v tan(u ),v sec(u ),v )2
paraboloid u : (  , )

Elliptic paraboloid
Elliptic cone Hyperboloid paraboloid

s1, s2 : 0.5 ~ 2.5(# 5)

superellipsoid
x 2 / s2
y 
2 / s2 s2 / s1
z 2 / s1
1
(coss1 (v )coss2 (u ),coss1 (v )sins2 (u ),
sins1 (v ))
v : (  / 2, / 2)
u : (  , )
Superhyperbo s2 / s1 (sec s1 (v )coss2 (u ),sec s1 (v )sins2 (u ), v : (  / 2, / 2)
loid of one x 2 / s2
y 2 / s2
 z 2 / s1
1 s
  tan 1 (v )) u : (  , )
sheet

superellipsoid

Superhyperboloid of one sheet

s1, s2 : 0.5 ~ 2.5(# 5) u : sheet # 2 : ( / 2,3 / 2)
u : sheet #1: ( / 2, / 2)

superhyperbo s2 / s1 (sec s (v )sec s (u ),sec s (v )tans (u ), v : (  / 2, / 2)
1 2 1 2

loid
 x 2 / s2  y 2 / s2   z2 / s1  1 s
  tan (v ))
1 u : sheet#1 or sheet#2

s2 / 2 ((d  coss1 (v ))coss2 (u ),
( x
2/s

2
y
2/s 2

  d)2/s1
v : (  , )
Supertorus (d  coss1 (v ))sins2 (u ),
u : (  , )
 z 2 / s1  1 sins1 (v ))

superhyperboloid

Supertorus

CG OpenGL 3D object representations-course 8

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