cg opengl 3d object representations-course 8

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

3D object representations

Chen Jing-Fung (2006/12/5) Assistant Research Fellow,

Digital Media Center, National Taiwan Normal University

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

Video Processing Lab

臺灣師範大學數位媒體中心視訊處理研究室

2

3D objects

• Regular Polyhedrons

– Solid

– Mesh surface



3

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



4

Useful design

• The constructive solid-geometry techniques are useful design – Aircraft wings, gear and other

engineering structures

Air France 747-200(Boeing)

landing gear wing loading

The F-15 Eagle has a large relatively lightly loaded wing (large and low..)

http://en.wikipedia.org/wiki/Image:USAF_F15.jpg

http://en.wikipedia.org/wiki/Image:540756.jpg



5

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



6

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..



7

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) ….



8

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

glutWireCube (edgeLength) glutSolidCube (edgeLength)

edgeLength must be pre-defined



9

Design notices

• Display function – Set viewing transformation

• gluLookAt()

• Reshape function – glMatrixMode(GL_PROJECTION)

• 3D projection – glFrustum() or glOrtho() ….

demo

ch8_Wire-framePolyhedron.exe



10

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?



11

Curved surfaces

• General coordinates – Cartesian coordinate

• Quadric surfaces – Spherical surface – Ellipsoid’s surface – Torus’s surface

y

x

z



12

Spherical Surface

• In cartesian coordinates, a spherical surface – radius r, any point (x,y,z)

• x2+y2+z2=r2

– Describe the spherical surface •

y

x

z (x,y,z)

θ

φ

sin

sincos

coscos

rz

ry

rx

/ 2 / 2



13

Think!!

• We change the angle φ of the point’s axis – the angle range will be

re-computed

– Want to re-range two angles in 0~1 • substituting

y

x

z (x,y,z)

θ

φ

sin

sincos

coscos

rz

ry

rx

0

0 2

Why?

, 2u v



14

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

• Side

– Construct it with rectangles – A surface

Top & floor means covered the surface



15

Top (+z) & floor (-z)

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

Δθ

sin

sincos

coscos

rz

ry

rx



16

Side

for(phi=-80.0; phi<=80.0; phi+=20.0){ glBegin(GL_QUAD_STRIP); for(thet=-180.0; thet<=180.0;thet+=20.0) { x=cos(c*phi)*cos(c*thet); y=cos(c*phi)*sin(c*thet); z=sin(c*phi); glVertex3d(x,y,z); x=cos(c*(phi+20.0))*cos(c*thet); y=cos(c*(phi+20.0))*sin(c*thet); z=sin(c*(phi+20.0)); glVertex3d(x,y,z); } glEnd(); }

Δθ

glBegin(GL_QUAD_STRIP); glVertex3d(p1)…. glVertex3d(p8) glEnd();

p1

p2 p3

p4

p6

p5

p7

p8

x2+y2+z2=r2

sin

sincos

coscos

rz

ry

rx



17

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



18

Display architecture

Host

Simple graphics architecture

Immediate mode

Host

Display processor

(DPU)

Display file (display list)

Display-processor architecture

Retained mode Ex : progressive

Ex : interlace

server server



19

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….



20

OpenGL display lists

• Display lists in OpenGL are defined similarly to geometric primitive

glNewList(BOX, flag);

glBegin(GL_POLYGON)

…

glEnd();

glEndList();

Object’s name in the list

GL_COMPILE: send list to server, not to display

GL_COMPILE_AND_EXECUTE: immediate display

Display function:

*Each time that we wish to draw: (Ex: do transformation or set up time control…)

glCallList(BOX);

Matrix and attribute stacks:

glPushAttrib(GL_ALL_ATTRIB_BITS);

glPushMatrix();

Display list function:

glPopAttrib();

glPopMatrix();



21

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



22

GLUT Cone functions

• 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

height

rBase



23

Ellipsoid’s surface

• Describe it an extension of a spherical surface – Three radii

– In Cartesian coordinates

– Representation by two angle (φ&θ)

y

x

z

rz

rx ry

cos cos

cos sin

sin

x

y

z

x r

y r

z r

/ 2 / 2

22 2

1x y z

x y z

r r r

This mesh is similar the spherical mesh



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• 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

φ r

0

(0,y,z)

y axis

z axis

raxial

Side view

0

θ

(x,y,z) y axis

x axis

Top view

Torus’s surface (1) raxial



25

Torus’s surface (2)

• In Cartesian equation

• Using a circular cross section

raxial

2 2 2 2 2( )axialx y r z r

( cos )cos

( cos )sin

sin

axial

axial

x r r

y r r

z r



26

• If generate a torus by rotating an ellipse & round circle about the z axis – In Cartesian equation

– The torus with an elliptical cross section

2 22 2

1axial

y z

x y r z

r r

( cos )cos

( cos )sin

sin

axial y

axial y

z

x r r

y r r

z r



27

GLUT torus 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

rin

rout



28

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.



29

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);



30

How to use? • Example: to draw the sphere

• 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)

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



31

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)



32

Triangle -> tetrahedron

void triangle( point a, point b, point c)// display one triangle using a line loop for wire frame { glBegin(GL_LINE_LOOP); glVertex3fv(a); glVertex3fv(b); glVertex3fv(c); glEnd(); }

void tetrahedron() { triangle(v[0],v[1],v[2]); triangle(v[3],v[2],v[1]); triangle(v[0],v[3],v[1]); triangle(v[0],v[2],v[3]); }

v0

v1 v2 v3



33

void normal(point p){ /* normalize a vector */ double sqrt(); float d =0.0; int i; for(i=0; i<3; i++) d+=p[i]*p[i]; d=sqrt(d); if(d>0.0) for(i=0; i<3; i++) p[i]/=d;}

)det(pVnor

p

void tetrahedron( int m){ /* Apply triangle subdivision to faces of tetrahedron */ divide_triangle(v[0], v[1], v[2], m); divide_triangle(v[3], v[2], v[1], m); divide_triangle(v[0], v[3], v[1], m); divide_triangle(v[0], v[2], v[3], m);}

void divide_triangle(point a, point b, point c, int m){ /* triangle subdivision using vertex numbers righthand rule applied to create outward pointing faces */ point v1, v2, v3; 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]; normal(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 */}

v0

v1 v2 v3



34

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



35

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



36

Homework 1

• Draw two object’s mesh (wire & solid) name Implicit function (F=0) Parametric form v range u range

Hyperboloid

Elliptic cone

Hyperbolic

superellipsoid

superhyperboloid

x2+y2-z2-1

x2+y2-z2

-x2+y2-z

( / 2, / 2)

( , )

( cos( ), sin( ), )v u v u v

0,( , )v 2( tan( ), sec( ), )v u v u v

any real nums,( , )

1112

22 /2//2/2

sssss zyx

(sec( )cos( ),sec( )sin( ),

tan( ))

v u v u

v

1 2 1 2

1

(cos ( )cos ( ),cos ( )sin ( ),

sin ( ))

s s s s

s

v u v u

v 1 2

( / 2, / 2),( , )

, : 0.5 ~ 3.0(#6)s s

2 12 2 1

/2 / 2 / 2 / 1

s ss s sx y z

1 2 1 2

1

(sec ( )sec ( ),sec ( )tan ( ),

tan ( ))

s s s s

s

v u v u

v

#1: ( / 2, / 2)sheet

#2: ( / 2,3 /2)sheet Superellipse demo

: ( / 2, / 2)

: sheet#1 or sheet#2

v

u

ch8_Surperellipse.exe



37

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



38

Appendix

name Implicit function (F=0) Parametric form v, u range

Ellipsoid

Hyperboloid of one sheet

Hyperboloid of two sheet

x2+y2+z2-1 (cos( )cos( ),cos( )sin( ),

sin( ))

v u v u

v

: ( / 2, / 2)

: ( , )

v

u

x2+y2-z2-1 (sec( )cos( ),sec( )sin( ),

tan( ))

v u v u

v

x2-y2-z2-1

: ( / 2, / 2)

: ( , )

v

u

(sec( )cos( ),sec( )tan( ),

tan( ))

v u v u

v

: ( / 2, / 2) v

sheet# 1: : ( , )2 2

3sheet# 2 : : ( , )

2 2

u

u

Ellipsoid Hyperboloid of one sheet Hyperboloid of two sheet

Ch9: Computer Graphics, Hill



39

name Implicit function (F=0) Parametric form v, u range

Elliptic cone

Elliptic paraboloid

Hyperboloid paraboloid

x2+y2-z2

x2+y2-z

-x2+y2-z

( cos( ), sin( ), )v u v u v

0

: ( , )

v

u

: any real

: ( , )

v

u

2( cos( ), sin( ), )v u v u v

2( tan( ), sec( ), )v u v u v0

: ( , )

v

u

Elliptic cone Elliptic paraboloid

Hyperboloid paraboloid



40

superellipsoid

Superhyperboloid of one sheet

1112

22 /2//2/2

sssss zyx1 2 1 2

1

(cos ( )cos ( ),cos ( )sin ( ),

sin ( ))

s s s s

s

v u v u

v

: ( / 2, / 2)

: ( , )

v

u

2 12 2 1

/2 / 2 / 2 / 1

s ss s sx y z

1 2 1 2

1

(sec ( )cos ( ),sec ( )sin ( ),

tan ( ))

s s s s

s

v u v u

v

1 2, : 0.5 ~ 2.5(#5)s s

: ( / 2, / 2)

: ( , )

v

u

superellipsoid

Superhyperboloid of one sheet



41

superhyperboloid

Supertorus 2

2 2 1

1

/ 22 / 2 / 2 /

2 /

( )

1

ss s s

s

x y d

z

1 2

1 2

1

(( cos ( ))cos ( ),

( cos ( ))sin ( ),

sin ( ))

s s

s s

s

d v u

d v u

v

1 2, : 0.5 ~ 2.5(#5)s s

: ( , )

: ( , )

v

u

superhyperboloid

Supertorus

2 12 2 1

/2 / 2 / 2 / 1

s ss s sx y z

1 2 1 2

1

(sec ( )sec ( ),sec ( )tan ( ),

tan ( ))

s s s s

s

v u v u

v

: sheet #1: ( / 2, / 2) u

: sheet #2 : ( / 2,3 /2) u

: ( / 2, / 2)

: sheet#1 or sheet#2

v

u