Computer Graphics (CS 4731) Computer Graphics (CS 4731) Lecture 7: - - PowerPoint PPT Presentation

Computer Graphics (CS 4731) Computer Graphics (CS 4731) Lecture 7: Building 3D Models (Part 1) Prof Emmanuel Agu

Computer Science Dept. Worcester Polytechnic Institute (WPI)

St d d U it V t Standard Unit Vectors

 

, , 1  i

Define y i k

 

 

, 1 ,  j

i k

 

1 , ,  k

j So that any vector,

 

k j i b b

z x

 

k j i v c b a c b a     , ,

C P d t (V t d t) Cross Product (Vector product)

 

z y x

a a a , ,  a

 

z y x

b b b , ,  b

If Then

k j i b a ) ( ) ( ) (

x y y x x z z x y z z y

b a b a b a b a b a b a       

Remember using determinant

a a a k j i

z y x z y x

b b b a a a

Note: a x b is perpendicular to a and b

C P d t Cross Product

Note: a x b is perpendicular to both a and b a x b a b a b

C P d t Cross Product

Calculate a x b if a = (3,0,2) and b = (4,1,8)

C P d t Cross Product

Calculate a x b if a = (3,0,2) and b = (4,1,8) a x b = -2i – 16j + 3k

Fi di V R fl d F S f Finding Vector Reflected From a Surface



a = original vector



a = original vector



n = normal vector



r = reflected vector j i f l



m = projection of a along n



e = projection of a orthogonal to n

n a Note: Θ1 = Θ2 m

• m

r

m a r m a e m e r 2       

Θ1 Θ2 e e

m a r 2   

Li

C id ll i t f th f

Lines

 Consider all points of the form

 P()=P0 +  d

Li S t f ll i t th t th h P i di ti

 Line: Set of all points that pass through P0 in direction

• f vector d

P t i F Parametric Form

T di i l f f li

 Two‐dimensional forms of a line

 Explicit: y = mx +h  Implicit: ax + by +c =0  Implicit: ax + by +c =0  Parametric:

x() = x0 + (1-)x1

P1 α P 1 - α

x() x0 (1 )x1 y() = y0 + (1-)y1

 Parametric form of line

Po Pα  More robust and general than other forms  Extends to curves and surfaces

C it Convexity

 An object is convex iff for any two points in the  An object is convex iff for any two points in the

• bject all points on the line segment between

these points are also in the object these points are also in the object

P P Q Q t convex not convex

Obj ti Objectives

 Introduce 3D set up  Introduce simple data structures for 3D models

 Vertex lists  Edge lists

 Deprecated OpenGL vertex arrays  Drawing 3D objects

3D A li ti 3D Applications

 2D points: (x y) coordinates  2D points: (x,y) coordinates  3D points: have (x,y,z) coordinates  In OpenGL 2D graphics are special case of 3D graphics  In OpenGL, 2D graphics are special case of 3D graphics

Setting up 3D Applications

 Programming 3D, not many changes from 2D

1.

Load representation of 3D object into data structure

Each vertex has (x,y,z) coordinates. Store as vec3, glUniform3f NOT vec2

2

Draw 3D object

2.

Draw 3D object

3.

Set up Hidden surface removal: Correctly determine

• rder in which primitives (triangles, faces) are

rendered (e.g Blocked faces NOT drawn)

3D C di t S t

 All vertex (x,y,z) positions are with respect to a

3D Coordinate Systems

 All vertex (x,y,z) positions are with respect to a

coordinate system

 OpenGL uses right hand coordinate system

p g y

Y Y x + z + z x

Left hand coordinate system Right hand coordinate system Tip: sweep fingers x‐y: thumb is z Left hand coordinate system

• Not used in OpenGL

G i 3D M d l GLUT M d l Generating 3D Models: GLUT Models

 One way of generating 3D shapes is by using GLUT 3D models

(Restrictive?)

 Note: Simply make GLUT 3D calls in OpenGL program to

generate vertices describing different shapes generate vertices describing different shapes

 Two types of GLUT models:



Wireframe Models



Solid Models

Solid models Wireframe models Wireframe models

3D M d li GLUT M d l 3D Modeling: GLUT Models

 Basic Shapes  Basic Shapes



Cone: glutWireCone( ), glutSolidCone( )



Sphere: glutWireSphere( ), glutSolidSphere( )



Cube: glutWireCube( ), glutSolidCube( )

 More advanced shapes:



Newell Teapot: (symbolic)

Torus Cone



Newell Teapot: (symbolic)



Dodecahedron, Torus

Sphere Sp e e

GLUT M d l l t i T t( ) GLUT Models: glutwireTeapot( )

 Famous Utah Teapot: unofficial computer graphics mascot  Famous Utah Teapot: unofficial computer graphics mascot glutWireTeapot(0 5) Create teapot of size 0 5 center positioned at (0 0 0) glutWireTeapot(0.5) - Create teapot of size 0.5, center positioned at (0,0,0) Also glutSolidTeapot( ) You need to apply transformations to position, scale and rotate it

3D M d li GLUT M d l 3D Modeling: GLUT Models

 Glut functions under the hood  Glut functions under the hood

 generate sequence of points that define a shape

E l l tWi C t f

 Example: glutWireCone generates sequence of

vertices, and faces defining cone and connectivity

 Generated vertices and faces passed to OpenGL for  Generated vertices and faces passed to OpenGL for

rendering

glutWireCone generates sequence of vertices, and faces defining cone OpenGL program receives vertices and Faces, renders them

P l l M h Polygonal Meshes

 Modeling with GLUT shapes (cube sphere etc) too restrictive  Modeling with GLUT shapes (cube, sphere, etc) too restrictive  Difficult to approach realism  Other (preferred) way is using polygonal meshes:

(p ) y g p yg

 Collection of polygons, or faces, that form “skin” of object  More flexible, represents complex surfaces better  Examples:  Human face  Animal structures  Furniture, etc

Each face of mesh is a polygon

P l l M h E l Polygonal Mesh Example

h d Mesh (wireframe) Smoothed Out with Shading (later)

P l l M h Polygonal Meshes

 Meshes now standard in graphics  Meshes now standard in graphics  OpenGL



Good at drawing polygons, triangles



Mesh = sequence of polygons forming thin skin around object

 Simple meshes exact. (e.g barn)

C l h i ( h f )

 Complex meshes approximate (e.g. human face)

M h t Diff t R l ti Meshes at Different Resolutions

Original: 424,000 triangles 60,000 triangles (14%). 1000 triangles (0.2%) (courtesy of Michael Garland and Data courtesy of Iris Development.)

R ti M h Representing a Mesh

 Consider a mesh

v6 v5 e3 e2 e

 Consider a mesh

v v8 v4 e1 e8

3

e11 e10 e e9 v1 v7 v3 e6 e7 e5 e4 e12

 There are 8 vertices and 12 edges

v2

 5 interior polygons  6 interior (shared) edges (shown in orange)

( )

 Each vertex has a location vi = (xi yi zi)

Si l R t ti Simple Representation

 Define each polygon by (x y z) locations of its vertices  Define each polygon by (x,y,z) locations of its vertices  OpenGL code

t [i] 3( 1 1 1) vertex[i] = vec3(x1, y1, z1); vertex[i+1] = vec3(x6, y6, z6); vertex[i+2] = vec3(x7, y7, z7); i+=3;

 Inefficient and unstructured

 Vertices shared by many polygons are declared multiple times  Vertices shared by many polygons are declared multiple times  Consider deleting vertex, moving vertex to new location  Must search for all occurrences

I i h Si l R i Issues with Simple Representation

 Declaring face f1

v5

 Declaring face f1

vertex[i] = vec3(x1, y1, z1); vertex[i+1] = vec3(x7, y7, z7); vertex[i+2] = vec3(x8, y8, z8);

v6 v8 v4 f1

 Declaring face f2

vertex[i+3] = vec3(x6, y6, z6);

v1 v7 v

vertex[i] = vec3(x1 y1 z1);

f2 f1

 Inefficient and unstructured

v2 v3

vertex[i] = vec3(x1, y1, z1); vertex[i+1] = vec3(x2, y2, z2); vertex[i+2] = vec3(x7, y7, z7);

 Inefficient and unstructured

 In example, vertices v1 and v7 repeated while declaring f1 and f2  Vertices shared by many polygons are declared multiple times

y y p yg p

 Consider deleting vertex, moving vertex to new location  Must search for all faces in which vertex occurs

G t T l Geometry vs Topology

 Better data structures separate geometry from topology

Better data structures separate geometry from topology

 Geometry: (x,y,z) locations of the vertices  Topology: How vertices and edges are connected  Example: a polygon is an ordered list of vertices with an edge

connecting successive pairs of vertices T l h ld if t h ( t )

 Topology holds even if geometry changes (vertex moves)

v6 v v5 v v v7 v8 v4 f f1

Example: even if we move (x,y,z) location of v1, v1 still connected to v6, v7 and v2

v1 v2 v3 f2 v1

P l T l C i Polygon Traversal Convention

 Use the right hand rule = counter clockwise encirclement  Use the right‐hand rule = counter‐clockwise encirclement

• f outward‐pointing normal

 OpenGL can treat inward and outward

p facing polygons differently

 The order {v1, v0, v3} and {v3, v2, v1} are

equivalent in same polygon, rendered same way rendered by OpenGL

 But order of {v1, v2, v3} is different  The first two describe outwardly facing

polygons

5 4 3

polygons

1 6 2

V t Li t Vertex Lists

 Vertex list: (x,y,z) of vertices (its geometry) are put in array  Use pointers from vertices into vertex list  Polygon list: vertices connected to each polygon (face)

x1 y1 z1

Topology example: Polygon P1 of mesh is connected to vertices (v1,v7,v6)

x2 y2 z2 x3 y3 z3 x y z

P1 P2 v1 v7 v6

x4 y4 z4 x5 y5 z5. x6 y6 z6

P3 P4 P5 v8 v5

Geometry example: Vertex v7 coordinates ( 7 7 7)

6 y6 6

x7 y7 z7 x8 y8 z8

P5 v5 v6

are (x7,y7,z7). Note: If v7 moves, changed once in vertex list

Sh d Ed Shared Edges

 Vertex lists draw filled polygons correctly  Vertex lists draw filled polygons correctly  If each polygon is drawn by its edges, shared edges are

drawn twice drawn twice

 Alternatively: Can store mesh by edge list  Alternatively: Can store mesh by edge list

Ed Li t Edge List

Simply draw each edges once

v6 v5 e3 e2

Simply draw each edges once E.g e1 connects v1 and v6

v7 v8 e1 e8

3

e11 e10 e e9 e1 e2 e3

x1 y1 z1 x2 y2 z2 x3 y3 z3

v1 v6 v1 v7 v3 e6 e7 e5 e4 e12 e4 e5 e6

x3 y3 z3 x4 y4 z4 x5 y5 z5.

v2 e6 e7 e8 9

x6 y6 z6 x7 y7 z7 x y z

Note polygons are not represented

e9

x8 y8 z8

not represented

M d li C b Modeling a Cube

• In 3D, declare vertices as (x,y,z) using point3 v[3]
• Define global arrays for vertices and colors

x y z

typedef vec3 point3; point3 vertices[] = {point3(-1.0,-1.0,-1.0), point3(1.0,-1.0,-1.0), point3(1.0,1.0,-1.0),

y z

point3(-1.0,1.0,-1.0), point3(-1.0,-1.0,1.0), point3(1.0,-1.0,1.0), point3(1.0,1.0,1.0), point3(-1.0,1.0,1.0)};

r g b

typedef vec3 color3; color3 colors[] = {color3(0.0,0.0,0.0), color3(1.0,0.0,0.0), color3(1.0,1.0,0.0),

g b

color(0.0,1.0,0.0), color3(0.0,0.0,1.0), color3(1.0,0.0,1.0), color3(1.0,1.0,1.0), color3(0.0,1.0,1.0});

D i i l f li f i di Drawing a triangle from list of indices

Draw a triangle from a list of indices into the array Draw a triangle from a list of indices into the array

vertices and assign a color to each index

void triangle(int a, int b, int c, int d) { vcolors[i] = colors[d]; position[i] = vertices[a];

a

vcolors[i+1] = colors[d]); position[i+1] = vertices[b]; vcolors[i+2] = colors[d]; position[i+2] = vertices[c]; i+=3; }

b c

Variables a, b, c are indices into vertex array Variable d is index into color array Note: Same face, so all three vertices have same color

N l V Normal Vector

 Normal vector: Direction each polygon is facing  Normal vector: Direction each polygon is facing  Each mesh polygon has a normal vector  Normal vector used in shading  Normal vector used in shading  Normal vector • light vector determines shading (Later)

D b f f Draw cube from faces

void colorcube( ) { quad(0,3,2,1); d(2 3 7 6)

5 6

quad(2,3,7,6); quad(0,4,7,3); quad(1,2,6,5); quad(4,5,6,7);

2 4 7 1

Normal vector q quad(0,1,5,4); }

7 3 3

Note: vertices ordered (counterclockwise) so that we obtain correct outward facing normals g

References

 Angel and Shreiner, Interactive Computer Graphics,

6th edition, Chapter 3 Hill d K ll C G hi i O GL 3rd

 Hill and Kelley, Computer Graphics using OpenGL, 3rd

edition