Modeling

Geometric Models 2D and 3D objects

Triangles, quadrilaterals, polygons Spheres, cones, boxes

Surface characteristics Color Texture

Composite objects Other objects and their relationships to each other

2

The basic idea Describe objects using surfaces, which are polygons

Triangles, quadrilaterals, etc. Important thing is that they are flat and convex

Convex : A line segment connecting any two points on the polygon is contained in the polygon.

Provide points in counterclock-wise order from the visible side

3

P

Q

P

convex concave

Outline

4

OpenGL Elementary Rendering (codes) Representation Appendix: Vector Operations *

OpenGL Elementary Rendering Geometric Primitives Building Models Managing OpenGL State

5

OpenGL Geometric Primitives All geometric primitives are specified by

vertices

GL_QUAD_STRIPGL_QUAD_STRIP

GL_POLYGONGL_POLYGON

GL_TRIANGLE_STRIPGL_TRIANGLE_STRIP GL_TRIANGLE_FANGL_TRIANGLE_FAN

GL_POINTSGL_POINTS

GL_LINESGL_LINES

GL_LINE_LOOPGL_LINE_LOOPGL_LINE_STRIPGL_LINE_STRIP

GL_TRIANGLESGL_TRIANGLES

GL_QUADSGL_QUADS

6

Simple Examplevoid drawRhombus( GLfloat color[] ){ glBegin( GL_QUADS ); glColor3fv( color ); glVertex2f( 0.0, 0.0 ); glVertex2f( 1.0, 0.0 ); glVertex2f( 1.5, 1.118 ); glVertex2f( 0.5, 1.118 ); glEnd();

}

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OpenGL Command Formats

glVertex3fv( glVertex3fv( vv ) )

Number ofNumber ofcomponentscomponents

2 - (x,y) 2 - (x,y) 3 - (x,y,z)3 - (x,y,z)4 - (x,y,z,w)4 - (x,y,z,w)

Data TypeData Typeb - byteb - byteub - unsigned byteub - unsigned bytes - shorts - shortus - unsigned shortus - unsigned shorti - inti - intui - unsigned intui - unsigned intf - floatf - floatd - doubled - double

VectorVector

omit omit ““vv”” for forscalar formscalar form

glVertex2f( x, y )glVertex2f( x, y )

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Specifying Geometric Primitives Primitives are specified using

glBegin( glBegin( primType primType ););

glEnd();glEnd();

primType determines how vertices are combined

GLfloat red, greed, blue;GLfloat red, greed, blue;Glfloat coords[3];Glfloat coords[3];glBegin( glBegin( primType primType ););for ( i = 0; i < nVerts; ++i ) { for ( i = 0; i < nVerts; ++i ) { glColor3f( red, green, blue );glColor3f( red, green, blue ); glVertex3fv( coords );glVertex3fv( coords );}}glEnd();glEnd();

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Shapes Tutorial

10

OpenGL Color Models * RGBA or Color Index color index mode

Display12

48

16

Red Green Blue

0123

24

2526

123 219 74

RGBA mode

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Building models Consider a mesh

There are 8 nodes and 12 edges 5 interior polygons 6 interior (shared) edges

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

v1 v2

v7

v6

v8

v5

v4

v3

e1

e8

e3

e2

e11

e6

e7

e10

e5

e4

e9

e12

Simple Representation Each polygon is defined by the geometric

locations of its vertices, which leads to OpenGL code such as

Inefficient and unstructured

glBegin(GL_POLYGON); glVertex3f(x1, y1, z1); glVertex3f(x7, y7, z7);

glVertex3f(x8, y8, z8); glVertex3f(x6, y6, z6); glEnd();

Vertex Lists Put the geometry in an array Use pointers from the vertices into this

array Introduce a polygon list x1 y1 z1

x2 y2 z2

x3 y3 z3

x4 y4 z4

x5 y5 z5.

x6 y6 z6

x7 y7 z7

x8 y8 z8

P1P2P3P4P5

v1

v7

v6

v8

v5

v6topology geometry

Example: Modeling a Cube Define global arrays for vertices and colorsGLfloat vertices[][3] = {{-1.0,-1.0,-1.0},{1.0,-1.0,-1.0},

{1.0,1.0,-1.0}, {-1.0,1.0,-1.0}, {-1.0,-1.0,1.0}, {1.0,-1.0,1.0}, {1.0,1.0,1.0}, {-1.0,1.0,1.0}};

GLfloat colors[][3] = {{0.0,0.0,0.0},{1.0,0.0,0.0},{1.0,1.0,0.0}, {0.0,1.0,0.0}, {0.0,0.0,1.0},

{1.0,0.0,1.0}, {1.0,1.0,1.0}, {0.0,1.0,1.0}};

Draw a quadrilateral from a list of indices into the array vertices and use color corresponding to first index

void polygon(int a, int b, int c , int d){ glBegin(GL_POLYGON); glColor3fv(colors[a]); glVertex3fv(vertices[a]); glVertex3fv(vertices[b]); glVertex3fv(vertices[c]); glVertex3fv(vertices[d]); glEnd(); }

Draw cube from faces Note that vertices are ordered so that we

obtain correct outward facing normals

void colorcube( ){ polygon(0,3,2,1); polygon(2,3,7,6); polygon(0,4,7,3); polygon(1,2,6,5); polygon(4,5,6,7); polygon(0,1,5,4);}

0

5 6

2

4 7

1

3

Efficiency The problem is that we must do many function

calls to draw the cube 6 glBegin, 6 glEnd 6 glColor 24 glVertex More if we use texture and lighting

* OpenGL provides vertex arrays to store

array data Six types of arrays supported

Vertices Colors Color indices Normals Texture coordinates Edge flags

Here we will need only colors and vertices

*

Initialization Using the same color and vertex data, first we

enableglEnableClientState(GL_COLOR_ARRAY);

glEnableClientState(GL_VERTEX_ARRAY); Identify location of arraysglVertexPointer(3, GL_FLOAT, 0, vertices);

glColorPointer(3, GL_FLOAT, 0, colors);

3d arrays stored as floats data contiguousdata array

* Form an array of face indices

Each successive four indices describe a face of the cube

GLubyte cubeIndices[24] = {0,3,2,1,2,3,7,6 0,4,7,3,1,2,6,5,4,5,6,7,0,1,5,4};

* Drawing the cube

Method 1:

Method 2:

for(i=0; i<6; i++) glDrawElements(GL_POLYGON, 4, GL_UNSIGNED_BYTE, &cubeIndices[4*i]);

format of index data start of index data

what to drawnumber of indices

glDrawElements(GL_QUADS, 24, GL_UNSIGNED_BYTE, cubeIndices);

Draws cube with 1 function call!!

OpenGL’s State MachineOpenGL’s State Machine All rendering attributes are encapsulated in

the OpenGL State rendering styles shading lighting texture mapping

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Manipulating OpenGL State Appearance is controlled by current state

for each ( primitive to render ) { update OpenGL state render primitive

} Manipulating vertex attributes is most

common way to manipulate stateglColor*() / glIndex*()glColor*() / glIndex*()

glNormal*()glNormal*()

glTexCoord*()glTexCoord*()

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Controlling current state Setting State

glPointSize( glPointSize( sizesize ); );

glShadeModel( glShadeModel( GLGL__SMOOTHSMOOTH ); );

Enabling FeaturesglEnable( glEnable( GLGL__LIGHTING LIGHTING ););

glDisable( glDisable( GL_TEXTURE_2D GL_TEXTURE_2D ););

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Outline

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OpenGL Elementary Rendering (codes) Representation Appendix: Vector Operations

Objectives

Introduce dimension and basis Introduce coordinate systems for

representing vectors spaces and frames for representing affine spaces

Discuss change of frames and bases Introduce homogeneous coordinates

Vectors Objects with length and direction Examples include

Force Velocity Directed line segments

We will think of a vector as a displacement from one point to another

We use bold typeface to indicate vectors

v

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Vectors as Displacements

29

As a displacement, a vector has length and direction, but no inherent location

You can drive one mile East in Zhuhai To represent vectors, we need firstly learn…

Linear Independence A set of vectors v1, v2, …, vn is linearly

independent if 1v1+2v2+.. nvn=0 iff 1=2=…=0 If a set of vectors is linearly independent, we

cannot represent one in terms of the others If a set of vectors is linearly dependent, as

least one can be written in terms of the others

Dimension In a vector space, the maximum number of

linearly independent vectors is fixed and is called the dimension of the space

In an n-dimensional space, any set of n linearly independent vectors form a basis for the space

Given a basis v1, v2,…., vn, any vector v can be written as

v=1v1+ 2v2 +….+nvn

where the {i} are unique

Coordinate Systems Consider a basis v1, v2,…., vn

A vector is written v=1v1+ 2v2 +….+nvn

The list of scalars {1, 2, …. n}is the representation of v with respect to the given basis Scalar: here an ordinary real (e.g., float)

number. Scalars alone have no geometric properties

We can write the representation as a row or column array of scalarsa=[1 2 …. n]

T=

n

2

1

.

Examples v=2v1+3v2-4v3

a=[2 3 –4]T

Note that this representation is with respect to a particular basis

v = 5v1-2v2 in a plane

Coordinate Systems Which is correct?

Both are because vectors have no fixed location

v

v

Frames A coordinate system is insufficient to

represent points If we work in an affine space we can add a

single point, the origin, to the basis vectors to form a frame

P0

v1

v2

v3

Representation in a Frame Frame determined by (P0, v1, v2, v3) Within this frame, every vector can be written

as v=1v1+ 2v2 +….+nvn

Every point can be written as P = P0 + 1v1+ 2v2 +….+nvn

Confusing Points and Vectors Consider the point and the vector

P = P0 + 1v1+ 2v2 +….+nvn

v=1v1+ 2v2 +….+nvn

They appear to have the similar representations

p=[1 2 3] T v=[1 2 3] T

which confuses the point with the vectorv

pv

Vector can be placed anywhere

point: fixed

A Single Representation If we define 0•P = 0 and 1•P =P then we can

write

v=1v1+ 2v2 +3v3 = [v1 v2 v3 P0] [1 2 3 0 ] T

P = P0 + 1v1+ 2v2 +3v3= [v1 v2 v3 P0] [1 2 3 1 ] T

Thus we obtain the four-dimensional homogeneous coordinate representation

v = [1 2 3 0 ] T

p = [1 2 3 1 ] T

Homogeneous Coordinates The homogeneous coordinates form for a three

dimensional point [x y z] is given asp =[x’ y’ z’ w] T =[wx wy wz w] T

We return to a three dimensional point (for w0) by

xx’/wyy’/wzz’/w

If w=0, the representation is that of a vector For w=1, the representation of a point is [x y z 1]

Note that homogeneous coordinates replaces points in three dimensions by lines through the origin in four dimensions

Homogeneous coordinates are key to all computer graphics systems All standard transformations (rotation,

translation, scaling) can be implemented with matrix multiplications using 4 x 4 matrices

Hardware pipeline works with 4 dimensional representations

For orthographic viewing, we can maintain w=0 for vectors and w=1 for points

For perspective we need a perspective division

Change of Coordinate Systems Consider two representations of a same

vector with respect to two different bases. The representations are

a=[1 2 3 ] T

b=[1 2 3] T

where

v=1v1+ 2v2 +3v3 = [v1 v2 v3] [1 2 3] T

=1u1+ 2u2 +3u3 = [u1 u2 u3] [1 2 3] T

Representing 2nd basis in terms of 1st Each of the basis vectors, u1,u2, u3, are

vectors that can be represented in terms of the first basis

u1 = 11v1+12v2+13v3

u2 = 21v1+22v2+23v3

u3 = 31v1+32v2+33v3

v

[u1 u2 u3] = [v1 v2 v3]MT

Simple example

43

1

1

2

*1

1

0

1

*1

1

1

2

*1

1

0

0

*3

0

1

0

*2

0

0

1

*5v

3213

3212

3211

*1*1*2

*1*0*1

*1*1*2

vvvu

vvvu

vvvu

Matrix Form The coefficients define a 3 x 3 matrix

and the representations can be related by

a=MTb

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M =

Change of Frames We can apply a similar process in

homogeneous coordinates to the representations of both points and vectors

Any point or vector can be represented in either frame

We can represent Q0, u1, u2, u3 in terms of P0, v1, v2, v3

Consider two frames:(P0, v1, v2, v3)(Q0, u1, u2, u3) P0 v1

v2

v3

Q0

u1u2

u3

Representing 2nd Frame in the 1st

u1 = 11v1+12v2+13v3

u2 = 21v1+22v2+23v3

u3 = 31v1+32v2+33v3

Q0 = 41v1+42v2+43v3 +44P0

Extending what we did with change of bases

defining a 4 x 4 matrix

M =

Working with Representations Within the two frames any point or vector

has a representation of the same form

a=[1 2 3 4 ]T in the first frameb=[1 2 3 4 ] T in the second frame

where 4 4 for points and 4 4 for vectors and

The matrix M is 4 x 4 and specifies an affine transformation in homogeneous coordinates

a=MTb

Representing geometry elements Points

Location in space Operations allowed between points and vectors

Point-point subtraction yields a vector Equivalent to point-vector addition P=v+Q

v=P-Q

48

Lines Consider all points of the form

P()=P0 + d

Set of all points that pass through P0 in the direction of the vector d

49

Rays and Line Segments If >= 0, then P() is the ray leaving P0 in the

direction d If we use two points to define v, then

P( ) = Q+v = Q + (R-Q)

= R + (1-)Q For 0<=<=1 we get all the points on the line

segment joining R and Q

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Planes A plane can be defined by a point and two

vectors or by three points

P(,)=R+u+v P(,)=R+(Q-R)+(P-R)

u

v

R

P

R

Q

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Barycentric Coordinates

52

We can reorder the terms of the equation:

with: This coordinate system is called barycentric

coordinates For any point P inside the triangle (a, b, c):

Point on an edge: one coefficient is 0 Vertex: two coefficients are 0, remaining one is 1

RPQ

RPQ

RPΡQRP

)1(

)()(),(

1

Outline

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OpenGL Elementary Rendering Representation Appendix: Vector Operations *

Vector Operations

54

Addition and multiplication by a scalar: a + b s*a

Examples:

Vector Addition

55

Head to tail: At the end of a, place the start of b.

Parallelogram rule:

Components: [a1 + b1, a2 + b2]

Vector Subtraction

56

To draw a - b, go forward along a and then backward along b.

Components: [a1 - b1, a2 - b2]

The Dot Product

57

Also called the inner product of two vectors Produces a number, not a vector In 2D: In 3D:

Symmetry: Linearity: Homogeneity:

Length of a Vector

58

The length (or norm) of a vector a is:

The dot product of a vector with itself yields the length squared:

Unit Vectors and Normalization

59

A unit vector is a vector whose length equals 1 Example: (1, 0, 0), (0, 1, 0), (0, 0, 1)

Divide any nonzero vector v by its length |v|. The result is a unit vector in the same direction as v.

This is called normalizing a vector:

Any vector can be written as its magnitude times its direction:

The Angle Between Vectors

60

The angle between vectors can be computed by:

Using normalized vectors:

The Sign of a•b

61

Imagine the cosine function:

Two vectors a and b are perpendicular, orthogonal, or normal if a•b= 0

The end

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Questions and answers