cse 167: introduction to computer graphics lecture #4...

CSE 167:

Introduction to Computer Graphics

Lecture #4: Vertex Transformation

Jürgen P. Schulze, Ph.D.

University of California, San Diego

Spring Quarter 2016

Announcements

� Project 1 due tomorrow at 2pm

� Grading window is 2-4pm

� Upload source code to TritonEd by 2pm

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Lecture Overview

� View Volumes

� Vertex Transformation

� Rendering Pipeline

� Culling

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View Volumes

� View volume = 3D volume seen by camera

World coordinates

Camera coordinates

Perspective view volume

World coordinates

Camera coordinates

Orthographic view volume

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Projection

matrix

Projection Matrix

Camera coordinates

Canonical view volume

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Image space

(pixel coordinates)

Viewport

transformation

Orthographic View Volume

� Specified by 6 parameters:

� Right, left, top, bottom, near, far

� Or, if symmetrical:

� Width, height, near, far

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Orthographic Projection Matrix

Portho(right,left,top,bottom,near, far) =

2

right − left0 0 −

right + left

right − left

02

top − bottom0 −

top + bottom

top − bottom

0 02

far − near

far + near

far − near

0 0 0 1

Portho(width,height,near, far) =

2

width0 0 0

02

height0 0

0 02

far − near

far + near

far − near

0 0 0 1

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In OpenGL:glOrtho(left, right, bottom, top, near, far)

No equivalent in OpenGL

Perspective View Volume

General view volume

� Defined by 6 parameters, in camera coordinates � Left, right, top, bottom boundaries� Near, far clipping planes

� Clipping planes to avoid numerical problems� Divide by zero� Low precision for distant objects

� Usually symmetric, i.e., left=-right, top=-bottom

Camera

coordinates

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Perspective View Volume

Symmetrical view volume

� Only 4 parameters

� Vertical field of view (FOV)

� Image aspect ratio (width/height)

� Near, far clipping planes

-z

FOV

y

z=-near

z=-far

y=top

aspect ratio=right − left

top − bottom=

right

top

tan(FOV / 2) =top

near

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Perspective Projection Matrix

� General view frustum with 6 parameters

Camera

coordinates

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In OpenGL:glFrustum(left, right, bottom, top, near, far)

Perspective Projection Matrix

� Symmetrical view frustum with field of view, aspect ratio, near and far clip planes

Ppersp (FOV ,aspect,near, far) =

1

aspect ⋅ tan(FOV / 2)0 0 0

01

tan(FOV / 2)0 0

0 0near + far

near − far

2 ⋅ near ⋅ far

near − far

0 0 −1 0

-z

FOV

y

z=-near

z=-far

y=top

Camera

coordinates

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In OpenGL:

gluPerspective(fov, aspect, near, far)

Canonical View Volume

� Goal: create projection matrix so that

� User defined view volume is transformed into canonical view volume: cube [-1,1]x[-1,1]x[-1,1]

� Multiplying corner vertices of view volume by projection matrix and performing homogeneous divide yields corners of canonical view volume

� Perspective and orthographic projection are treated the same way

� Canonical view volume is last stage in which coordinates are in 3D

� Next step is projection to 2D frame buffer

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Viewport Transformation

� After applying projection matrix, scene points are in normalized viewing coordinates

� Per definition within range [-1..1] x [-1..1] x [-1..1]

� Next is projection from 3D to 2D (not reversible)

� Normalized viewing coordinates can be mapped to image (=pixel=frame buffer) coordinates

� Range depends on window (view port) size:[x0…x1] x [y0…y1]

� Scale and translation required:

D x0 , x1, y0 , y1( )=

x1 − x0( ) 2 0 0 x0 + x1( ) 2

0 y1 − y0( ) 2 0 y0 + y1( ) 2

0 0 1 2 1 2

0 0 0 1

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Lecture Overview

� View Volumes

� Vertex Transformation

� Rendering Pipeline

� Culling

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Complete Vertex Transformation

� Mapping a 3D point in object coordinates to pixel coordinates:

� M: Object-to-world matrix

� C: camera matrix

� P: projection matrix

� D: viewport matrix

Object space

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Object space

World space







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Object space

World space

Camera space







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Object space

World space

Camera space








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Object space

World space

Camera space

Image space








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Pixel coordinates:

The Complete Vertex Transformation

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Model Matrix

Camera Matrix

Projection Matrix

Viewport Matrix

Object Coordinates

World Coordinates

Camera Coordinates

Canonical View Volume Coordinates

Window Coordinates

Complete Vertex Transformation in OpenGL






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OpenGL GL_MODELVIEW matrix

OpenGL GL_PROJECTION matrix

Complete Vertex Transformation in OpenGL

� GL_MODELVIEW, C-1M� Defined by the programmer.

� Think of the ModelView matrix as where you stand with the camera and the direction you point it.

� GL_PROJECTION, P� Utility routines to set it by specifying view volume:

glFrustum(), gluPerspective(), glOrtho()

� Think of the projection matrix as describing the attributes of your camera, such as field of view, focal length, etc.

� Viewport, D

� Specify implicitly via glViewport()

� No direct access with equivalent to GL_MODELVIEW or GL_PROJECTION

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Introduction to OpenGL

� Using slides from SIGGRAPH course:

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OpenGL and GLFW Overview

� What is OpenGL & what can it do for me?

� OpenGL in windowing systems

� Why GLFW

� A GLFW program template

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What Is OpenGL?

� Graphics rendering API

� high-quality color images composed of geometric and image primitives

� window system independent

� operating system independent

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OpenGL as a Renderer

� Geometric primitives

� points, lines and polygons

� Image Primitives

� images and bitmaps

� separate pipeline for images and geometry

� linked through texture mapping

� Rendering depends on state

� colors, materials, light sources, etc.

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Related APIs

� GLU (OpenGL Utility Library)

� part of OpenGL

� NURBS, tessellators, quadric shapes, etc.

� GLFW (OpenGL Utility Toolkit)

� portable windowing API

� not officially part of OpenGL

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Preliminaries

� Headers Files� #include <GL/gl.h>

� #include <GL/glu.h>

� #include <GLFW/glfw3.h>

� Libraries

� Enumerated Types

� OpenGL defines numerous types for compatibility� GLfloat, GLint, GLenum, etc.

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GLFW Basics

� Application Structure

� Configure and open window

� Initialize OpenGL state

� Enter event processing loop

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Sample Program#include <GLFW/glfw3.h>

int main(void)

{

GLFWwindow* window;

/* Initialize the library */

if (!glfwInit()) return -1;

/* Create a windowed mode window and its OpenGL context */

window = glfwCreateWindow(640, 480, "Hello CSE 167", NULL, NULL);

if (!window)

{

glfwTerminate();

return -1;

}

/* Make the window's context current */

glfwMakeContextCurrent(window);

/* Initialize OpenGL here */

/* Loop until the user closes the window */

while (!glfwWindowShouldClose(window))

{

/* Render here with OpenGL */

/* Swap front and back buffers */

glfwSwapBuffers(window);

/* Poll for and process events */

glfwPollEvents();

}

glfwTerminate();

return 0;

}

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OpenGL Initialization

� Set up whatever state you are going to use

void init( void )

{

glClearColor( 0.0, 0.0, 0.0, 1.0 );

glClearDepth( 1.0 );

glEnable( GL_LIGHT0 );

glEnable( GL_LIGHTING );

glEnable( GL_DEPTH_TEST );

}

Elementary Rendering

� Geometric Primitives

� Managing OpenGL State

� OpenGL Buffers

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OpenGL Geometric Primitives

� All geometric primitives are specified by vertices

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GL_QUAD_STRIP

GL_POLYGON

GL_TRIANGLE_STRIP GL_TRIANGLE_FAN

GL_POINTS

GL_LINES

GL_LINE_LOOPGL_LINE_STRIP

GL_TRIANGLES

GL_QUADS

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( v )

Number of

components

2 - (x,y)

3 - (x,y,z)

4 - (x,y,z,w)

Data Type

b - byte

ub - unsigned byte

s - short

us - unsigned short

i - int

ui - unsigned int

f - float

d - double

Vector

omit “v” for

scalar form

glVertex2f( x, y )

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Specifying Geometric Primitives

� Primitives are specified usingglBegin( primType );

glEnd();

� primType determines how vertices are combined

GLfloat red, greed, blue;Glfloat coords[3];

glBegin( primType );for ( i = 0; i < nVerts; ++i ) {

glColor3f( red, green, blue );glVertex3fv( coords );

}glEnd();

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Shapes Tutorial

OpenGL’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 statefor each ( primitive to render ) {

update OpenGL staterender primitive

}

� Manipulating vertex attributes is mostcommon way to manipulate stateglColor*() / glIndex*()

glNormal*()

glTexCoord*()

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Controlling current state

� Setting StateglPointSize( size );

glLineStipple( repeat, pattern );

glShadeModel( GL_SMOOTH );

� Enabling FeaturesglEnable( GL_LIGHTING );

glDisable( GL_TEXTURE_2D );

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