Modeling 101

• For the moment assume that all geometry consists of points, lines and faces

• Line: A segment between two endpoints• Face: A planar area bounded by line segments

– Any face can be triangulated (broken into triangles)

Modeling and OpenGL

• In OpenGL, all geometry is specified by stating which type of object and then giving the vertices that define it

• glBegin(…) …glEnd()• glVertex[34][fdv]

– Three or four components (regular or homogeneous)

– Float, double or vector (eg float[3])

• Chapter 2 of the red book

Rendering

• Determine where each object should go in the image

• Determine which object is in front at each pixel• Determine what color it is

Graphics Pipeline Watt Ch 5 and 6

• Graphics hardware employs a sequence of coordinate systems– The movement of geometry through these spaces is

considered a pipeline

Local Coordinate Space

World Coordinate Space

View Space 3D Screen Space

Display Space

Local Coordinate Space

• It is easiest to define individual objects in a local coordinate system– For instance, a cube is easiest to define with faces

parallel to the coordinate axis

• Key idea: Object instantiation– Define an object in a local coordinate system

– Use it multiple times by copying it and transforming it into the global system

Global Coordinate System

• All the objects in the world are transformed into one coordinate system - the global coordinate system

• Lighting is defined in this space• The camera is defined with respect to this space• Some higher level operations, such as advanced

visibility computations, can be done here

View Space

• Associate a set of axes with the image plane– One normal to the image plane

– One up in the image plane

– One right in the image plane

• Some camera parameters are easiest to define in this space– Focal length, image size

3D Screen Space

• Transform view space into a cube– Parallel sides make things easier

• Task to do:– Rasterization - decide which pixels are covered

– Hidden surface removal - decide what is in front

– Shading - decide what color things are

Display Space

• Convert the virtual screen into real screen coordinates– Drop the depth coordinates and translate

• The windowing system takes care of this

OpenGL and Transformations

• OpenGL combines all the transformations up to view space into the MODELVIEW matrix

• View space to Screen Space is done with the PROJECTION matrix

• Matrix calls multiple some matrix M onto the current matrix C: CM– Set view transformation first, then set transformations

from local to world space

Defining View Space

• View space is defined by location of three mutually perpendicular axes in world space– Translation, rotation and scaling can take points in world space to

points in view space

• Typically defined by:– Center of the image plane in world space: View Reference Point

(VRP)

– The normal to the image plane: View Plane Normal (VPN)

– A vector in world space that should be “up” in view space (VUP)

x y

z

World coordinates

View plane normal

View referencepoint

Up vector

u

v

n

View reference pointand view plane normalspecify film plane.

Up vector gives an “up”direction in the film plane.vector v is projection of up vector into film plane= (n x vup) x n

u is chosen so that (u, v, n) is a right handed coordinatesystem; i.e. it is possible torotate (x->u, y->v, z->n) (and we’ll do this shortly).

VRP, VPN, VUP must be inworld coords

World to View Space

• Translate by subtracting VRP

• Rotate by amount that aligns camera axes with world axes:

• All done for you in OpenGL:– gluLookAt takes the VRP, a point along the VPN, and VUP

– Multiplies the required transformation onto the current transformation (normally the identity)

n

v

u

Default OpenGL Camera

• The default OpenGL image plane has u aligned with the x axis, v aligned with y, and n aligned with z– Means the default camera looks along the negative z

axis

– Makes it easy to do 2D drawing (no need for any view transformation)

Left vs Right Handed View Space

• You can define u as right, v as up, and n as toward the viewer: a right handed system uv=n– Advantage: Standard mathematical way of doing things

• You can also define u as right, v as up and n as into the scene: a left handed system vu=n– Advantage: Bigger n values mean points are further

away

• OpenGL is right handed

Projection

• The conversion from view space to screen space is called projection

• Two general classes:– Orthographic, or parallel, projection

– Perspective projection

Orthographic Projection

• Points project along rays perpendicular to the image plane

• Just drop the n coordinate, and maybe scale and translate

• OpenGL: glOrtho(…)– Sets the appropriate projection matrix

Orthographic projection

Perspective Projection

• Abstract camera model - box with a small hole in it

• Pinhole cameras work in practice - camera obscura, etc

Distant Objects Are Smaller

Parallel lines meetcommon to draw film planein front of the focal point

Vanishing points• Each set of parallel lines (=direction) meets at a different

point: The vanishing point for this direction

• Sets of parallel lines on the same plane lead to collinear vanishing points: the horizon for that plane

• Easy examples – corridor

– higher = further away

• Good way to spot faked images