Parallel Projection

transcript

Shandong University Software College

Instructor: Zhou YuanfengE-mail: yuanfeng.zhou@gmail.com

Objectives

• Introduce the classical views• Compare and contrast image formation by computer with how images have been formed by architects, artists, and engineers

• Learn the benefits and drawbacks of each type of view

• Introduce the mathematics of projection• Introduce OpenGL viewing functions• Look at alternate viewing APIs

We call the pictures for observing objects with different ways are views.

View results are related with the shape and size of scene objects, the position and direction of viewpoint.

Classical Viewing

• Viewing requires three basic elements One or more objects A viewer with a projection surface Projectors that go from the object(s) to the projection

surface• Classical views are based on the relationship among

these elements The viewer picks up the object and orients it how she

would like to see it• Each object is assumed to constructed from flat

principal faces Buildings, polyhedra, manufactured objects

Planar Geometric Projections

• Standard projections project onto a plane• Projectors are lines that either

converge at a center of projection are parallel

• Such projections preserve lines but not necessarily angles

• Nonplanar projections are needed for applications such as map construction

3D space

2D image

Projection

YpWindow coordinate

Screen coordinate systemWorld coordinate system

Viewport

Window

Far plane

Near plane

Viewpoint

Image plane

Viewing Volume

Classical Projections

Perspective vs Parallel

• Computer graphics treats all projections the same and implements them with a single pipeline

• Classical viewing developed different techniques for drawing each type of projection

• Fundamental distinction is between parallel and perspective viewing even though mathematically parallel viewing is the limit of perspective viewing

Taxonomy of Planar Geometric Projections

parallel perspective

Axonometric multivieworthographic

oblique

Isometric Dimetric trimetric

2 point1 point 3 point

planar geometric projections

Perspective Projection

Parallel Projection

Orthographic Projection

Projectors are orthogonal to projection surface

Arguably the simplest projectiono Image plane is perpendicular to one of the coordinate axes;o Project onto plane by dropping that coordinate; o All rays are parallel.

Multiview Orthographic Projection

• Projection plane parallel to principal face• Usually form front, top, side views

isometric (not multivieworthographic view)

sidetop

in CAD and architecture, we often display three multiviews plus isometric

Advantages and Disadvantages

•Preserves both distances and angles Shapes preserved Can be used for measurements

• Building plans• Manuals

•Cannot see what object really looks like because many surfaces hidden from view Often we add the isometric

Axonometric Projections

Allow projection plane to move relative to object

classify by how many angles ofa corner of a projected cube are the same

none: trimetrictwo: dimetricthree: isometric

Types of Axonometric Projections

• Lines are scaled (foreshortened) but can find scaling factors

• Lines preserved but angles are not Projection of a circle in a plane not parallel to the

projection plane is an ellipse• Can see three principal faces of a box-like object• Some optical illusions possible

Parallel lines appear to diverge• Does not look real because far objects are scaled the same as near objects

• Used in CAD applications

Oblique Projection

Arbitrary relationship between projectors and projection plane

• Can pick the angles to emphasize a particular face Architecture: plan oblique, elevation oblique

• Angles in faces parallel to projection plane are preserved while we can still see “around” side

• In physical world, cannot create with simple camera; possible with bellows camera(波纹管相机 ) or special lens (architectural)

Perspective Projection

Projectors converge at center of projection

Naturally we see things in perspectiveo Objects appear smaller the farther away they are;o Rays from view point are not parallel.

Vanishing Points

• Parallel lines (not parallel to the projection plan) on the object converge at a single point in the projection (the vanishing point)

• Drawing simple perspectives by hand uses these vanishing point(s)

vanishing point

Three-Point Perspective

• No principal face parallel to projection plane• Three vanishing points for cube

Two-Point Perspective

• On principal direction parallel to projection plane• Two vanishing points for cube

One-Point Perspective

• One principal face parallel to projection plane• One vanishing point for cube

• Objects further from viewer are projected smaller than the same sized objects closer to the viewer (diminution)

Looks realistic• Equal distances along a line are not projected

into equal distances (nonuniform foreshortening)• Angles preserved only in planes parallel to the

projection plane• More difficult to construct by hand than parallel

projections (but not more difficult by computer)

Computer Viewing

•There are three aspects of the viewing process, all of which are implemented in the pipeline,

Positioning the camera• Setting the model-view matrix

Selecting a lens• Setting the projection matrix

Clipping• Setting the view volume

The OpenGL Camera

• In OpenGL, initially the object and camera frames are the same

Default model-view matrix is an identity

•The camera is located at origin and points in the negative z direction

•OpenGL also specifies a default view volume that is a cube with sides of length 2 centered at the origin

Default projection matrix is an identity

OpenGL code

•Remember that last transformation specified is first to be applied

glMatrixMode(GL_MODELVIEW)glLoadIdentity();glTranslatef(0.0, 0.0, -d);glRotatef(-90.0, 0.0, 1.0, 0.0);

Default Projection

Default projection is orthogonal

clipped out

Moving the Camera Frame

• If we want to visualize object with both positive and negative z values we can either

Move the camera in the positive z direction• Translate the camera frame

Move the objects in the negative z direction• Translate the world frame

•Both of these views are equivalent and are determined by the model-view matrix

- Want a translation (glTranslatef(0.0,0.0,-d);)- d > 0

Moving Camera back from Origin

default frames

frames after translation by –d d > 0

glTranslatef(0.0,0.0,-d);

Moving the Camera

•We can move the camera to any desired position by a sequence of rotations and translations

•Example: side view Rotate the camera Move it away from origin Model-view matrix C = TR

glTranslatef(0.0, 0.0, -d);glRotatef(-90.0, 0.0, 1.0, 0.0); Important!!!

The LookAt Function

• The GLU library contains the function gluLookAt to form the required modelview matrix through a simple interface

• Note the need for setting an up direction• Still need to initialize

Can concatenate with modeling transformations• Example: isometric view of cube aligned with

axesglMatrixMode(GL_MODELVIEW):glLoadIdentity();gluLookAt(1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0., 1.0. 0.0);

gluLookAtglLookAt(eyex, eyey, eyez, atx, aty, atz, upx, upy, upz)

Other Viewing APIs

•The LookAt function is only one possible API for positioning the camera

•Others include View reference point, view plane normal, view

up (PHIGS, GKS-3D) Yaw偏转 , pitch倾斜 , roll侧滚 Elevation, azimuth, twist（仰角、方位角、扭转角） Direction angles

Yaw, pitch, roll

侧滚倾斜偏转

PHIGS, GKS-3D

VPN & VUP

•VPN: Normal of projection face•One direction cannot decide the camera• Camera can rotate around the VPN;•+ VUP can decide the camera frame

•Can not be parallel with Projection face.Project VUP to view plane get v.

方位角

仰角

OpenGL Orthogonal Viewing

void glOrtho(GLdouble left, GLdouble right, GLdouble bottom, GLdouble top, GLdouble near, GLdouble far);

near and far measured from camera

OpenGL Perspective

void glFrustum(GLdouble left,GLdouble Right,GLdouble bottom,GLdouble top,GLdouble near,GLdouble far);

Using Field of View

• With glFrustum it is often difficult to get the desired view

•void gluPerspective(GLdouble fovy, GLdouble aspect, GLdouble zNear, GLdouble zFar);

• often provides a better interface

aspect = w/h

Projection Matrix

,Q CConnecting We get line:

CCQCP zz

) , ,( QQQ zyxQ) , ,( CCC zyxC

)0 , ,( PP yxP

Combining with image plane equation

Solving equations

Projection Matrix

We can get

Similarly

Substitute one point of 3D object into above, we can get the projection point on plane

CCQCP zz

) , ,( QQQ zyx0z) ,( PP yxP

fromQC

CCQCP zz

Projection Matrix

( ) C Q Q CCP C Q C

C Q C Q

x z x zzx x x xz z z z

( ) C Q Q CCP C Q C

C Q C Q

y z y zzy y y y

z z z z

Homogeneous coordinate expression:

C Qq z z

Normalization

•Rather than derive a different projection matrix for each type of projection, we can convert all projections to orthogonal projections with the default view volume

•This strategy allows us to use standard transformations in the pipeline and makes for efficient clipping

Pipeline View

modelviewtransformation

projectiontransformation

perspective division

clipping projection

nonsingular4D 3D

against default cube 3D 2D

Orthogonal Normalization

glOrtho(left,right,bottom,top,near,far)

normalization find transformation to convertspecified clipping volume to default

Orthogonal Matrix

• Two steps Move center to origin

T(-(left+right)/2, -(bottom+top)/2,(near+far)/2)) Scale to have sides of length 2

S(2/(left-right),2/(top-bottom),2/(near-far))

0 0 0 1

right leftright left right left

top bottomtop bottom top bottom

far nearnear far far near

P = ST =

Final Projection

• Set z =0 • Equivalent to the homogeneous coordinate

transformation

• Hence, general orthogonal projection in 4D is

1000000000100001

Morth =

P = MorthST

Simple Perspective

Consider a simple perspective with the COP at the origin, the near clipping plane at z = -1, and a 90 degree field of view determined by the planes

x = z, y = z

Perspective Matrices

Simple projection matrix in homogeneous coordinates

Note that this matrix is independent of the far clipping plane

0100010000100001

Generalization

0100βα0000100001

after perspective division, the point (x, y, z, 1) goes to

x’’ = x/zy’’ = y/zz’’ = -(+/z)

which projects orthogonally to the desired point regardless of and

Picking and

If we pick

nearfarfarnear

farnearfarnear2

the near plane is mapped to z = -1the far plane is mapped to z =1and the sides are mapped to x = 1, y = 1

Hence the new clipping volume is the default clipping volume

Normalization Transformation

original clipping volume

original object new clipping volume

distorted objectprojects correctly

Projection Matrix

Summary

Perspective Projection and Parallel Projection• Parallel projection – orthographics projectionInfinite viewpointProjection rays are parallelProjection has the same size of original objectApplications: architecture, computer aided design, etc• Perspective projectionFinite viewpointConverging projection point: center of projectionObjects appear smaller the father away they areApplications: animation, visual simulation, etc

• glViewport(0,0,Width,Height); • glMatrixMode(GL_PROJECTION);• glLoadIdentity();• gluPerspective• ( 45.0f, • (GLfloat)Width/(GLfloat)Height,• 0.1f,• 3000.0f• );

Parallel Projection

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