3D viewing -b.pdf

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3D Viewing

Chapter 7

Projections

3D clipping

OpenGL viewing functions and clipping planes

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Projections

Parallel

Coordinates are transformed

along parallel lines

Relative sizes are preserved

Parallel lines remain parallel

Perspective

Projection lines converge at a

point

More distant objects are rela-

tively smaller

Closer to the way we actually see

things

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Orthogonal Projections

An orthogonal projection is a parallel projection where the transformation

lines are perpendicular to the view plane.

Orthogonal projections are used for engineering drawings front and side

elevations and plan view.

Axonometricprojection lets you see multiple faces of an object at once.

An isometricprojection has the view direction along a (1,1,1) direction of the

world coordinate system.

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If the view plane is perpendicular to the z-axis,

xp = x

yp = y

The z coordinate is used only for visibility calculations.

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Clipping

The view volume for an orthogonal projection is a rectangular parallelepiped

with a cross-section the same size as the clipping window.

This will be normalized to a cube.

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Normalization

The normalization transformation involves translation and scaling in all 3directions.

M ortho,norm

2xwmaxxwmin

0 0 xwmax+xwminxwmaxxwmin

0 2xwmaxxwmin

0 xwmax+xwminxwmaxxwmin

0 0 2znearzfar

znear+zfarznearzfar

0 0 0 1

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Oblique Parallel Projections

For this kind of projection, projection paths are not perpendicular to the view

plane.

Use two angles to specify the projection.

Point (x,y,z) projects to (xp,yp, zvp).

xp = x + L cos

yp = y + L sin

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Oblique Parallel Projections

the projection line makes an angle with the line between the projected point

and the orthogonal projection point.

tan =zvp z

Lx

An orthogonal projection has = 90.

Oblique projections correspond to a shearing transformation.

Can use a projection vector to specify the projection.

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Cabinet and Cavalier Projections

Typical choices for are 30 and 45.

A cavalier projection has = 45

A cabinet projection has tan = 2 so 63.4.

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Oblique Transformations

The view volume is a parallelepiped.

Use a shear transformation to convert this volume into a rectangular paral-

lelepiped.

Moblique =

1 0 V pxVpz

zvpVpxVpz

0 1 V pyVpz

zvpVpyVpz

0 0 1 0

0 0 0 1

Then transform as for an orthogonal projection.

M oblique, norm = Mortho,norm Moblique

This is applied after MWC, V C.

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Perspective Projections

Parallel projections are easy but not very realistic looking.

Perspective transformations are more realistic.

Points are projected along lines that meet at a reference point.

P1

P2

(xprp, yprp, zprp)

ProjectionReferencePoint

yview

zview

More distant objects appear relatively smaller in a perspective projection.

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Transformations for Perspective Projections

Use parametric equations for positions along a projection line.

x = x (x xprp )u

y = y (y yprp)u

z = z (z zprp)u

At (x,y,z), u = 0 and at (xprp, yprp , zprp), u = 1.

If the view plane is at z = zvp, then

u =zvp z

zprp z

Then, x and y can be calculated from u.

Graphics packages often restrict the projection point to either be at the origin

or along the zview axis which makes the expressions for x and y simpler.

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Vanishing Points

In a perspective projection, parallel lines that arent parallel to the view point

converge at a vanishing point.

Each direction has its own vanishing point; principal vanishing points are for

lines parallel to one of the coordinate axes.

Depending on the orientation of the view plane, there may be 1, 2 or 3 principal

vanishing points.

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View Volumes

The view volume for perspective projections is an infinite pyramid formed

from lines starting at the perspective reference point and going through the

corners of the view plane.

Adding near and far clipping planes reduces this to a frustum.

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Perspective Transformations

The values of xp and yp calculated earlier have coefficients that depend on z.

Need to use a transformation to homogeneous coordinates of the form

xp =x(zprpzvp)+xprp(zvpz)

zprpz

yp =y(zprpzvp)+yprp(zvpz)

zprpz

This converts the description to homogeneous parallel-projection coordinates.

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Viewport Transformation

A perspective transformation can be either symmetric or oblique.

Use a shear transformation to convert an oblique perspective to a symmetric

one.

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OpenGL Viewing Functions

Specify viewing parameters with

gluLookAt( x0, y0, z0, xref, yref, zref, vx, vy, vx)

where

P0 = (x0, y0, z0) is the viewing origin

Pref = (xref, yref, zref) is the look-at point

V = (V x, vy , Vz) is the view-up vector

The positive zview-axis is in the di-

rection

N = P0 Pref

Default parameters are

P0 = (x0, y0, z0) = (0, 0, 0)

Pref = (0, 0,1)

V = (0, 1, 0)

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An orthogonal projection is specified with

glMatrixMode( GLPROJECTION);

glOrtho( xwmin, xwmax, ywmin, ywmax, dnear, dfar);

Default parameters are

xwmin = ywmin = dnear = -1

xwmax = ywmax = dfar = 1

Note that znear is behind the viewing position.

gluOrtho2D is equivalent to gluOrtho with dnear = -1.0 and dfar = 1.0.

To do oblique parallel projections you need to either rotate the scene to get

the effect you want or set up the matrices manually.

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Field of View

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Perspective Transformations

For a symmetric perspective transformation, use

gluPerspective( theta, aspect, dnear, dfar)

For a general perspective transformation, use

glFrustum( xwmin, xwmax, ywmin, ywmax, dnear, dfar);

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3D Clipping Algorithms

Often do the clipping after normalization to make it more efficient. This also

allows clipping to be done in hardware.

Several algorithms

Region Clipping

Polygon clipping

Arbitrary Clip planes

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3D Region Codes

Now have clipping planes instead of lines.

Extend the region codes to have two extra bits.

bit 6 bit 5 bit 4 bit 3 bit 2 bit 1

Far Near Top Bottom Right Left

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3D Region Codes

Then use the same basic rules as in the 2D algorithm for elimination and calcu-

lating intersections.

011001 011000 011010

010001 010000 010010

010101 010100 010110

001001 001000 001010

000001 000000 000010

000101 000100 000110

101001 101000 101010

100001 100000 100010

100101 100100 100110

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Polygon Clipping

Since most graphics packages deal with objects constructed from polygons,

clipping is basically a problem of clipping polygons in 3D.

Coordinate extents (bounding boxes) can be used for trivial rejection or ac-

ceptance.

Then clip edges to modify and create new vertex lists.

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OpenGL Clipping Planes

Sometimes need to specify additional clipping planes. Use

glClipPlane( id, planeParameters);

glEnable( id);

The values of id are GLCLIP_PLANE0, GL_CLIP_PLANE1

The plane parameters are given as a vector with the 4 coefficients from the

equation of the plane.

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3D viewing -b.pdf