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Topic 3 : View ports & clipping

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WINDOWS, VIEWPORTS & CLIPPING

THE WORLD

THE WINDOWTHE DEVICE

THE VIEWPORT

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YMAX

YMIN

vMAX

VMIN

VY

UX

.(x,y) .(u,v)

UMIN UMAXXMIN XMAX

By simple proportion for the x co-ordinate :

x xx x

u uu u

min

max min

min

max min

u uu ux x

x x

minmax min

max minmin.Rearrange

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axis(xmin,xmax,ymin,ymax,zmin,zmax,colmin,colmax)

axis square

hold on (or off)

MATLAB programming of the view port

Topic 4

3D Transforms &

Projection

Translation & Scaling

1000000000000

scalingfor and

1000100010001

ation For transl z

y

x

z

y

x

SS

S

TTT

Rotation in 3D

R

Rx

Ry

Rz

Z axis rotation

1000010000)()(00)()(

CosSinSinCos

RZ

X & Y axis rotation

10000)(0)(00100)(0)(

10000)()(00)()(00001

CosSin

SinCos

RCosSinSinCos

R YX

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function CubeProject

%To illustrate normal perspective projection

%First define the basic cube

V = [2 -2 -2 1; 2 -2 2 1; 2 2 2 1; 2 2 -2 1; -2 -2 -2 1; -2 -2 2 1; -2 2 2 1; -2 2 -2 1];

F = [1 2 3 4; 3 7 8 4; 7 6 5 8; 6 2 1 5; 2 6 7 3; 4 8 5 1];

VT = V';

%Define suitable rotation angles to orientate the cube

thetaY = 60*pi/180;

thetaX = 30*pi/180;

%Set up a rotation matrix for the Y axis

YROT = [cos(thetaY) 0 -sin(thetaY) 0 ; 0 1 0 0; sin(thetaY) 0 cos(thetaY) 0; 0 0 0 1];

XROT = [1 0 0 0 ; 0 cos(thetaX) -sin(thetaX) 0; 0 sin(thetaX) cos(thetaX) 0; 0 0 0 1];

%Now do the rotations

VT = XROT*YROT*VT;

%Strip off the homogeneous co_ordinate to use the patch command

VP = zeros(3,8);

VP=VT(1:3,:);

patch('Vertices',VP','Faces',F,'facecolor','red')

%axes('Ambientlightcolor',[1 1 1];

axisarray = [-5 5 -5 5 -5 5 0 1];

axis(axisarray)

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-10,-20,0

-10,-20,20

0,-20,30

10,-20,20

10,-20,0

-10,20,0

-10,20,20

0,20,30

10,20,20

10,20,0

1,2,3,4,5

5,4,9,10,0

6,7,2,1,0

4,3,8,9,0

7,8,3,2,0

10,9,8,7,6

1,5,10,6,0

Disc file data for the model house

Vertex data Face data

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function HouseDraw

%Draws a simple house

%Data on nodes is read from a text file NHouse.txt

%Data on patches is read from a text file PHouse.txt

Node = dlmread('NHouse.txt');

%Now read in the uncorrected patch data

UPatch = dlmread('PHouse.txt');

%First find how many patches we need

%We don't care about the other dimension X

[numpatches,X]=size(UPatch);

%Now lets sort into 4 node & 5 node faces

num4 = 0;

for k = 1:numpatches

if UPatch(k,5)==0

num4 = num4 + 1;

end

end

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Face4 = zeros(num4,4);

Face5 = zeros(numpatches-num4,5);

j = 1;

i=1;

for k = 1:numpatches

if UPatch(k,5)==0

Face4(j,:)=UPatch(k,1:4);

j=j+1;

else

Face5(i,:)=UPatch(k,:);

i = i+1;

end

end

patch('Vertices',Node,'Faces',Face4,'FaceVertexCData',hsv(5),'FaceColor','flat')

patch('Vertices',Node,'Faces',Face5,'FaceVertexCData',hsv(2),'FaceColor','flat')

axis_data = [-40 40 -40 40 -40 40 0 1];

axis(axis_data)

axis square

rotate3d on

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Running housedraw

C

Parallel Perspective

Figure 5.1 : Parallel & Perspective projections

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The medieval world did not understand perspective

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The medieval world did not understand perspective

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The classical Greek world used perspective

Co-ordinate axes for projection

Figure 5.2 : Co-ordinate system for projection

O

Y

Z

X

Definition of projection

X

Y

Z

XP

YP

R

P

Q

General parallel projection

100000000)sin(.100)cos(01

RR

P {x,y,z}

C - d - - z -

(u,v)

Normal perspective projection

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vd

yz d

( )

zSy

dzyv .

1

Calculating the transform by similar triangles

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S zS z

( )( )0 0 0

0 0 00 0 0 00 0 0 1

1

1

dz

zS

Normal perspective projection as matrix multiplication

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10

1

1

10

11100000000100001

dzydzx

dz

yx

zyx

d

An alternative (and better) matrix representation

Single vanishing pointVP

Figure 5.5 : A Single Vanishing Point

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X

Y

Z

Y axis

vanishing point

X axis

vanishing point

Two point perspective projection

Transverse Mercator

P

Q

Prime survey meridean

Figure 5.6 : Transverse Mercator Projection

Secondary survey meridean

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t = Tanφ

ω = λ – λ0

 E’ = ( K0νωCosφ ){ 1 + term1 + term2 + term3 }

Term1 = ( ω2 / 6 )Cos2φ( ψ – t2)

Term2 = ( ω4 / 120 )Cos4φ[ 4ψ3( 1 – 6t2 ) + ψ2( 1 + 8t2 ) – ψ2t2 + t4 ]

Term3 = ( ω6 / 5040 )Cos6φ( 61 – 479t2 + 179t4 – t6 )

E = E’ + False Easting

 

N’ = K0{ m + Term1 + Term2 + Term3 + Term4 }

Term1 = ( ω2 / 2 )νSinφ Cosφ

Term2 = ( ω4 / 24 )νSinφ Cos3φ( 4ψ2 + ψ – t2 )

Term3 = ( ω6 / 720 )νSinφ Cos5φ[ 8ψ4( 11 – 24t2 ) – 28ψ3( 1 – 6t2 ) + ψ2( 1 – 32t2 ) – ψ( 2t2 ) + t4 ]

Term4 = ( ω8 / 40320 )νSinφ Cos7φ( 1385 – 3111t2 + 543t4 – t6 )

N = N’ + False Northing

The Redfearn Equations

http://www.surveyplanet.com/Resources/standards

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